Leslie C. Woods
Theory of Tokamak Transport
Theory of Tokamak Transport. Leslie Colin Woods
Copyright © 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
ISBN: 3-527-40625-5
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Leslie C. Woods
Theory of Tokamak Transport
New Aspects for Nuclear Fusion Reactor Design
WILEY-VCH Verlag GmbH & Co. KGaA
The Author
Prof. Dr. Leslie Colin Woods, Oxford,
Great Britain
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ISBN-13: 978-3-527-40625-8
ISBN-10: 3-527-40625-5
Contents
Preface
XI
Lists of physical constants, plasma parameters and frequently used symbols
XV
1
The quest for fusion power
1.1
Tokamak machines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1.1 Topology and ignition . . . . . . . . . . . . . . . . . . . . . . . . .
1.1.2 Some early tokamaks . . . . . . . . . . . . . . . . . . . . . . . . .
1.1.3 Toroidal current . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2
Basic tokamak variables . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.1 Aspect ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.2 Beta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.3 Safety factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.4 Z-effective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3
Global confinement times . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3.1 Energy confinement time . . . . . . . . . . . . . . . . . . . . . . .
1.3.2 Electron-energy confinement time . . . . . . . . . . . . . . . . . .
1.3.3 Particle confinement time . . . . . . . . . . . . . . . . . . . . . . .
1.3.4 Momentum confinement time . . . . . . . . . . . . . . . . . . . . .
1.4
Heating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4.1 Ohmic heating . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4.2 Neutral beam heating . . . . . . . . . . . . . . . . . . . . . . . . .
1.4.3 Radio-frequency heating . . . . . . . . . . . . . . . . . . . . . . .
1.5
Electron energy confinement time . . . . . . . . . . . . . . . . . . . . . . .
1.5.1 Ohmically-heated tokamaks . . . . . . . . . . . . . . . . . . . . . .
1.5.2 Auxiliary heated plasmas . . . . . . . . . . . . . . . . . . . . . . .
1.5.3 Profile shapes and energy losses . . . . . . . . . . . . . . . . . . .
1.5.4 Disruptive instabilities . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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2
Tokamak magnetic fields
2.1
Axisymmetric toroidal equilibrium . . . . . . . . . . . . . . . . . . . . . .
2.1.1 Grad–Shafranov equation . . . . . . . . . . . . . . . . . . . . . . .
2.1.2 First integral constraint . . . . . . . . . . . . . . . . . . . . . . . .
2.1.3 Second integral constraint . . . . . . . . . . . . . . . . . . . . . . .
2.1.4 Diffusion velocity . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Theory of Tokamak Transport: New Aspects for Nuclear Fusion Reactor Design. Leslie Colin Woods
Copyright © 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
ISBN: 3-527-40625-5
VI
Contents
2.2
Equilibrium in a circular torus . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Shafranov geometry . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.2 Solution of the Grad–Shafranov equation . . . . . . . . . . . . . . .
2.2.3 Magnetic fields and electric currents . . . . . . . . . . . . . . . . .
2.3
Particle trapping in magnetic fields . . . . . . . . . . . . . . . . . . . . . .
2.3.1 Magnetic bottles . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.2 Fraction of trapped particles . . . . . . . . . . . . . . . . . . . . .
2.4
Trapping in tokamak magnetic fields . . . . . . . . . . . . . . . . . . . . .
2.4.1 Tokamak mirrors . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.2 Trapped particles . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.3 Bounce time in a tokamak field . . . . . . . . . . . . . . . . . . . .
2.4.4 Trapped particle resistivity . . . . . . . . . . . . . . . . . . . . . .
2.5
Diffusivity of trapped particles . . . . . . . . . . . . . . . . . . . . . . . .
2.5.1 Energy sinks at magnetic mirrors . . . . . . . . . . . . . . . . . . .
2.5.2 Physics of diffusivity . . . . . . . . . . . . . . . . . . . . . . . . .
2.5.3 Parallel diffusivity due to trapped particles . . . . . . . . . . . . . .
2.5.4 Thermal pumping . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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3 Energy transport in Tokamaks
3.1
Banana orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.1 Drifts due to variations in the magnetic field . . . . . . . . . . . . .
3.1.2 Gyro-averages . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.3 Banana width . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.4 Neoclassical diffusivity . . . . . . . . . . . . . . . . . . . . . . . .
3.2
Thermal conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.1 Neutral gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.2 Magnetoplasma . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.3 Fluid shear and transport . . . . . . . . . . . . . . . . . . . . . . .
3.2.4 Heat flux, second-order in Knudsen number . . . . . . . . . . . . .
3.3
Classical treatment of particle transport . . . . . . . . . . . . . . . . . . . .
3.3.1 Equilibrium currents . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.2 Pfirsch–Schlüter current . . . . . . . . . . . . . . . . . . . . . . . .
3.3.3 Mass diffusivity . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4
Neoclassical theory and its validity . . . . . . . . . . . . . . . . . . . . . .
3.4.1 Banana and plateau regimes . . . . . . . . . . . . . . . . . . . . . .
3.4.2 Testing neoclassical theory . . . . . . . . . . . . . . . . . . . . . .
3.4.3 Bootstrap current . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5
Second-order transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5.1 Electron thermal diffusivity . . . . . . . . . . . . . . . . . . . . . .
3.5.2 Cylindrical coordinates . . . . . . . . . . . . . . . . . . . . . . . .
3.5.3 Physical mechanism for heat flux . . . . . . . . . . . . . . . . . . .
3.5.4 Role of turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5.5 Knudsen number constraint . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Contents
VII
4
Energy losses from tokamaks
4.1
Low poloidal beta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.1 Empirical profiles . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.2 Radial distribution of thermal diffusivity . . . . . . . . . . . . . . .
4.1.3 Electron energy confinement time . . . . . . . . . . . . . . . . . .
4.1.4 Comparison of theory with observation . . . . . . . . . . . . . . . .
4.2
High poloidal beta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.1 Oscillatory temperature profiles . . . . . . . . . . . . . . . . . . . .
4.2.2 Thermal diffusivity . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.3 Electron energy confinement time . . . . . . . . . . . . . . . . . .
4.3
The L- and H-modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.1 Role of boundary conditions . . . . . . . . . . . . . . . . . . . . .
4.3.2 Energy confinement in the L- and H-modes . . . . . . . . . . . . .
4.4
Thermal transport in the ion fluid . . . . . . . . . . . . . . . . . . . . . . .
4.4.1 Thermal diffusivity . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4.2 Ambipolar constraint . . . . . . . . . . . . . . . . . . . . . . . . .
4.5
Comparison of experiment and theory . . . . . . . . . . . . . . . . . . . .
4.5.1 Neutral beam injection . . . . . . . . . . . . . . . . . . . . . . . .
4.5.2 Confinement times for L- and H-modes . . . . . . . . . . . . . . .
4.5.3 Loop voltage . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5.4 Steady state with ohmic heating . . . . . . . . . . . . . . . . . . .
4.5.5 Internal transport barriers . . . . . . . . . . . . . . . . . . . . . . .
4.6
Profile instabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.6.1 Safety factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.6.2 Thermal instability . . . . . . . . . . . . . . . . . . . . . . . . . .
4.6.3 Review of electron thermal transport . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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5
Plasma flow and loop voltage
5.1
Flow of plasma across strong magnetic fields . . . . . . . . . . . . . . . . .
5.1.1 Plasma particle confinement . . . . . . . . . . . . . . . . . . . . .
5.1.2 Viscous stress tensor in cylindrical geometry . . . . . . . . . . . . .
5.1.3 Radial diffusion velocity . . . . . . . . . . . . . . . . . . . . . . .
5.1.4 Ambipolar flow . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2
Particle transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.1 Particle diffusivity and the pinch velocity . . . . . . . . . . . . . .
5.2.2 Particle confinement time . . . . . . . . . . . . . . . . . . . . . . .
5.2.3 Plasma source term . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.4 Observations of particle confinement . . . . . . . . . . . . . . . . .
5.3
The toroidal current and voltage relationship . . . . . . . . . . . . . . . . .
5.3.1 Loop (induced) voltage . . . . . . . . . . . . . . . . . . . . . . . .
5.3.2 Lorentz voltage . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.3 Loop voltage instability . . . . . . . . . . . . . . . . . . . . . . . .
5.3.4 Lorentz current . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.5 Determining Zeff from current and loop voltage . . . . . . . . . . .
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VIII
Contents
5.4
Toroidal velocities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4.1 Role of second-order viscosity . . . . . . . . . . . . . . . . . . . .
5.4.2 Angular momentum diffusivity . . . . . . . . . . . . . . . . . . . .
5.4.3 Comparison of theory and observation . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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6 Thermal Instabilities
6.1
Sawtooth oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.1.1 Some observations of temperature and density sawteeth . . . . . . .
6.1.2 Kadomtsev’s model of sawtooth oscillations . . . . . . . . . . . . .
6.1.3 Sawtooth ramp phase . . . . . . . . . . . . . . . . . . . . . . . . .
6.1.4 Sawtooth period . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.1.5 Theory v. observation for the sawtooth period . . . . . . . . . . . .
6.2
Disruptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2.1 Description of major disruptions . . . . . . . . . . . . . . . . . . .
6.2.2 Precursor waves . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2.3 Collapse phase . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3
MHD instabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3.1 Ideal and resistive instabilities . . . . . . . . . . . . . . . . . . . .
6.3.2 Theory of the ballooning stability limit . . . . . . . . . . . . . . . .
6.3.3 Some observations of limiting betas . . . . . . . . . . . . . . . . .
6.4
L ⇒ H transition, ELMS, Snakes, PEPS, and MARFES . . . . . . . . . . .
6.4.1 The L ⇒ H transition . . . . . . . . . . . . . . . . . . . . . . . . .
6.4.2 Edge Localized Modes . . . . . . . . . . . . . . . . . . . . . . . .
6.4.3 Snakes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.4.4 Pellet enhanced performance mode (PEP) . . . . . . . . . . . . . .
6.4.5 MARFES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.5
Minimum reactor size for ignition . . . . . . . . . . . . . . . . . . . . . . .
6.5.1 Stability constraints . . . . . . . . . . . . . . . . . . . . . . . . . .
6.5.2 Minimum dimensions . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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A Plasma Physics Notes
A.1 Equations of fluid motion . . . . . . . . . . . . . . . . . . . . . . . . . . .
A.2 Collision intervals and Spitzer resistivity . . . . . . . . . . . . . . . . . . .
A.3 Energy in the electron and ion fluids . . . . . . . . . . . . . . . . . . . . .
A.4 Cyclotron frequencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A.5 Dimensional analysis applied to energy confinement time . . . . . . . . . .
A.6 Divergence and curl in cylindrical coordinates . . . . . . . . . . . . . . . .
A.7 Tensorial form for Ohm’s law . . . . . . . . . . . . . . . . . . . . . . . . .
A.8 Constants of the motion of gyrating particles . . . . . . . . . . . . . . . . .
A.9 Equilibrium velocity distribution function . . . . . . . . . . . . . . . . . .
A.10 Escape time for trapped particles . . . . . . . . . . . . . . . . . . . . . . .
A.11 Motion of a fluid element . . . . . . . . . . . . . . . . . . . . . . . . . . .
A.12 Kinetic equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Contents
A.13 Drift kinetic equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A.14 Guiding center drifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A.15 Convection and diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . .
A.16 The decomposition of second-order tensors . . . . . . . . . . . . . . . . . .
A.17 Div and curl in local toroidal coordinates . . . . . . . . . . . . . . . . . . .
A.18 Knudsen numbers and local thermodynamic equilibrium . . . . . . . . . . .
A.19 Onsager’s reciprocal relations in neoclassical transport . . . . . . . . . . . .
A.20 Putative role of turbulence in transport . . . . . . . . . . . . . . . . . . . .
A.21 Solution of a vector equation . . . . . . . . . . . . . . . . . . . . . . . . .
A.22 Viscous stress tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A.23 Solution of a tensor equation . . . . . . . . . . . . . . . . . . . . . . . . .
A.24 MHD instabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A.25 The Catherine wheel fallacy . . . . . . . . . . . . . . . . . . . . . . . . . .
A.26 Limitations of Boltzmann’s kinetic equation . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Index
IX
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Preface
The world-wide demand for energy is growing exponentially. In the middle of the nineteenth
century mankind’s energy consumption was less than half a Q/century (1
1021 Joules), in
1851–1950 it had increased to 4Q/century, and in the half century 1951–2000 it was 15 Q.
Estimates of the world reserves of fossil fuels keep changing — perhaps these reserves are
100 Q, and even if this is a substantial underestimate, it is likely that they would be depleted
within a century, particularly when the rapidly increasing demand for energy from the recently
industrialized nations of the East is included in the reckoning. Reactors based on the fusion
of light elements may provide an almost unlimited supply of energy in the future.
There are other considerations that make the development of fusion reactors a worth-while
task. As remarked recently in the New Scientist, “Burning fossil fuels and using the atmosphere as an open sewer has turned out to be a recipe for disaster. The Earth is warming and
the pace is quickening.” Fission reactors are likely to provide the short-term replacement for
oil and gas and the development of renewable energy sources, like wind and wave power is
progressing, but much too slowly. It seems unlikely that the latter will be sufficient in the long
run and the supply of U235 is even more limited than fossil fuels, not to mention the problems of storing radioactive waste and of proliferating bomb-making capacity. Fast-breeder
reactors, consuming the much more common U238 , could provide a long-term solution, but
these reactors are potentially more vulnerable to accidents and would produce large amounts
of plutonium that could be used in nuclear weapon production.
The fusion of light nuclei such as deuterium and tritium offers an alternative energy supply
without the disadvantages of the fossil and fission sources. While a fusion reactor would
generate some radioactive waste, this is believed to be largely short-lived and manageable.
However, the serious problem with fusion is the enormous temperature required to overcome
the repulsive force between colliding charged particles. The nuclei have to clash together
with the speeds achieved at temperatures about 12 times hotter than the centre of the Sun,
which also operates on fusion, but at densities some 1012 times greater than reactor values. At
these enormous temperatures confining the gas long enough for appreciable fusion reactions to
occur is a major problem. Strong magnetic fields provide the only possible constraint over the
motions of such energetic particles, and the most successful device employing this principle
is known as a tokamak.
A tokamak (Toroidal Kamera Magnitnaya, invented in the Soviet Union in the late 1950s)
is a toroidal chamber carrying a strong toroidal magnetic field to trap a high temperature
plasma. For a tokamak containing deuterium and tritium in equal parts to become a fusion
reactor, temperatures exceeding 2 ×108 K are required. The Joint European Torus (JET) at
Culham Laboratory, Oxfordshire, UK, has reached more than half of the required temperature, but the triple product of the ion number density i , the energy confinement time E and
temperature , still falls well short of the value 3 ×1021 s m−3 keV required for ignition; in
some D-T fusion experiments in JET a value of 8 7 ×1020 s m−3 keV has been attained.
Theory of Tokamak Transport: New Aspects for Nuclear Fusion Reactor Design. Leslie Colin Woods
Copyright © 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
ISBN: 3-527-40625-5
XII
Preface
A survey of the situation in the journal Nuclear Fusion, Vol. 39, no. 12, December, 1999,
commenced with the words:
“Magnetic fusion energy research has reached the point where a tokamak burning
plasma facility in which the thermonuclear heating balances (or is comparable to)
transport and radiation losses for periods of 1000 s or longer can be seriously contemplated as an appropriate next step. Achieving this goal would be a major step
forward, both in science and in technology, towards the ultimate goal of magnetic
fusion generation of electric power with significant environmental advantages.”
This volume of Nuclear Fusion was entirely devoted to explaining the background science and technology involved in the design of the International Thermonuclear Experimental
Reactor (ITER), of which there are two versions: Ignition ITER , which has a major radius
of 8 14 m and an estimated cost of 5870 million (1989) dollars and a less ambitious tokamak
called High-Q ITER , with a major radius of 6 2 m, costing 2755 million (1989) dollars and
for which ignition is not the main goal.
On 28th June, 2005, it was announced that High-Q ITER would be constructed at
Cadarache in the south of France; it has the aim of achieving an extended burn, with a ratio
of fusion power to auxiliary heating power of at least 10, and it is expected to begin operating
by 2015. The parties involved are China, the European Union, Japan, the Russian Federation,
South Korea, and the United States. The group emphasized “the importance of exploring the
long-term potential of fusion energy as a virtually limitless, environmentally acceptable and
economically competitive source of energy" and said they advocated “wide international cooperation in developing this source of energy for all mankind". It is forecast that terrestrial
fusion energy is likely to become a practical energy source by 2045. Presently, there are
more than 44 experimental tokamaks in laboratories around the globe, so the theory of these
machines is of continuing interest and seems likely to remain so for some decades.
Whether or not the project is practicable is difficult to judge at this stage, but in view of the
impending long-term energy crisis, it is important to continue the research and development,
which dates from the early 1950s. Also, apart from their likely relevance to the looming energy
crisis, tokamaks are useful apparatuses for a variety of experiments involving high energy
phenomena, radiation, and for obtaining a better understanding of the behaviour of plasmas,
which constitute more than 95% of the universe. One obvious gap in the tokamak literature
concerns the economics of fusion reactors, not merely their cost in relation to competitive
energy sources, but more importantly the energy investment required in their construction and
the time over which a reactor would need to operate to recover this investment. When the
basic physics and technology are better understood, this gap will need to be filled.
The last 100 pages of my text on the Principles of Magnetoplasma Dynamics (Clarendon
Press, Oxford, 1987) were devoted to the theory of tokamak machines and since then a number
of books have appeared on the subject, most notably the treatise entitled Tokamaks (Clarendon
Press, Oxford, 3rd ed, 2004) by John Wesson and some of his colleagues working at Culham
Laboratory. My aim here is to present an improved and enlarged version of my original
treatment of tokamak theory, to make more comparisons of the theory with observations and to
give explanations of some recently discovered phenomena. Although my theoretical approach
is quite different from the accepted treatments, it has the merit of yielding good agreement
with a wide range of observations and of being a ‘complete’ theory, in that the empirical input
Preface
XIII
is negligible. When and why it departs from received tokamak theory, as set out for example
in Wesson’s treatise, is noted appropriately in the text, which is mainly concerned with the
complexities of thermal and particle transport in toroidal geometry; for an introduction to
the more straightforward MHD calculations of stability, etc., and some of the technical issues
involved, besides Wesson’s text there is the volume of Nuclear Fusion cited above, and a work
by Miyamoto entitled Fundamentals of Plasma Physics and Controlled Fusion, (Iwanami
Book Service Center, Tokyo, 1997).
The physical principle that underlies most of the theory in this text is as follows. By
Fourier’s law the heat flux vector q is related to the temperature gradient by q =
∇ ,
where is the thermal conductivity. If the gradient ∇ is orthogonal to the magnetic field
)2 , where Q is the particle charge,
B = b , then is proportional to 1 ( c )2 = 1 (Q
−1
is the particle mass, and ( ) is the particle collision frequency. In tokamaks it is found
that electrons are mainly responsible for the energy loss and the electron parameter, 1 ( ce e ),
is typically 10−7 ; thus the heat flux vector across the magnetic field, q⊥ , is a mere 1 1014
times its value in the absence of a magnetic field, a circumstance that should have allowed
thermonuclear temperatures to have been easily reached with ohmic or other forms of heating.
However, in a strong magnetic field there is a transverse heat flux, q∧ =
∧b × ∇ ,
in which ∧ 1 ( ce e ), making q∧ about 107 times larger than q⊥ . But this heat, being
at right angles to the temperature gradient, normally circulates around the minor axis of the
tokamak torus and makes no difference to energy confinement within the tokamak, and all
would be well except for the presence of fluid shear. Shear is well-known to deflect any heat
flux vector through a small angle and to create what is called a second-order heat flux at right
angles to the primary, or first-order heat flux. The ‘order’ here refers to the Knudsen number
N , which in the tokamak application is e ∇ve , where ve is the electron fluid velocity and
the gradient ∇ve is a measure of its shear. Validity of macroscopic transport theory requires
that N
1, and in tokamaks N is typically 0 01. On comparing the first-order heat flux
2
2
with the deflected second-order heat flux q∧d
( ce e ), we see
q⊥
N ( ce e )
N
that the combination of shear and transverse diffusion removes energy from tokamaks at a
rate 105 times more rapidly than the early expectations, which were based on the first-order
theory. Curiously, this dominant process is still ignored in the tokamak literature, despite the
passage of more than twenty years since its discovery.
The deflected second-order heat flux will be directed either up or down the temperature
gradient depending on whether the radial gradient of the toroidal current density, ϕ , is antiparallel or parallel to the temperature gradient. The knowledge that there are circumstances in
which heat can flow up the temperature gradient, allows many strange tokamak observations
to be understood. Incidentally, it is very likely that this phenomenon is responsible for the
extremely hot solar corona, explaining how it is possible for thermal energy to flow up plasma
loops from the relatively cool 6 000 K photosphere to the 2 ×106 K corona. Although the
primary concern of this book is with fusion reactors, most of the transport theory developed
in the earlier chapters has applications to solar physics, for example to plasma loops, spicules,
flares and corona heating.
A similar treatment of the viscous force acting in tokamak magnetoplasmas enables the
radial flow velocity r to be determined from the second-order formula for this force, and
hence the rate at which plasma is lost to the tokamak walls can be calculated. The resulting
toroidal electric field, ϕ
r θ , where
θ is the poloidal component of the magnetic
XIV
Preface
field, drives a non-inductive current — called a Lorentz current in the text — that is additional to the induced current; substantial non-inductive currents in agreement with the Lorentz
current prediction have been observed and are important for the stability and heating of the
plasma.
For tokamaks there is a modified first-order theory called “neoclassical" transport, which
by allowing for non-local particle excursions over large ‘banana’ orbits, increases q⊥ by a
factor of several hundred, but this adjustment is still far too small to explain the observations. The usual approach is to speculate that turbulence is responsible for the unexpectedly
large thermal transport, and the experimental results from many tokamaks operating in a variety of conditions are assembled into best-fit, empirical curves, which, while practicable for
interpolation, provide no understanding of the physical mechanisms involved. The design
calculations for ITER are based on a single, straight-line extrapolation by a factor of more
than two beyond the highest points on the empirical curve for the energy confinement time
E . However, the presumption that turbulence is responsible for thermal transport is wrong,
as is easily inferred from the observation that the voltage drop around the torus is close to its
classical (non-turbulent) value.
Plasma physics is an exceedingly complex branch of macroscopic physics, especially
when applied in the domain of tokamak toroidal geometry. In this situation it is too easy
to allow formal equations to dominate and to impede a physical grasp of the convective and
diffusive mechanisms of transport upon which the success or failure of the tokamak enterprize depends. There is no single master equation from which deductive analysis will yield
good estimates of the losses of plasma energy from tokamaks. For example, the ‘shearedtransverse-diffusion’ transport described above and which is the basis of much of this book,
cannot be deduced from Boltzmann’s famous kinetic equation, which is generally supposed to
cover all transport possibilities. As Eddington once remarked in a lecture at a stage where he
was stressing the importance of a proper background to the analysis he was about to present:
“I regard the introductory part of the theory as the more difficult, because we have
to use our brains all the time. . . . Afterwards we can use mathematics instead.”
In tokamak physics the situation is particularly demanding, for excepting some stretches
of straightforward deductive analysis, physical modelling is required as an essential guide
throughout.
To make the account nearly self-contained for graduate students with some experience
in continuum physics, most of the background knowledge required in plasma physics, kinetic
theory and thermodynamics is either provided in the text or collected as ‘plasma physics notes’
in the Appendix.
I am grateful to Mr D. E. T. F. Ashby, ex-Culham Laboratory, for his constructive criticism
and generous help in the drafting of this book and to Dr Grant Deane of Scripps Institution of
Oceanography, who took time from his research to revisit his tokamak background to give me
many helpful comments.
Finally, I record with pleasure my appreciation of the help and ready support given me by
the officers of the Wiley-VCH Press.
L. C. Woods
Oxford, 20 July, 2005
Lists of physical constants, plasma parameters and
frequently used symbols
In SI units, the constants required in plasma theory are:
Physical Quantity
Electron mass
Proton mass
Electron charge
Boltzmann constant
Permittivity (Free Space)
Permeability (Free Space)
Speed of light (Vacuum)
Proton/electron mass ratio
Temperature at 1 eV
Planck constant
Stefan-Boltzmann constant
Gas constant
Symbol
Value
units
me
mp
e
kB
0
µ0
c
mp /me
e/kB
h
σ
R
9.1095 ×10−31
× −27
kg
kg
C
J K−1
F m−1
H m−1
m s−1
1.6726 10
1.6022 ×10−19
1.3807 ×10−23
8.8542 ×10−12
4π ×10−7
2.9979 ×108
1.8362 ×103
1.1605 ×104
6.6262 ×10−34
5.6703 ×10−8
8.3144
K
Js
W m−2 K−4
J K−1 mol−1
The important plasma parameters are:
Parameter
Resistivity
Cyclotron frequency (electrons)
Thermal speed
Larmor radius
Coulomb logarithm
Collision intervals
Thermal conductivity (B = 0)
Magnetic diffusivity
Magnetic Reynolds number
Plasma frequency
Collisionless skin-depth
Debye length
Symbol
η
ωce
C
rL
ln Λ
τe , τi
κ
ξ
Rm
ωpe
δe
λD
Formula
page
2
180
185
185
185
182
124
61
199
–
179
–
179
αme /(e ne τe )
eB/m
e
2kB T /m
C/ωc
γkB pτ /m
η/µ0
U
L/ξ
ne e2 /0 me
c/ω
pe
0 kB Te /ne e2
Theory of Tokamak Transport: New Aspects for Nuclear Fusion Reactor Design. Leslie Colin Woods
Copyright © 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
ISBN: 3-527-40625-5
XVI
Lists of physical constants, plasma parameters and frequently used symbols
Frequently used Tokamak symbols
Symbol
Definition
a
b
Bϕ
D
E
g
H
j
M
pe , pi
q0 , qa
Qr
minor radius
= B/|B|
toroidal field
convective derivative
electric field
passing particle fraction
velocity gradient
current density
magnetic moment
pressures
central & surface q
radial heat flux
= Bθ /Bϕ
internal energy
loop voltage drop
= (r/a)2
1
182
179
43
45
179
38
12
8
11
8
11
107
87
profile parameter
plasma beta
poloidal beta
profile parameter
= r/R
resistivity
magnetic diffusivity
density
energy confinement time
collision times
thermal diffusivity
88
7
8
89
8
182
199
11
11
124
61
S
u
V
y
α
β
βp
γ
ε
η
ξ
τE
τe , τ i
χ
page
4
Symbol
Definition
page
B
Bθ
c
magnetic field (induction)
poloidal field
peculiar velocity
rate of strain tensor
trapped fraction
specific enthalpy
plasma current
radiation rate
number densities
safety factor
particle charge
major radius
temperature
fluid velocity
Lorentz voltage
Z-effective
3
1
181
192
41
11
1
183
2, 3
8
e
fT
h
Ip
L
ne , ni
q
Q
R0 , R
T
v
VL
Zeff
αE
βt
βN
δ
κ
η , η⊥
σ
τ∗E
τϕ
ωc
profile parameter
toroidal beta
normalized beta
profile parameter
thermal conductivity
parallel & perpendicular
viscosity tensor
electrical conductivity
energy replacement time
momentum confinement time
cyclotron frequency
4, 94
11
132
9
89
8
8
89
61
182, 183
184
183
11
14
185
We shall often deviate from SI units with temperature, number density and plasma current
thus:
Temperature:
Number density:
Electric current:
T K
n m−3
Ip
=
=
=
1.1605 ×107 T̂ ,
1019 n19 ,
106 Îp ,
T̂ in keV,
n19 in 1019 units per m−3 ,
Îp in MA.
To reference particular equations forming part of a group, we shall adopt the notation
(a.b)(n) to indicate the n-th equation of the set (a.b).
1 The quest for fusion power
This chapter introduces the basic physics and associated variables. Except for those variables
cited at the foot of page XVI, SI units are almost always adopted. Pages XV and XVI have
lists of physical constants, plasma parameters and frequently used symbols.
1.1 Tokamak machines
1.1.1 Topology and ignition
A tokamak is a toroidal chamber which uses a strong toroidal magnetic field, Bϕ , to contain a
high temperature plasma within the torus. Charged particles cannot easily move across strong
magnetic fields and if the fields are closed into nested surfaces, then deuterium and tritium ions
trapped in this way and colliding with sufficient energy to overcome their repulsive Coulomb
potential, will fuse and liberate energy. The toroidal field is produced by external electric
currents flowing in coils wound around the torus, as shown in Fig. 1.1. Superimposed on the
toroidal field is a much weaker poloidal field, Bθ , generated by an electric current Ip flowing
in the plasma around the torus. The plasma forms the secondary circuit of a transformer,
so that Ip is induced by changing the magnetic flux BT passing through the torus, which is
usually carried by an iron core as indicated in the figure.
external poloidal
current producing
B field
iron core
plasma
B
BT
Ip
Ip
B
Ip
(a)
(b)
Figure 1.1: Tokamak currents and fields: (a) toroidal plasma current induced by transformer,
(b) primary winding
Theory of Tokamak Transport: New Aspects for Nuclear Fusion Reactor Design. Leslie Colin Woods
Copyright © 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
ISBN: 3-527-40625-5
2
1 The quest for fusion power
In a plasma consisting of deuterium, or deuterium mixed with tritium, the fusion reactions
He3 + n1 + 3.27 MeV
2
2
D +D →
T3 + H1 + 4.03 MeV
and
D2 + He3 → He4 + H1 + 18.3 MeV
D2 + T3 → He4 + n1 + 17.6 MeV
will occur frequently if the ion temperature, Ti , and the ion number density, ni , are large
enough. Furthermore, in a fusion reactor these high values of Ti and ni must be maintained
long enough for the energy liberated by fusion to more than balance the energy losses due
to radiation, conduction, convection and neutron flux. Let τE be the time it takes these loss
processes to remove all the energy from the system, then for a given value of ni τE there
is a minimum temperature at which the plasma is said to ignite, i.e. at which the liberated
fusion energy is just adequate to balance all losses. As D-D plasmas require considerably
higher temperatures to achieve ignition, almost all reactor proposals have concentrated on
D-T fusion.
Figure 1.2: Ignition curve for a D-T plasma
Figure 1.2 shows the ignition curve for a D-T plasma. It has a minimum at a temperature
of about 30 keV, where for ignition we need ni τE > 1.5 ×1020 m−3 s. A slightly lower bound
(ni τE > 6 ×1019 m−3 s) known as Lawson’s criterion (Lawson 1957) is obtained if a continuous power supply from outside the system is used to compensate transport and radiation
losses. Combining the ne τE value with T̂ ∼ 10 keV, we obtain
τE ni T̂ > 3 ×1021 s m−3 keV ,
(1.1)
1.1 Tokamak machines
3
which is based on the assumption that the number density and temperature profiles across the
minor radius are flat. When allowance is made for typical profile shapes, and the constraint is
applied to the peak values, T0 and ni0 of the temperature and number density profiles, (1.1) is
replaced by
τE ni0 T̂0 > 5 ×1021 s m−3 keV .
Observations show that electron energy loses are dominant and in a pure D-T plasma, by
charge neutrality, ni = ne , and so to a good approximation the left-hand side of (1.1) can be
replaced by τEe ne T̂e .
Let B denote the strength of the magnetic field1 , then for a reason explained in the first of
the plasma physics notes in the Appendix, B 2 /2µ0 is called the magnetic pressure, where µ0
is the free-space permeability. An important parameter in plasma physics is the ratio of the
plasma pressure p to the magnetic pressure, which is known as the plasma beta,
2µ0 p
.
(1.2)
β≡
B2
The power output for a given magnetic field and plasma assembly is proportional to the square
of beta, and for an adequate return on an energy investment in magnetic fields, it has been
estimated that in a reactor β should exceed 0.1.
Figure 1.3: The Joint European Torus (JET)
1 Strictly the magnetic induction, but the misnomer ‘field’ is commonly adopted in plasma physics.
4
1 The quest for fusion power
1.1.2 Some early tokamaks
The advantage of the Russian tokamak machine over similar toroidal devices that were being
developed in the United States and Great Britain at the same time, lay in the better stability
obtained by using much stronger toroidal magnetic fields. ‘Stability’ in this context means
no more than the persistence of the magnetic fields and electric currents — at least in the
earlier machines — for times of the order of milliseconds. The British ZETA machine, which
received much publicity in the 1950s, was so-called ‘stable’ for less than about 5 milliseconds,
whereas the discharge in comparable tokamaks lasted over ten times longer.
In his review of the history of tokamak research from 1955 to 1980, Rutherford (1980)
noted that this confinement device was responsible for more than half the articles published
in the specialist journal Nuclear Fusion. The first substantial tokamak was T-3, built at the
Kurchatov Institute, Moscow in the 1960s. It had a minor radius of 15 cm, a major radius of
100 cm, a toroidal magnetic field of 15 kG, and carried a plasma current of 100−250 kA. In the
standard notation (see Fig. 1.4), a = 0.15 m, R0 = 1 m, Bϕ = 1.5 T, Îp = 0.1 − 0.25 MA.
Some twenty years later the Joint European Torus (JET) was constructed at a cost of
around £200 M on the Culham site at Abingdon, England, and this is currently the largest
tokamak in the world. The cross-section of the torus in JET is D-shaped, with a (horizontal) width of 2.4 m and a height of 4.2 m. Its parameters are: a = 1.2 × 2.1 m, R0 = 3 m,
Bϕ = 3.5 T, Îp = 5 MA. Whereas T-3 reached electron temperatures ∼ 0.4 − 1.0 keV and
ion temperatures ∼ 0.2 keV at average electron number densities of n̄e ∼ 2 ×1019 m−3 and
energy confinement times of only a few milliseconds, by 1986 JET had achieved Te ∼ 6 keV,
Ti ∼ 12 keV, n̄e ∼ 3.5 ×1019 m−3 and τE ∼ 0.9 s, although not simultaneously. However,
from (1.1) increases by factors of 3 in Ti and 5 in ni τE were still required for ignition.
Wesson (2004) gives details of forty-four tokamaks built up to 1985 in England, France,
Germany, Italy, Japan, USA, and USSR; Table 1.1 lists those built since 1975. Notice that
under the column of the minor radius, DOUBLET III and JET have two lengths written as
a × b where b is the half-height of the plasma and a is the minor radius, or half-width of the
plasma; these lengths serve as a rough specification of D-shaped cross sections (e.g. JET’s
Table 1.1: Typical values of tokamak parameters (not simultaneous)
Machine
year
(m)
R0
a
Bϕ
(m)
(T)
DITE
1975 1.17
0.26
PLT
1975 1.3
0.40
T-10
1975 1.5
0.37
DOUBLET III
1979 1.43 0.44 × 0.75
TFTR
1982 2.4
0.80
JET
1983 3.0
1.2 × 2.1
TEXTOR
1983 1.75
0.46
JT-60
1985 3.0
0.9
DIII-D
1986 1.67
0.67
ASDEC (upgrade) 1991 1.65
0.50
2.7
3.5
4.5
2.4
5.0
3.5
2.0
4.5
2.1
3.9
Îp
n̄e
(MA) 10−19 m−3
0.2
0.6
0.5
0.9
2.2
5.0
0.4
2.0
5.0
1.4
5
5
4
10
4
3.5
3
7
8
11
T̂e0
T̂i0
τE
(keV) (keV) (ms)
0.7
3
1.4
4
2
6
1.2
3
26
0.6
3
0.7
4
8
8
0.8
5
20
14
40
50
100
200
500
40
100
160
1.2 Basic tokamak variables
5
vacuum vessel shown in Fig. 1.3). The elongation of the cross-section follows from a solution
of the MHD equilibrium equations, which determine the magnetic field structure appropriate
for a given choice of pressure and current profiles (Section 2.1). However, in this text to
simplify the analysis with relatively little impact on general conclusions concerning transport,
the ‘elongation’ variable, κ = b/a, will be taken to be unity.
1.1.3 Toroidal current
There is one evident disadvantage in the tokamak design as illustrated in Fig. 1.1, namely that
its operation is necessarily pulsed because resistivity will gradually dissipate the inductive
current and switch off the discharge. Quite apart from its role in heating the plasma through
ohmic dissipation, a toroidal current is essential to maintain an elongated toroidal system in
equilibrium, for without the Bθ field that it generates, there is a vertical instability that causes
the plasma to drift in the direction of elongation. The force driving this instability results from
the interaction of the poloidal field coil currents (see Fig. 1.1) and the plasma current. In some
cases feedback control circuitry is necessary to maintain the plasma’s position (see Wesson,
2004, p. 342).
Early tokamaks, which relied entirely on inductive currents for both heating and stabilization, were therefore designed for pulsed operation in the hope that the pulse time could
be made sufficiently long for fusion to be effective; but these times are measured in seconds
rather than minutes and are too short for reactor operation.
Finding other ways of continuously heating the plasma and of maintaining the stabilizing
toroidal current, has been an important quest in recent tokamak research. Steady currents can
be driven around the torus with radio-frequency (RF) waves and also with neutral beam injection (NBI), but there are limits to this type of ‘current drive’ that make it unable to generate
all of the current required for a stable reactor. One such constraint, called the ‘Greenwald’
limit, is concerned with the avoidance of major disruptions (Section 6.2.1). For a survey of
NBI current drive the reader is referred to ITER team (1999, p. 2527).
However, there is another mechanism that generates non-inductive toroidal currents. It is
widely believed that a large current of this type, termed a ‘bootstrap’ current, can be generated
simply by the existence of radial gradients in the plasma density and temperature. Observations certainly support the presence of a non-inductive current, but its origin is not the bootstrap phenomenon, for as shown in Section 3.4.3, such a current does not satisfy Ampère’s
law and cannot exist. In Section 5.3.2 we show that the observed non-inductive current is a
result of the toroidal electric field generated by the radial flow of the plasma across the Bθ
magnetic field.
Let vD be the radial velocity of the plasma flowing across the tokamak magnetic field, then
the toroidal electric field, say EϕLR , driving the non-inductive current is proportional to the
product vD Bθ , so the ‘price’ of this potentially steady current is the continual loss of plasma
from the torus. Regular refueling by beam injection near the minor axis is therefore required
to maintain the current, a process with its own limitations (see Section 1.4.2).
6
1 The quest for fusion power
Figure 1.4: Cylindrical and local coordinates for a tokamak machine
1.2 Basic tokamak variables
1.2.1 Aspect ratio
Figure 1.4 shows the coordinate systems for a tokamak of circular cross-section. The local
radial dimension lies in the range 0 < r < a, where a is the maximum radius of the plasma. In
order to prevent the plasma reaching the vacuum vessel, either a material limiter or a magnetic
divertor is used, as shown in Fig. 1.5. Most tokamaks have limiters, but divertors have the
merit of reducing the influx of ionized impurities into the interior of the plasma by diverting
them into an outer “scrape-off” layer.
The tokamak aspect ratio, R0 /a, usually lies between 3 and 5 and as we shall see later, it
has an important role in plasma energy confinement.
Figure 1.5: Separation of plasma from wall by (a) a limiter, (b) a divertor
1.2 Basic tokamak variables
7
Figure 1.6: Nested magnetic surfaces confining a plasma
1.2.2 Beta
Several forms of the ratio of the average plasma pressure to the magnetic field pressure2 arise
in tokamak theory. For simplicity we shall assume that the magnetic surfaces have concentric,
circular cross-sections and that conditions are independent of the value of the toroidal variable,
ϕ, defined in Fig. 1.4. To obtain the volume-averaged pressure p, we integrate over a crosssection ϕ = const.,
a
2
p(r)r dr .
(1.3)
p =
p dS
dS = 2
a 0
From the ϕ̂-component of the differential form of Ampère’s law relating the magnetic
field vector B to the electric current density j, viz. ∇ × B = µ0 j, we get
1 ∂ µ0 r
rBθ = µ0 jϕ ,
Bθ =
jϕ (r )r dr
(1.4)
r ∂r
r 0
and
Ip = 2π
a
0
jϕ r dr = 2πaBθa /µ0 ,
(1.5)
where Ip is the total current flowing around the torus and Bθa is the poloidal magnetic field
at the limiter, r = a. In the following we shall assume that small variations in Bϕ across the
plasma cross-section can be ignored.
In Section A.1 it is shown that in equilibrium configurations, B and j lie on constant
pressure surfaces, which if closed, appear as continuous windings of intersecting magnetic
field and current lines; these are said to lie on ‘magnetic surfaces’ and p is termed a ‘surface
quantity’. Figure 1.6 shows a set of nested surfaces, with a limit line at their center, known as
the ‘magnetic axis’. If p increases towards the axis, its negative gradient is balanced by the
j × B force directed inwards; the plasma is thus confined by the magnetic force.
2 See Section A.1, the first of the Plasma Physics Notes, collected in the Appendix and mostly intended
for readers not familiar with the equations of plasma physics. The Notes are referenced in the text as Section A.1, Section A.2 . . . and the equations are numbered consecutively throughout the Appendix: (A.1), (A.2),. . . ,
(A.100),. . . , etc.
8
1 The quest for fusion power
Functions of importance in tokamak theory are the toroidal beta βt and the poloidal beta
βp , which are defined by
βt =
2µ0 p
,
Bϕ2
βp =
2µ0 p
8π 2 a2 p
=
.
2
Bθa
µ0 Ip2
(1.6)
In Section 1.1 we mentioned the connection between βt and the economic viability of a
tokamak reactor, which expressed as a percentage, is βt ≥ 10%; this is only a rough estimate
of the economic constraint — higher values may be required.
On the other hand, ideal MHD stability imposes an upper limit on βt . The type of instability involved is termed a ‘ballooning mode’ (see Section 6.3.2), and the outcome are the
approximate β-limits,
a
R0 qa
aBϕ qa ≡
,
βp ≤ 0.15
,
βt ≤ 0.15
R0 qa
a
R0 Bθa
or
R0 qa
a
βN ≡ 20βt
= 20βp
≤ 3.5 ,
(1.7)
a
R0 qa
where βN is called the ‘normalized’ beta and qa is the safety factor defined in the following
section.
1.2.3 Safety factor
The safety factor is another important parameter, so named because of its association with
stability, as explained in Section A.24. In a large aspect ratio tokamak with a circular crosssection, this parameter is defined by
q(r) =
ε
rBϕ
= ,
R0 Bθ
S
where
r
ε≡ ,
R0
Bθ
µ0
S ≡
=
Bϕ
Bϕ r
(1.8)
r
0
jϕ (r )r dr .
(1.9)
In tokamaks S is much smaller than unity.
At the limiter by (1.5) and (1.18) q has the value
qa =
aBϕ
2πa2 Bϕ
5a2 Bϕ
=
=
,
R0 Bθa
µ0 Ip R0
Îp R0
Îp in MA .
(1.10)
Hence the average current density, jϕ = Ip /πa2 , is
µ0 jϕ =
2Bϕ
.
R0 qa
(1.11)
By expanding jϕ in the form jϕ = jϕ0 + O(r2 ), where jϕ0 is the current density on the minor
axis, we find from (1.8) and (1.9) that on the magnetic axis (r = 0), the safety factor has the
value
2Bϕ
.
(1.12)
q0 =
µ0 jϕ0 R0
1.2 Basic tokamak variables
9
From (1.11) and (1.12) we obtain
qa /q0 = jϕ0 /jϕ ,
(1.13)
hence large values of qa /q0 correspond to peaked current profiles.
The general definition of q is
Bϕ
ds ,
q=
R0 Bθ
where the integral is along a closed path enclosing the minor axis and lying on a specific
magnetic surface; thus q is a surface quantity.
1.2.4 Z-effective
Tokamaks usually have several types of ion in their plasmas, due mainly to impurities entering from the torus walls, and a convenient measure of the extent to which the plasma is
contaminated is the function known as ‘Z-effective’, defined by
ns Zs2
ne Zeff =
s
ne =
ns Z s ,
s
where Zs is the charge number for the s-type ion. In a pure hydrogen plasma, Zeff = 1, but few
tokamaks achieve values even near this ideal. Pfeiffer and Waltz (1979) list 118 observations
on 11 early tokamaks. Many of these machines were heavily contaminated, the average Zeff
being about 5. Initially the JET tokamak had Zeff lying in a range extending from above 2 to
about 10 (Christiansen et al. 1985). More recently this has dropped to a range from just below
2 to about 3.5.
Figure 1.7, from the JET Team (1990), illustrates the importance of the choice of boundary
0.7
materials in limiter tokamaks. An empirical law for JET of the type Zeff ∝ 1/(n0.9
19 qa ),
Figure 1.7: Zeff as a function of density with either graphite or beryllium limiters
10
1 The quest for fusion power
where n19 = n/1019 , has been found (Cordey et al. 1985b), while Matthews et al. (1997)
have compiled a multi-machine data base showing that Zeff depends on the radiated energy,
the plasma surface area and n2e for all divertor tokamaks, independent of geometry.
Impurity concentrations may be determined by analyzing resonance line intensities in the
vacuum UV, supplemented by measurements of soft X-ray spectra; this data, coupled with a
theory for ionization rates, enables Zeff to be estimated. Another method determines Zeff from
the visible bremsstrahlung radiation. In JET the two methods yield values for Zeff that are
usually within ±1 of each other. The main impurities in JET are C (2–3 per cent), O (1–4 per
cent), Cl and Ni (Denne et al. 1985).
A further method of estimating Zeff relies on an application of Spitzer’s (1962) formula for
the parallel resistivity (see Section A.2). Measurements of the plasma current Ip , the ‘loop’
voltage V around the torus, and assumptions about the radial distribution of the variables,
enables Zeff to be calculated from the integral
a
Ip =
jϕ dS = 2π
ϕ̂ · σ · E + v × B dr ,
0
where σ is the conductivity tensor, E is the electric field, and v is the plasma velocity. We
also need the equation for the electron collision interval (see (A.16) in Section A.2),
τe =
3/2
2.75 ×105 Te
,
ln Λ ne Zeff
(1.14)
and the relation
Vt /2πR0 = ϕ̂ · E + v × B = Eϕ + vr Bθ ,
(1.15)
defining the total voltage Vt . It is usual to omit the term vr Bθ compared with Eϕ , but this can
result in appreciable errors, as will be explained in Section 5.3.2.
An important modification to this method (Christiansen et al. 1985) replaces the parallel
conductivity σ (see (A.45)) by the so-called neoclassical (Section 1.5.1) conductivity, one
formula for which is (Wesson 2004, p. 174)
σ̂ = gσ ,
1 2
g ≈ 1 − ε2 ,
ε = r/R0 .
(1.16)
As will be explained in Section 2.4.4, the factor g is due to the trapping of particles between
magnetic mirrors in the tokamak field, which reduces the number of electrons available to
conduct electric currents. (In the rest of this text, we shall use σ̂ and η̂ = η/g to denote the
‘trapped particle’ values of the parallel conductivity and parallel resistivity.)
1.3 Global confinement times
Overall measures of the confinement properties of tokamaks are provided by the times taken
for the whole of their mass, momentum, and energy to be lost in the absence of replacements.
In the following we shall ignore the toroidal curvature, treating the cross-sections as having
axial symmetry about the minor axis. Alternately, we could take poloidal averages to remove
1.3 Global confinement times
11
the θ-dependence of the variables, but to first-order in ε = r/R0 the results are the same. The
most frequently used and important global confinement time is that for the plasma thermal
energy. Before defining it, we need an appropriate form of the energy equation.
From the equation of plasma motion (see (A.3)),
∂
+ v · ∇ v + ∇p = j × B ,
∂t
where
is the plasma density and v is the fluid velocity, we find that
∂v
v · ∇p − j × B = v ·
+ v · ∇v .
∂t
(1.17)
Let vD denote the radial velocity of the plasma, which with good plasma confinement, we
expect to
In tokamaks the force lies in the radial direction and (1.17) shows
be quite small.
that v · ∇p − j × B is O(vD2 ), small enough to be removed from the plasma energy equation
defined in (A.29). Also the poloidal average of jθ Eθ is zero, whence
∂( u) 1 ∂ +
r hvD + Qr = jϕ Eϕ − L ,
∂t
r ∂r
(1.18)
where u (= 32 p) and h (= 52 p) are the internal energy and enthalpy densities, Qr is the
sum of the electron and ion heat fluxes and L is the rate at which energy lost by radiation.
1.3.1 Energy confinement time
The total thermal energy in the torus is proportional to
a
1
3
3
W =
2 p dS =
2 kB (ne Te + ni Ti ) r dr ,
2π
0
(1.19)
so if (1.18) is integrated over a plasma cross-section orthogonal to the minor axis, the result
can be expressed
1
1
1
∂
ln W +
= ∗− R,
∂t
τE
τE
τE
(1.20)
5
r( 2 pvD + Qr ) r=a ,
(1.21)
jϕ Eϕ r dr ,
(1.22)
L r dr .
(1.23)
where
τE ≡ W
τE∗ ≡ W
and
τER ≡ W
a
0
a
0
These expressions define the energy confinement time τE , the energy replacement time τE∗ ,
and the radiation loss time τER . In deriving (1.22) it is assumed that τE∗ is due only to ohmic
heating, jϕ Eϕ . With other methods of supplying thermal energy, the denominator on the right
12
1 The quest for fusion power
hand side of (1.22) is modified to give the total power input. An apparent difficulty in the
definition of τE is that the denominator is evaluated at the limiter, where the variables will be
sensitive to boundary conditions. A method of avoiding this strong local dependence will be
given in Section 4.1.3.
The radiation losses vary considerably from one tokamak to the next, depending on the
amount and type of impurities that have entered from the walls. With a relatively clean plasma
the radiated power will lie between 10 and 20 per cent of the input power, but with contaminated plasmas, Zeff can be 5 or larger, resulting in some 50 per cent or more of the input power
being radiated. Impurity radiation typically peaks at temperatures less than 100 eV (Ashby and
Hughes 1981), so that clean, hot plasmas radiate mostly from the peripheral regions. In these
cases the radiation term in (1.18) can be neglected almost up to the limiter position. If steady
conditions can be assumed, (1.20) gives
τE =
τE∗ τER
.
τER − τE∗
(1.24)
Values of τE∗ and, with more difficulty, τER , can be deduced from observations, and a theory of
the transport of energy and mass in tokamaks would enable τE to be calculated. Allowing for
the uncertainty in the observations, a satisfactory theory should yield values of τE agreeing
with the right-hand side of (1.24) to within a factor of about 2 for a wide range of tokamak
conditions.
Besides giving correct values for the confinement time, a tokamak transport theory must
also pass the more difficult test of giving the correct radial dependence for dependent variables
like Te and ne . When the mass and thermal diffusivities are themselves complicated, nonlinear functions of these variables, a particularly severe test for the theory is that the radial
dependencies that it predicts for these diffusivities agree with the experimental distributions
of these quantities, a issue to which we shall return in Section 4.1.2.
1.3.2 Electron-energy confinement time
In many tokamak experiments the ion temperature and density are poorly known and in these
cases it is usual to introduce the electron analogues of τE and τE∗ . The energy equation for the
electron gas is (see (A.28)):
∂
e ue + ∇ ·
e he ve + Qe = j · E + v · ∇pe − j × B + Qei − Le ,
∂t
where we have used the approximation ve = v − j/ene , which follows from the definition of
j given in (A.11), the relation v = (mi vi + me ve )/(mi + me ) and me
mi .
It follows from ∇ · j = 0 and the assumed geometry that ∂jr /∂r = 0, so that
vir = ver = vD (known as the ambipolar condition). From (1.17) and Dalton’s law,
p = pi + pe , we find that for the electron gas (1.18) is replaced by:
∂pi
∂( e ue ) 1 ∂ +
r e he vD + Qer = jϕ Eϕ − vD
+ Qei − Le .
∂t
r ∂r
∂r
We also need the perfect gas law, pe = ne kB Te , where kB is Boltzmann’s constant3 .
3 There is a list of physical constants on page XV.
(1.25)
1.3 Global confinement times
13
The electron time scales are:
τEe ≡ We r( 52 pe vD + Qer ) r=a ,
τE∗e ≡ We
and
τERe ≡ We
where
We =
a
0
a
(1.26)
jϕ Eϕ r dr ,
(1.27)
Le r dr ,
(1.28)
3
2 kB ne Te r dr .
(1.29)
0
a
0
However, the appearance of −vD ∂pi /∂r and Qei on the right-hand side of (1.25) requires a
knowledge of ni (r) and Ti (r) for an accurate determination of τE ; these terms are usually
neglected, which is justified if approximate estimates are sufficient.
1.3.3 Particle confinement time
Balance of electron numbers gives
∂ne
1 ∂ +
rne vD = Se (r) ,
∂t
r ∂r
(1.30)
where Se (r) is the electron number density source term. In the steady state this equation
yields
τp = τp∗ ,
where
a
(1.31)
rne vD r=a ,
(1.32)
is the plasma confinement time and
a
a
ne r dr
Se r dr .
τp∗ ≡
(1.33)
τp ≡
0
0
ne r dr
0
is the plasma replacement time.
The main problem in using (1.31) to test a theory of mass diffusivity lies in finding an
estimate for Se (r). In clean plasmas Se is largely due to the ionization of the working gas,
which raises the problem of the distribution of the neutrals. In highly contaminated plasmas
ionization of impurities is the main source of electrons, so a theory giving their distribution is
required. Estimates of τp∗ can be made from observations of the response of the discharge to
a brief puff of neutral gas admitted through a fast-acting valve. These difficulties are reflected
in the fact that relatively few data are available for the particle replacement time; some of the
early observations have been listed by Hugill (1983).
14
1 The quest for fusion power
1.3.4 Momentum confinement time
Let the plasma be subject to a force density Fb due to a beam of particles being injected from
an outside source, then the momentum equation for the plasma as a whole reads (see (A.7))
∂ v + ∇ · vv + ∇p + ∇ ·
∂t
= j × B + Fb .
The forces j × B and ∇p lie along r̂, where r̂ is unit vector in the radial direction, therefore
in the axi-symmetric geometry described earlier, this equation has the toroidal component,
1 ∂ ∂
vϕ +
r vr vϕ + ∇ ·
∂t
r ∂r
· ϕ̂ = Fb · ϕ̂ ,
(1.34)
where ϕ̂ is unit vector in the toroidal direction. The toroidal momentum confinement and
replacement times are:
a
−1
τϕ = Hϕ
∇ · · ϕ̂ r dr + r vr vϕ a
,
(1.35)
0
and
τϕ∗ = Hϕ
where
Hϕ ≡
a
a
0
0
Fb · ϕ̂ r dr =
vϕ r dr ,
2π 2 R02 a2 Hϕ
,
Beam torque
(1.36)
(1.37)
and the beam torque is about the major axis. Similar definitions can be given for the poloidal
momentum time-scales. In the steady-state (1.34) has the integral τϕ = τϕ∗ .
Collisions ensure that all ions have much the same toroidal speed, so that the Doppler shift
of spectroscopic lines from various impurities can be used to determine vϕ . Estimates of the
beam torque supplied to the plasma can be obtained by applying Monte Carlo methods to the
beam particles, and then values of τϕ∗ given by (1.36) may be used to check any theory yielding
values for τϕ . Also, an approximate value of τϕ may be obtained directly by switching off
the beam and determining the e-folding time, τsϕ , for vϕ to decay to ohmic collisional levels.
With linear viscosity, we would expect τsϕ ≈ τϕ , although this proves to be inaccurate (see
Section 5.4.3).
1.4 Heating
A brief account of the various methods of heating tokamak plasmas is appropriate at this
stage, since the central problem that will concern us later is the loss of this thermal energy
at rates many times greater than initially predicted by the usual theories. Figure 1.2 indicates
the magnitude of the heating task. Three types of heating are commonly used — ohmic
heating (OH), neutral beam injection (NBI) and radio-frequency heating (RFH); unfortunately
to date the temperatures achieved by these methods are somewhat lower than those required
for ignition.
1.4 Heating
15
1.4.1 Ohmic heating
The experimental evidence in the early experiments (Hugill 1983) appeared to support the
Spitzer formula for the parallel conductivity, which in a hydrogen plasma is (see (A.18))
σ = 1.98 e2ne τe /me .
(1.38)
The validity of the neoclassical4 factor g appearing in (1.16) is difficult to test in small tokamaks, but in Section 4.5.3 it will shown that it increases the resistivity in JET by a factor of
∼ 2.86 and ohmic heating is similarly enhanced.
Provided the transformer action illustrated in Fig. 1.1 occurs on a time-scale long enough
to permit the electric field to penetrate the plasma, it may be assumed that Eϕ is approximately
constant across the plasma cross-section. In this case the current profile can be deduced from
the temperature profile, since by Ohm’s law (jϕ = σ Eϕ ), (1.14) and (1.38) it follows that
jϕ ∝ Te3/2 /Zeff .
(1.39)
The initial heating in tokamaks is due to ohmic dissipation of the toroidal current, which
occurs at the rate,
PΩ = η̂ jϕ2
(1.40)
η̂ ≡ η /g
per unit volume. While this is sufficient to achieve temperatures up to 1 keV or so, because η
−3/2
is proportional to Te
, it becomes inefficient at higher temperatures. An estimate for PΩ at
the center of the plasma can be found from (1.12) and the approximation q0 ≈ 1; thus with
g ≈ 0.35 for JET (see Section 4.5.3),
2
2
2Bϕ
−3/2 ≈ 0.20 T̂e0
(1.41)
PΩ0 = η̂
Bϕ /R0 Zeff MW m−3 ,
µ0 R0
where T̂e0 is the central temperature in keV.
This input power first heats the electrons, consequently for equilibrium it should balance
the loss rate PL ∼ 32 kB ne0 Te0 /τEe . Later (Section 4.1.3) we shall show that in low βp plasmas,
τEe ≈ 9.4 ×10−22 ne aR02 qa T̂e −1/2 ,
where for circular cross-sections,
a
2
ne ≡ 2
ne (r) r dr ,
a 0
a
and
2
ne (r) Te (r) r dr .
Te ≡
ne a2 0
(1.42)
(1.43)
(1.44)
With typical profiles (see Section 4.1.1), ne0 ≈ 2.25ne and Te0 ≈ 2.1Te , therefore
PL = 0.41
3/2
T̂e0
aR02 qa
MW m−3 .
(1.45)
4 For a brief description of neoclassical transport see Section 1.5.1; a fuller account is given in Section 3.4.
16
1 The quest for fusion power
Equating PL and PΩ0 we arrive at the approximate relation
1/3 2/3
Bϕ Zeff keV .
T̂e0 ≈ 0.79 aqa
(1.46)
In JET, under typical ohmic heating conditions (a = 1.2, qa = 3, Bϕ = 3), this formula
gives T̂e0 ∼ 2 keV. A typical pre-1980 tokamak (Pfeiffer and Waltz 1979) has a = 0.2,
qa = 5, Bϕ = 3, and by (1.46), T̂e0 ∼ 1.6 keV; these temperatures are similar to those obtained in experiments. The central temperatures are subject to considerable variations because
of MHD instabilities and impurities, so (1.46) is not expected to be accurate, but at least it is
sufficient to indicate the limitations of ohmic heating.
1.4.2 Neutral beam heating
When a beam of high-velocity neutral particles is injected into a tokamak plasma, it becomes
ionized by charge exchange and particle collisions. The fast ions that result are then slowed
down by Coulomb collisions, transferring most of their energy into electron thermal energy.
Let mb , vb , and ξb = 12 mb vb2 denote the beam ion mass, velocity, and energy. The drag
force Fbe that the beam particles experience due to collisions with the electrons is mb vb /τbe
where τbe is the slowing-down time for beam particles. The rate at which particle momentum
is lost is proportional to the masses involved, thus τb /τe = mb /Zme . Hence
τbe =
mb
τe
Zme
3/2
2.75 ×105 Te τe =
.
ln Λ Zne
(1.47)
The collision interval is only weakly dependent on the Coulomb logarithm ln Λ and in evaluating τe for application to tokamaks, we shall adopt the value ln Λ = 17 as being is sufficiently
accurate for typical temperatures and densities (see (A.17)).
The rate of energy loss is Fbe vb = 2ξb /τbe and therefore the electrons are heated at the
rate Pe = 2ξb /τbe . Evaluating the constant we get
Pe = 1.71 ×10−18
ne ξb
3/2
Ab T̂e
keV s−1
T̂e , ξb in keV, Ab ≡
mb ,
mp
(1.48)
per beam ion.
Similarly, we find that the plasma ions are heated at the rate
1
Pi =
ni Ab2
2ξb
mb
≈ 0.97 ×10−17
1 ,
mb + mi τbi
Ai ξ 2
(1.49)
b
where the energy has been divided between the beam ions and the plasma ions inversely as
their masses (cf. (A.24)) and the slowing-down time for a beam colliding with ions is given
by (Spitzer 1962),
τbi =
4π 02 mb mi vb3
mb
.
mb + mi ni e4 ln Λ
(1.50)
1.4 Heating
17
Therefore
1
Pi = 0.97 ×10−16
ni Ab2
1
2
Ai ξb
keV s−1
ξb in keV .
(1.51)
Let
14.8Ab T̂e
ξc ≡ 3/2 ,
Zi Ai
(1.52)
then the sum of (1.48) and (1.51), i.e. the total plasma heating per beam ion, can be expressed
ξ 32 ne ξb
c
P = 1.71 ×10−18
(1.53)
1
+
keV s−1 .
3/2
ξ
b
Ab T̂e
When ξb = ξc , the electron and ion heating rates are equal.
Injection energies are usually greater than ξc , so at first the electrons are preferentially
heated; as the beam ions slow down and ξb falls below ξc , it is the ions that receive most of
the energy. The net effect is that the total electron heating and ion heating are comparable; ion
temperatures over 15 keV have been achieved in JET by NBI.
To produce a neutral beam it is first necessary to charge the particles by ionization so
that they can be accelerated by an electric field. Following this, they are neutralized by charge
exchange. But there is a balance between the rates at which they are neutralized and re-ionized
by collisions, so a completely neutral beam is not possible. Unfortunately, the ionized fraction
in the beam increases rapidly with increasing beam energy, and since these beam ions would
not penetrate the tokamak field, but would be deflected on to the walls of the injection port,
they are removed magnetically from the beam and dumped; thus the beam efficiency falls off
rapidly with beam energy. A reactor plasma might be over 2 m in radius (see Section 6.5.2),
so for the beam to penetrate far enough to deposit the energy in the central regions implies a
very inefficient beam.
1.4.3 Radio-frequency heating
Radio-frequency (RF) heating depends on the transfer of energy from electromagnetic waves
generated by an external source to particles at suitable resonance frequencies. Resonance
absorption of wave energy does not involve collisions and unlike ohmic heating, the process
becomes more efficient with increasing temperature. A multi-species plasma in a magnetic
field has several resonance frequencies capable of absorbing the energy of incident waves,
and gradients in the number density and temperature mean that these resonances occur in
narrow regions, admitting the possibility of localized heating and hence of some control over
the temperature and current profiles across the minor cross-section. The cyclotron frequencies
are defined in (A.33), viz. ωc = QB/m, where Q is the particle charge and m is its mass.
Ion-cyclotron resonance heating (ICRH) (ω ∼ ωci ), lower hybrid resonance heating
(LHRH), (ωce < ω < ωci ) and electron-cyclotron resonance heating (ECRH) (ω ∼ ωce )
have proved to be the most successful of the RF experiments, and temperatures have been
raised substantially (up to 5 keV). Lower hybrid resonance has been used in JET to modify
the current distribution by what is termed “current drive”. The waves are directed along the
18
1 The quest for fusion power
field lines and absorption takes place by Landau damping (e.g. see Woods 2004, p. 123) of
those plasma electrons that have a parallel velocity similar to the phase velocity.
The theory of these high-frequency waves and their absorption by Landau damping is an
extensive and much researched subject, but falls outside the range of this introductory text;
Porkolab (1979) has written a general survey, and Wesson (2004) gives a review with many
references.
1.5 Electron energy confinement time
1.5.1 Ohmically-heated tokamaks
A large number of early experiments concerned with transport in tokamaks has been reported,
mainly in the journal Nuclear Fusion. Hugill’s review lists 237 papers and deals almost entirely with ohmically-heated discharges (Hugill 1983). The observations reveal two regimes,
corresponding to low and high beta plasmas with continuous variation between. In the pre1980 and mainly low beta tokamaks, the empirical scaling laws inferred from observations
were simple, with confidence about the linear dependence of τEe on the line averaged density
n̄e defined in (1.56), but not much else; this situation has changed and now there is general
agreement about the dependence of τEe on all the major plasma parameters in the low beta
regime.
Neoclassical transport
We shall refer to ‘neoclassical’ transport several times before reaching Section 3.1.4 and Section 3.4, where the physical basis of the phenomenon will be discussed in detail. For the
present the following remarks will serve to identify the distinction implied by the prefix ‘neo’.
By ‘classical transport’ is meant the diffusion of some property through the plasma carried
by individual ions or electrons moving under the usual Lorentz force, without any disturbance
of their orbits by turbulence or instabilities. Fourier’s law for the diffusive transport of energy
is a good example:
q = −κ · ∇T ,
(1.54)
where κ is the thermal conductivity tensor, whose structure is described in Section A.7. The
classical value of κ can be derived from kinetic theory. Particles move through a mean free
path (the displacement between successive collisions) and then pass on their excess energy
by colliding with particles that have arrived from a cooler part of the plasma. In a direction
normal to strong magnetic fields, the mean free paths are just twice the Larmor radius (see
Fig. A.2), so transport is considerably inhibited by the limited displacements possible.
Diffusive transport is very different from convective transport in which it is the bodily
movement of fluid elements that moves (convects) the energy through the plasma. Both kinds
of transport are evident in equation (1.26), in which the term 52 pe vD is due to the convection
of electron energy, while Qer represents the diffusion of electron thermal energy.
Neoclassical transport differs from classical transport in that for many particles rather
large displacements are possible during their transit between collisions. These particles are
1.5 Electron energy confinement time
19
trapped in the tokamak magnetic fields and as a consequence trace rather large, banana-shaped
orbits whose widths are many times greater than a Larmor radius; this phenomenon increases
the cross-field transport of heat and momentum to values several hundred times the classical
value. Neoclassical transport was once considered to be the explanation for the rapid loss of
heat from tokamaks, which occurs hundreds of times faster than early expectations based on
the classical theory. However, tokamak losses exceed those predicted by neoclassical theory
by roughly two orders of magnitude, so attention has turned to turbulent transport to explain
both energy and particle losses. We shall discuss these problems in more detail in Chapter 3.
(i) Low beta regime
With ohmic heating it became standard practice to express τEe in the form
τEe = 10−α n̄eαn aαa R0αR qaαq Te αT Zeff αZ . . . ,
(1.55)
where the indices α, αn , αa , . . . are chosen to obtain the best statistical fit for a wide range
of observations. The density-averaged temperature Te used above is defined in (1.44); for
density, instead of the volume-averaged density defined in (1.43), it is usual to adopt the lineaveraged density defined by
1 a
ne dr ,
(1.56)
n̄e ≡
a 0
which is more closely related to actual observations.
Of course there is no a priori reason why (1.55) should be the correct form and later (in
Section 4.2) we shall find from a theoretical approach that a sum of two terms is required to
explain the functional dependence of τEe . The statistical approach predates the existence of a
reliable theory and in fact now provides a useful test that any proposed theory should pass.
One variable surprisingly absent from (1.55) is the magnetic field strength B, but out of a
dozen empirical scaling laws of this type reported by Hugill (1983), only one involved B, and
in any case (1.10), viz. qa = 5a2 Bϕ /Îp R0 , could have been used to remove Bϕ in favor of
the plasma current and the variables already appearing in (1.55).
To determine the indices is not straightforward, since it is rarely possible to vary the parameters one at a time. Furthermore, with steady-state, ohmically-heated tokamaks, the temperature cannot be externally controlled, and as both τEe and τE ∗e in (1.26) and (1.27) depend
on Te , the scaling of τE ∗e with Te masks the confinement time scaling. In principle this ambiguity could be overcome with the help of additional non-ohmic heating, but if this additional
heating is dominant, a new independent variable, the input power P , must be added to the
list and again the temperature dependence is obscured. However, if the radiation losses are
negligible, the value of αT in (1.55) can be deduced by dimensional analysis. For this we need
the theorem given in Section A.5, which allows us to write (1.55) in the form,
αR
αn αT
Ba5/4 qaαq R0 /a)
Zeff αZ a(αa +αR −2αn −αT /2−5/4)
Te a1/2
BτEe ∝ n̄e a2
and since the dimensional term a(··· ) cannot appear, we deduce that
αT = 2 αa + αR − 2αn − 54 .
(1.57)
20
1 The quest for fusion power
Table 1.2: Power law indices for τEe
Experiment†
1
2
3
4
5
6
7
‘ideal’
α
αn
αa
αR
αq
αZ
αT
α∗T
19.02 0.90 0.98 1.63 − 0.23 −
−0.88
18.44
1
2
− 0.75 −
−
−
20.46 1 0.25 2.75 1
− −0.5 −0.5
−
1
1
2
1
−
−
−0.5
20.3
1
2
1
0.5
−
−
−0.5
−
1.15 −
−
0.9
−
−
−
21
1 1.04 2.04 0.5
−
−
−0.34
−
1
1
2
1
− −0.5
†1. Pfeiffer & Waltz (1979); 118 observations on 11 tokamaks. 2. Ejima et al. (1982); Doublet III.
3. Merezhkin (see Lenov et al. (1980)); T11. 4. Efthimion et al. (1984); TFTR. 5. Equipe TFR (1980);
mainly TFR. 6. Cordey et al. (1985a); JET. 7. Goldston (1984); results combined from 12 tokamaks.
Table 1.2 lists the values of the indices obtained for a wide range of tokamak variables.
When an integer value was clearly indicated by the observations, this was chosen by some
authors even though not quite statistically optimal. Only one group ventured a value for αT ;
this was obtained indirectly, via an experimental determination of the thermal diffusivity. In
the earlier experiments the dependence on qa was not clear, but recent JET measurements give
αq = 0.9 ± 0.1, supporting the value of unity obtained on T11 and TFTR shown in Table 1.2.
The values of α∗T in the last column were not given in the papers quoted; they are our
dimensional analysis values given by (1.57). The earlier tokamaks, featured in Pfeiffer and
Waltz’s numerical study, lost about half their energy by radiation, which accounts for their
relatively high adverse scaling with temperature (α∗T = −0.88). At the bottom of Table 1.2,
the row marked ‘ideal’ gives the values of the indices that we would expect to appear in an
exact theory of electron thermal transport, at least for the density range represented in the
table. The fact that statistical analysis yields numbers for the indices close to integer values
suggests the existence of an under-pinning theory that is unlikely to involve the chaos of
turbulence. It is this theory that shall be developed in later chapters. No ‘ideal’ value for the
index α of the numerical coefficient is possible, for as we will see later, this number depends
on the temperature and density profiles.
The above description applies to τEe , but τE is bound to follow a similar pattern, being typically about 50% or so longer. In early experiments the ions reached about half the
electron temperature, which implied that although the electrons provided the dominant loss
mechanism, the losses through the ions were also somewhat larger than predicted by the early
theories. Convective energy losses are complications that will be treated later; it is usual to
treat these losses as being negligible, but the experimental evidence for this is not clear.
(ii) High beta regime
At high densities it was discovered that the empirical law τE ∝ n̄e overestimated τE , and a
weaker dependence was required (Gaudreau et al. 1977, Equipe TFR 1980). And at higher
1.5 Electron energy confinement time
21
densities still, τE reaches a flat maximum and then starts to fall as n̄e is increased (Ejima et al.
1982). Figure 1.8 shows an example of the ‘saturation’ of τE with increasing values of n̄e qa
in TFTR (Efthimion et al. 1984).
Since higher density means an increase in collision frequency, it was presumed (Alladio
et al. 1982) that neoclassical transport — in particular ion conductivity — was responsible
for the saturation of τE . But some observations had ion conduction losses several times larger
than neoclassical values (Ejima et al. 1982). It is not clear from the observations that ion
transport is the cause of the additional losses. Goldston (1984) noted similarities in the energy
confinement between the high beta regime and the L-mode (see Section 1.5.2) for neutral beam
heating. He correctly speculated that the same transport processes might well be operating in
each case and as the losses in beam-heated plasmas are known to be dominated by electron
transport, electron losses should also be dominant in the high beta regime.
In Section 4.3.2 it will be shown that in the L-mode
τE e =
µ0 e2 n̄e aR02 qa
0.5
,
1 + 2.13βp (2me ) 12 kB Te 12
where from (1.6) and (1.10)
βp =
2R02 µ0 p 2
n̄e q 2
qa ∝ 2 a .
2
2
a
Bϕ
a
Hence at a fixed values of Bϕ and temperature,
τE e ∝
(a2 /qa )βp
,
1 + 2.13βp
Figure 1.8: Total energy confinement time in TFTR
22
1 The quest for fusion power
showing that the electron energy confinement time, considered as a function of poloidal beta,
saturates when βp
0.47, which implies the existence of a similar constraint on n̄e qa as
indicated in Fig. 1.8. An important conclusion is that we cannot expect to find an accurate
single term formula like (1.55) for τEe over the whole of the accessible βp range.
1.5.2 Auxiliary heated plasmas
By ‘auxiliary’ heating is meant either neutral beam injection (NBI) or radio-frequency heating
(RFH). One might expect the transport of energy from a magnetoplasma to be independent
of the method of heating, but it appears that this is not so in tokamaks. As the auxiliary
heating is increased from zero to levels much higher than the ohmic heating (OH), the energy
confinement time τE changes from the function in (1.55) to a rather different one; furthermore,
with NBI the electrons remain the dominant energy loss channel. The implication is that either
the electron thermal conductivity depends on the method of heating, or more likely, that some
other mechanism involving electrons becomes important. Compared with ohmic heating, RF
heating has the advantages of providing the off-axis current drive required to maintain plasma
stability, and of giving direct ion heating; it also has the merit of generating small ELMs (see
Section 6.4.2). For a review of this topic see ITER team (1999), Chapter 6.
A surprising distinction between tokamaks with divertors and those with limiters was discovered (Wagner et al. 1982a,b), namely that with NBI those discharges with divertors were
able to contain particles and energy for about twice as long as was possible in the same conditions with normal ‘limiter’ discharges; this first regime is termed the ‘H’ (high) mode of
operation while the second usual limiter discharge is referred to as the ‘L’ (low) mode. Limiter discharges have also been made to perform in the H-mode by injecting a small amount
of neon (termed ‘neon puffing’) (Lazarus et al. 1985). It is evident that confinement with
auxiliary heating is quite sensitive to the boundary conditions; it is now accepted that the essential feature for H-mode operation is that there is a reduction in neutral recycling in the main
plasma. Why this should increase τE will be discussed shortly.
(i) The L-mode
In the L-mode the observations from several tokamaks are in broad agreement with the empirical law:
τE = 3.7 ×10−5 Ipv Pbw R0x ay ,
(1.58)
where Ip is the plasma current and Pb is the total beam power absorbed by the plasma. From
the relatively few observations available at the time, Goldston (1984) obtained the estimates
v = 1, w = −0.5, x = 1.75, y = −0.37.
(1.59)
Note that the energy replacement time for Pb is (cf. (1.19) and (1.22)),
τE∗ = 3π 2 R0 a2 kB ni Ti + ne Te /Pb .
(1.60)
Dimensional analysis yields the relation
1/3
BτE = F n̄a2 , Ip /Pb , Ip /(aB), β, R0 /a, Zeff
(1.61)
1.5 Electron energy confinement time
23
for the energy confinement time, and when this is applied to (1.58) the constraints
v + 3w + 1 = 0,
x+y = 1,
(1.62)
are obtained. Considering the possible errors involved, Goldston’s values are satisfactory.
Neilson et al. (1983) found that for the ISX tokamak at Oak Ridge, USA, v = 2/3 and
w = −2/3, values that are similar to Goldston’s.
With auxiliary heating there appears to be little, if any, dependence of τE on either n̄e or
B. Since the number of Coulomb collisions per unit path length — termed the ‘collisionality’
— scales as n̄e T −2 , collisions are clearly not the cause of the loss of energy. If turbulence
is assumed to be responsible the process must be independent of n̄e , which rules out several
types of turbulence.
With OH plasmas τE depends on n̄e , whereas with NBI plasmas it does not; therefore
when both forms of heating are present two separate processes are required to explain the
phenomenon.
(ii) The H-mode
There is no consensus about the scaling law in the H-mode. Some research groups find that
τE scales as in the L-mode, except that its magnitude is increased substantially. Others have
found scalings similar to OH plasmas, or intermediate scalings involving both n̄e and Ip . A
successful tokamak reactor will probably need to operate in the H-mode, although the improvement in confinement is offset by an increase in impurity level and by the appearance of
an instability known as an edge localized mode (ELM) explained in Section 6.4.2.
One clue to the H-mode phenomenon is the observation that limiter plasmas can be
switched into the H-mode by neon-puffing, and that this increases both τE and the particle confinement time τp (Lazarus et al. 1985). It appears that convection is being inhibited, and that
the boundaries are being partially thermally insulated from the body of the plasma. The collision cross-section between the plasma ions and the introduced impurities is relatively high, so
the neon impedes their radial flow, especially near the boundary; with divertors convection is
naturally lower because of the absence of neutrals recycling into the tokamak plasma. These
two observations suggest that the distinction between L-mode and H-mode plasmas depends
on the thermal boundary condition at the edge of the magnetoplasma; the L-mode requires
good thermal contact, whereas the H-mode depends on this contact being somewhat reduced.
The continuous injection of small, frozen hydrogen isotope pellets is the favored method of
particle refueling for the next generation of tokamaks, since this allows both deeper refueling
and better profile control than with gas puffing. It is found that the plasma that results after
pellet injection has different transport properties from the initial plasma (Hugon et al. 1992),
and the tokamak operates in what is termed a pellet enhanced performance (PEP) mode. For
example pellet injection can switch a limited L-mode plasma into an H-mode and increase the
energy confinement time by a factor ∼ 3 (see Section 6.4.4).
1.5.3 Profile shapes and energy losses
Changes in the shape of the temperature profile can be effected by adding metallic impurities,
and increasing the radiation losses. With sufficient impurities hollow profiles are obtained,
and the resulting values of τE are quite low. A moderate impurity level gives broad profiles
24
1 The quest for fusion power
and improved values for τE , whereas low impurity levels give peaked profiles and the highest
values of τE . These changes in τE occur when the gross parameters of the discharge are
similar. Factors up to 4.6 in τE due to profile alteration alone, have been reported (Meservey
et al. 1976). Profiles that are found to be in fair agreement with observations at low poloidal
beta are:
αt
αn
Te = Te0 1 − y
, ne = ne0 1 − y
y = (r/a)2 ,
where the constants αn and αt usually fall in the ranges (0.6, 1.5) and (1.5, 3) respectively.
Pfeiffer and Waltz’s (1979) list of observations for ohmically heated plasmas have average
values for αt and αn of 2.5 and 1.25 with a considerable spread.
With strong NBI heating, the additional heating and refueling in the central regions tends
to steepen both the density and temperature profiles. A distinction can be made between
tangential co-injection (beam parallel to the toroidal current and tangential counter-injection
(beam anti-parallel).
In purely OH-discharges it is found that in the central region the plasma mass flows in a direction opposing the current, and in the peripheral region it flows with the current. (Suckewer
et al. 1981; Brau et al. 1983). This description applies to the ion component of the plasma,
hence, with co-injection the velocity of the beam particles relative to the plasma particles will
be less in the central regions than with counter-injection. By (1.48) and (1.49) co-injection
will result in more rapid heating of the central plasma and hence steeper temperature profiles.
To anticipate Table 5.3, this means smaller values of τp with co-injection than with counterinjection, a phenomenon that has been observed on the ISX-B tokamak (Scott et al. 1985).
1.5.4 Disruptive instabilities
There is one remarkable phenomenon that should be mentioned in this introductory chapter. It
is the quite sudden changes that can occur in the basic macroscopic variables like temperature,
number density and the safety factor. By “sudden” is meant substantial changes that can occur
in times of the order of a few electron collision intervals, which by (1.14) for the typical JET
values: Zeff = 2, ne = 2 ×1019 m−3 , Te = 2 − 6 keV is 45 − 200 µs. This means that local
thermodynamic equilibrium is almost lost during these aptly named disruptions.
There are two main types of disruption: first there is a minor disruption from which the
temperature is restored to its original value, evolving along a ‘ramp phase’ that for JET takes
about 40 to 100 micro-seconds to complete. In this case the profile has a sawtooth appearance, with the ramp phase about 500 times longer than the collapse phase. These sawtooth
oscillations appear so regularly that they are interpreted as an indication that the discharge is
behaving normally. Figure 1.9 shows three distinct types of collapse precursors; in Fig. 1.9(a)
the oscillations preceding the sudden collapse have period of about 120 µs and the collapse
itself occurs on the same time-scale, so there appears to be a close relationship between the
‘over-stable’ precursor oscillations and the final collapse.
In certain circumstances, described as being near the density limit, there is a sudden collapse from which recovery does not occur. In this case the sawtooth oscillations that usually
precede a minor disruption do not occur, and the phenomenon is termed a major disruption,
which releases a lot of electromagnetic energy in a chaotic fashion that could seriously damage the tokamak structure and therefore they are usually avoided. As will be shown in Sec-
References
25
Figure 1.9: Three types of minor disruption in JET (in (a) the ordinate is proportional to Te )
tion 6.2.1, there are two circumstances that give rise to these severe instabilities, (i) there is a
‘low qa ’ limit and (ii) a ‘density limit’. This upper bound to ne affects the ignition condition
in (1.1) and could make tokamaks economically unviable.
References
Abbreviations for conference proceedings:
Plasma Physics and Controlled Nuclear Fusion Research
1em (Proc. 7th Int. Conf., Innsbruck, 1978) IAEA Vienna
(Proc. 8th Int. Conf., Brussels, 1980) IAEA Vienna
(Proc. 10th Int. Conf., London, 1984) IAEA Vienna
Current Disruption in Toroidal Devices
Proc. IAEA Tech. Committee Meeting, Garching; Feb. 1979, Rep. IPP-3/51
European Conference on Controlled Fusion and Plasma Physics
(Proc. 7th European Conf., Lausanne, 1975)
(Proc. 12th European Conf., Budapest, 1985)
(Proc. 11th Int. Conf., Kyoto, 1986)
(Proc. 13th Int. Conf., Washington, 1990)
(Proc. 16th Int. Conf., Montreal, 1996)
O
I
II
III
IV
V
VI
VII
VIII
26
References
Ashby, D. E. T. F. & Hughes, M. H. (1981). Nuclear Fusion, 21(8), 911–26.
Brau, K. (1983). Nuclear Fusion, 23(12), 1643.
Christiansen, J. P. et al. (1985). V, Pt I, 327.
Cordey, J. G. et al. (1985a). V, Pt 1, 167.
Cordey, J. G. et al. (1985b). V, Pt 1, 26.
Denne, B. et al. (1985). V, Pt 1, 379.
Efthimion, P. C. et al. (1984). II, Paper A-I-2.
Ejima, S. et al. (1982). Nuclear Fusion, 22(12), 1627–49.
Equipe TFR (1980). Nuclear Fusion, 20(10), 1227–45.
Gaudreau, M. et al. (1977). Phys. Rev. Lett., 39(20), 1266–70.
Goldston, R. J. (1984). Plasma Physics and Controlled Fusion, 26(1A), 87.
Hugill, J. (1983). Nuclear Fusion, 23(3), 331–73.
ITER team (1999). Nuclear Fusion, 39(12), Ch. 6.
JET team (1990). Plasma physics and controlled fusion, 32, 837.
Lawson, J. D. (1957). Proc. Phys. Soc., B 70, 6.
Lazarus, B. A. et al. (1985). Nuclear Fusion, 25(2), 135–49.
Lenov, V. M. et al. (1980). I, Vol. I, 393–403.
Matthews, G.F. et al. (1997). J. Nucl. Mater., 241–243, 450.
Meservey, E. B., Bretz, N., Dimock, D. L., & Hinnov, E. (1976). Nuclear Fusion, 16, 593.
Neilson, G. H. et al. (1983). Nuclear Fusion, 23(3), 285–94.
Pfeiffer, W. & Waltz, R. E. (1979). Nuclear Fusion, 19, 51.
Porkolab, M. (1979). In Theory of confined plasmas. Pergamon Press, Oxford.
Rutherford, P. H. (1980). Nuclear Fusion, 20(9), 1086–92.
Scott, S. D. et al. (1985). Private communication, PPL, Princeton, N.J.
Spitzer, L. (1962). Physics of fully ionized gases, 2nd edn. Interscience, New York.
Suckewer, S. et al. (1981). Nuclear Fusion, 21(10), 1301–09.
Wagner, F. et al. (1982a). Phys. Rev. Lett., 49, 1408.
Wagner, F. et al. (1982b). Plasma physics and controlled nuclear fusion research, IAEA-CN41/A-3. IAEA, Vienna.
Wesson, J.A. (2004). Tokamaks, 3rd edn. Oxford University Press.
Woods, L.C. (2004). Physics of plasmas, Wiley-VCH Verlag GmbH & Co., KGaA, Weinheim.
2 Tokamak magnetic fields
Sections 2.1 to 2.4.3 of this chapter are concerned with the topology of the equilibrium magnetic field, which has a dominant influence on the transport of mass and energy towards the
tokamak boundary. Charged particles become trapped between regions of increasing field
strength called magnetic mirrors and more than half of the electrons and ions oscillate between these mirrors; this is standard tokamak theory.
The remaining sections of the chapter describe how, under the combined influence of the
radial temperature gradient and electron fluid shear, the trapped particles transport thermal
energy out of the tokamak plasma at rates orders of magnitude larger than predicted by either
classical or neoclassical transport theory. The mathematical treatment of this transport theory,
called second-order because it involves two gradients instead of one, is presented in the next
chapter; our purpose here is to make the physical mechanisms involved as clear as possible.
The mechanism of thermal diffusivity described in Section 2.5 and first published over 20
years ago (Woods, 1983) is the corner stone of the author’s treatment of tokamak transport; and
in Chapters 4 to 6 it is shown that this theory explains a wide range of tokamak phenomena.
2.1 Axisymmetric toroidal equilibrium
The basic equations for the static equilibrium of a magnetoplasma are given at the end of
Section A.1. They are:
∇p = j × B,
µ0 j = ∇ × B,
∇ · B = 0,
∇·j = 0.
(2.1)
Therefore
B · ∇p = 0,
and
j · ∇p = 0 ,
1 2
1
∇ p+
B =
B · ∇B .
2µ0
µ0
(2.2)
(2.3)
The constraint ∇ · B = 0 allows us to introduce a vector potential A, defined by
B = ∇×A.
(2.4)
We shall also need the formula for the curl operator in cylindrical coordinates, which is given
in Section A.6.
Theory of Tokamak Transport: New Aspects for Nuclear Fusion Reactor Design. Leslie Colin Woods
Copyright © 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
ISBN: 3-527-40625-5
28
2 Tokamak magnetic fields
Z
Z
R
R
Figure 2.1: Cylindrical and local coordinates
2.1.1 Grad–Shafranov equation
To derive the equilibrium condition for a toroidal magnetoplasma, it is convenient to employ
the two coordinate systems shown in Fig. 2.1, namely the cylindrical coordinates (R, Z, ϕ),
and the ‘local’ coordinates (ψ, ξ, ϕ). The corresponding triads of orthogonal unit vectors
are (R̂, Ẑ, ϕ̂) and (ψ̂, ξ̂, ϕ̂). As we shall only be concerned with axisymmetric systems,
∂/∂ϕ = 0, making ϕ an ignorable coordinate.
It follows from an application of the curl operator (see (A.38)) to the vector potential A
and to the magnetic induction B (set r = R, θ = −ϕ, z = Z) that
∂A
∂AR 1 ∂ ∂Aϕ
Z
+ ϕ̂
−
− Ẑ
(2.5)
RAϕ ,
B = ∇ × A = R̂
∂Z
∂R
∂Z
R ∂R
∂B
and
∂BR 1 ∂ ∂Bϕ
Z
+ ϕ̂
−
− Ẑ
RBϕ .
(2.6)
µ0 j = ∇ × B = R̂
∂Z
∂R
∂Z
R ∂R
Functions ψ(R, Z) and F (R, Z) are defined by
ψ ≡ RAϕ ,
F ≡ RBϕ ,
(2.7)
then from (2.5) and (2.6) we obtain
RB = R̂
and
∂ψ
∂ψ
− Ẑ
+ F ϕ̂ = ∇ψ × ϕ̂ + F ϕ̂ ,
∂Z
∂R
(2.8)
∂F
∂F
− Ẑ
+ Rµ0 jϕ ϕ̂ = ∇F × ϕ̂ + Rµ0 jϕ ϕ̂ .
∂Z
∂R
(2.9)
Rµ0 j = R̂
As ϕ̂ · ∇ψ = 0, it follows from (2.8) that B · ∇ψ = 0, therefore (see Fig. 1.6)
ψ = constant is a magnetic surface. Functions that depend only on the value of ψ are termed
“surface quantities”. As there is no pressure gradient in the toroidal direction, ϕ̂ · ∇p = 0.
From (2.3)
µ0 B · ∇p = B · ∇B · B − 12 B · ∇B 2 = 0 ,
2.1 Axisymmetric toroidal equilibrium
29
and (2.8) gives: 0 = R B · ∇p = ∇p × ∇ψ · ϕ̂, from which it follows that ∇p and ∇ψ are
parallel. Therefore the magnetic surfaces are also isobars, which makes p = p(ψ) a surface
quantity.
Turning to the local coordinates, we choose ξ̂ to be orthogonal to ϕ̂ and to lie in the
magnetic surface, so that
B = Bξ ξ̂ + Bϕ ϕ̂ ,
(2.10)
where Bϕ and Bξ are called the toroidal and poloidal components of B. It follows from (2.8)
that
(2.11)
RBξ = −ξ̂ · ϕ̂ × ∇ψ = −ψ̂ · ∇ψ ,
where ψ̂ is the unit vector ξ̂ × ϕ̂. The function ψ(R, Z) is called the poloidal flux, or sometimes by analogy with classical fluid dynamics, the stream function. Similarly
Rµ0 jξ = −ψ̂ · ∇F .
(2.12)
The equilibrium condition ∇p = j × B in (2.1)1 (the first equation of the set (2.1)) can
now be written as
ψ̂ · ∇p = ψ̂ · (jξ ξ̂ + jϕ ϕ̂) × (Bξ ξ̂ + Bϕ ϕ̂) = jξ Bϕ − jϕ Bξ
=
1
1
jϕ ψ̂ · ∇ψ − Bϕ ψ̂ · ∇F ,
R
µ0
or since the gradients are parallel to ψ̂,
jϕ = Rp (ψ) +
1
F (ψ)F (ψ) ,
µ0 R
(2.13)
where the dash denotes ∂/∂ψ. By (2.5) to (2.8)
Rµ0 jϕ = −R
∂B
Z
∂R
−
∂BR ∂ 1 ∂ψ
∂2ψ
−
= −R
,
∂Z
∂R R ∂R ∂Z 2
so that ψ satisfies the relation usually known as the Grad–Shafranov equation (Grad and Rubin
1959, Shafranov 1957):
R
∂ 1 ∂ψ
∂2ψ
dF
dp
+
−F
.
= −µ0 R2
2
∂R R ∂R ∂Z
dψ
dψ
(2.14)
This is a non-linear elliptic equation, to solve which it is necessary to prescribe functions p(ψ)
and F (ψ) and suitable boundary conditions for ψ, usually in the form of a given boundary
curve ψB (R, Z) = constant.
Many solutions of (2.14) have been computed for toroidal fusion machines. However, as
we shall see below, Ohm’s law imposes two (integral) constraints on p(ψ) and F (ψ), and
the main difficulty in equilibrium theory is how to chose these functions so as to satisfy the
constraints.
30
2 Tokamak magnetic fields
2.1.2 First integral constraint
Let dσ be an element of magnetic surface, and let d be the distance between the surfaces
ψ = const. and ψ + dψ = const. (see Fig. 2.2), then the volume element is dτ = dσ d. The
pressure change between these surfaces is
dp = ∇p · ψ̂ d = −|∇p| d,
where p decreases along ψ̂. By (2.11), RBξ = |∇ψ|, so that |∇p| = RBξ |p |, where
p = ∂p/∂ψ, and the volume element can be written
dσ dp
|∇p| = RBξ |p | .
dτ = −
|∇p|
Figure 2.2: Magnetic surfaces
The electric field may be expressed in the form
ϕ̂ · ∇φ = 0 ,
E = E0 ϕ̂ − ∇φ
(2.15)
where E0 is the externally imposed field around the torus. A general form for Ohm’s law
reads (see (A.47))
1 j × B − ∇pe + δ · ∇T .
(2.16)
η ·j = E + v×B +
ene
The scalar product of this equation with B gives
B · η · j + B · ∇φ − Bϕ Eϕ − B · δ · ∇T = 0 .
Integrating (2.17) over the volume V within a magnetic surface, p = const., we get
dσ dp B · η · j + B · ∇φ − Bϕ Eϕ − B · δ · ∇T = 0.
V (p) |∇p|
We now use the fact that φ is a single-valued function:
ψ̂ · Bφ dσ = 0,
B · ∇φ dτ =
∇ · (Bφ) dτ =
V (p)
V (p)
Σ(p)
where Σ(p) is the surface of V (p) and the integral is zero because ψ̂ · B = 0.
(2.17)
(2.18)
2.1 Axisymmetric toroidal equilibrium
Similarly
V (p)
31
B · δ · ∇T dτ =
V (p)
δ B · ∇T dτ =
V (p)
δ ∇ · BT dτ.
In a strong magnetic field δ is a constant, so we are able to use the divergence theorem as
before to prove that the integral vanishes. Equation (2.18) is thus reduced to
dσ
B · η · j − Bϕ Eϕ dp = 0.
|∇p|
p
Σ(p)
This holds for any range of p and consequently the integral over Σ(p) must vanish, i.e.
dσ
B · η · j − Bϕ Eϕ = 0 ,
(2.19)
Σ(p) |∇p|
which is the first integral constraint.
2.1.3 Second integral constraint
To allow for the ionization of particles coming from the walls, it is convenient to define a
source function N by
∇ · (pv) = N.
(2.20)
By adding a source Q to the right-hand side of the continuity equation (A.4),
∂
+∇· v = 0,
∂t
and using the thermodynamic relation p = R T , we find that N and Q are related by
N = RT Q + R v · ∇T.
(2.21)
Integrating (2.20) over the volume within a magnetic surface we get
∇p N dτ =
∇ · (pv) dτ =
p ψ̂ · v dσ = − p
· v⊥ + v dσ,
V
V
Σ
Σ |∇p|
as ψ̂ = −∇p/|∇p| and v⊥ , v are the components of the velocity v perpendicular and
parallel to the magnetic surfaces.
Solving Ohm’s law (2.16) for v⊥ , using (2.17) and ∇p = ∇ pe + pi = j × B, we find
that
v⊥ = B × L/B 2 ,
L ≡ η · j − Eϕ ϕ̂ + ∇φ +
1
∇pi − δ · ∇Te .
ene
(2.22)
32
2 Tokamak magnetic fields
Now ∇p · v = 0 and −∇p · v⊥ = −∇p × B · L B −2 = j⊥ · L = j · L, since by (2.17),
B · L = 0. As the pressure is constant on the surface Σ, we now have
dσ
.
(2.23)
N dτ = p
j·L
|∇p|
V
Σ
Consider the integral containing j · ∇φ in (2.23). Integrating over p we have
dσ dp
= − j · ∇φ dτ = − ∇ · (j φ) dτ = − ψ̂ · j φ dσ = 0,
j · ∇φ
|∇p|
p Σ
V
V
Σ
and since this holds for all ranges of p, we conclude that
dσ
= 0.
j · ∇φ
|∇p|
Σ
As ∇p is parallel to j × B, (2.23) is reduced to
dσ
N dτ = p
j · η · j − jϕ Eϕ − j · δ · ∇Te ,
V
Σ |∇p|
(2.24)
which is the second integral constraint. It is not possible to prove that the integral involving
j · δ · ∇T is zero.
As tokamaks are nearly axisymmetric, we may adopt the approximation
· · · dσ = 2π
· · · R ds,
Σ
ψ
where s is the distance measured around a magnetic surface at right-angles to the magnetic
axis. The integrals may therefore be written,
R ds
B · η · j − Bϕ Eϕ = 0,
H(ψ) ≡ 2π
(2.25)
|∇p|
ψ
R ds
N dτ − 2πp
(2.26)
j · η · j − jϕ Eϕ − j · δ · ∇T = 0.
G(ψ) ≡
V (ψ)
ψ |∇p|
To sum up, the functions p(ψ) and F (ψ) must be chosen so as to make the integrals H and
G vanish. Because of the complexity of the relationships, an iterative, numerical treatment has
been developed to a high degree of computational refinement.
2.1.4 Diffusion velocity
Although not part of magnetostatics, the velocity at which plasma diffuses out of the equilibrium magnetic field is conveniently calculated at this point. We shall restore the electron
viscosity tensor to Ohm’s law, that is replace ∇pe by ∇ · pe and also include the thermoelectric term, which is associated with the electron temperature Te . Thus (2.16) becomes
v×B = η·j−E−
1
1
∇ · pe − δ · ∇Te +
j× B.
ene
ene
(2.27)
2.1 Axisymmetric toroidal equilibrium
33
Now ϕ̂ · v × B = ϕ̂ · vξ ξ̂ + vϕ ϕ̂ + v⊥ ψ̂ × Bξ ξ̂ + Bϕ ϕ̂ = Bξ v⊥ and
Bϕ
j + η⊥ jϕ ,
ϕ̂ · η · j = η ϕ̂ · j + η∧ ϕ̂ · b × j + η⊥ ϕ̂ · (j − j ) = (η − η⊥ )
B
where j B = jϕ Bϕ +jξ Bξ . In calculating the thermoelectric term, we shall assume that there
is no temperature gradient around the torus, ie. that ∇ Te is zero. Then by an expansion for
δ of the same form as that given for η in (A.46), we obtain
Bξ
ϕ̂ · δ · ∇Te = δ∧ ϕ̂ · b × ∇Te + δ⊥ ϕ̂ · ∇⊥ Te = −δ∧ ψ̂ · ∇Te .
B
2
2
2
As B = Bϕ + Bξ , the ϕ̂-component of (2.27) yields
v⊥ =
Bϕ
η⊥ Eϕ
∇· e
δ∧
jϕ Bξ − jξ Bϕ +
η j −
−
· ϕ̂ − ψ̂ · ∇Te ,
2
B
BBξ
Bξ
ene Bξ
B
where we have introduced the viscous stress tensor e (see (A.6)) and used the fact that pe is
a surface quantity, i.e. that ϕ̂ · ∇pe = 0. From the ψ̂-component of the equilibrium condition
∇p = j × B, viz. ψ̂ · ∇p = jξ Bϕ − jϕ Bξ , we obtain
v⊥ = −
Bϕ
η⊥
Eϕ
∇· e
δ∧
ψ̂ · ∇p +
η j −
−
· ϕ̂ − ψ̂ · ∇Te .
B2
BBξ
Bξ
ene Bξ
B
(2.28)
Using (2.11) to write (2.12) in the form µ0 jξ = Bξ F , we find from (2.13) that
j =
B
Bϕ 1
Bϕ
ψ̂ · ∇p +
Rp (ψ) +
BF (ψ) = −
jξ .
B
µ0
BBξ
Bξ
(2.29)
A more convenient form for j is obtained by taking the scalar product of (2.27) with the unit
vector b ≡ B/|B|:
1
∇· e ·b.
(2.30)
ene
The ‘diffusion velocity’, vD , is defined to be the value of v⊥ averaged over a magnetic
surface. Thus, by axisymmetry,
1
dσ =
v⊥ R ds,
A≡
R ds .
(2.31)
vD = v⊥ dσ
A ψ
p
p
ψ
η j = E +
It now follows from (2.28) and (2.30) that
δ∧
1
η⊥
1
1 ψ̂
·
∇p
−
ψ̂
·
∇T
vD = −
R
ds
+
Bϕ Eξ − Eϕ Bξ R ds
e
A ψ B2
B
A ψ B2
1
1
(2.32)
−
∇ · e · Bξ ϕ̂ − Bϕ ξ̂ R ds.
A ψ ene B 2
In a cylindrical plasma with axial symmetry Eξ is zero; further, if the field lines lie parallel
to the axis, Bξ is also zero and the second term in vD disappears. In strong magnetic fields with
singly-ionized ions, δ∧ = 1.5kB ne η⊥ /B (e.g. see Woods, 2004, p. 163), so the thermoelectric
term in vD reduces the diffusion velocity. In this case, if electron viscosity is discounted,
the dominant process in the diffusion of plasma from the cylinder is the electrical resistivity
coupled with the pressure gradient. The roles of the second term in vD and of the electron
viscosity in diffusion from toroidal (tokamak) plasmas, will be discussed in Section 5.1.
34
2 Tokamak magnetic fields
2.2 Equilibrium in a circular torus
2.2.1 Shafranov geometry
By a ‘circular’ torus is meant one that has a circular boundary in the poloidal plane, that is a
plane containing the major axis of the torus. Most of the early tokamaks had this shape. It
was found that the MHD stability of the doughnut-shaped magnetoplasma could be enhanced
by distorting the circular boundary into a shape resembling the letter ‘D’, the elongation being
parallel to the major axis. For example, the cross-section of JET is 2.1 meters in the ‘vertical’
direction (parallel to the major axis) and 1.2 meters in the plane passing through the minor
axis of the torus.
So far as transport is concerned, relatively little error is introduced by treating a D-shaped
cross-section as being equivalent to a circular cross-section of the same area, an approximation
that permits an analytical treatment of much of the theory. A further important simplifying
assumption concerns the magnitude of the aspect ratio of the torus. The major and minor radii
of the magnetoplasma are denoted by R0 and a, and the aspect ratio is R0 /a. The ‘large aspect
ratio’ approximation treats εa ≡ a/R0 as being a small parameter and terms smaller than ε2a
are neglected. The radius r = a determines the edge of the plasma; the chamber wall is at
r = b and a < r < b is a narrow region of very low pressure that may be assumed to be a
vacuum.
Figure 2.3: Shafranov shift
Figure 2.3 shows the local polar coordinates, (r, θ), in the poloidal plane; the origin
for r is the magnetic axis and θ = 0 is the line joining this axis to the ‘outside’ point,
R = R0 + a, Z = 0. As we shall show shortly, the magnetic surfaces are quite well approximated by a family of non-concentric circles having a limit point (the magnetic axis) at
R = R0 + ∆s (Shafranov 1966). The boundary surface is a circle of radius a, centered on the
minor axis at R = R0 . The outwards displacement ∆s is known as the ‘Shafranov shift’. To
verify this description and to obtain an expression for ∆s , we start from the Grad–Shafranov
equation, (2.14), expressed in local coordinates.
2.2 Equilibrium in a circular torus
35
The transformation from (R, Z) to (r, θ) is effected by
R = R0 + r cos θ,
Z = r sin θ,
(2.33)
from which we find
∂
sin θ ∂
∂
= cos θ
−
,
∂R
∂r
r ∂θ
∂
∂
cos θ ∂
= sin θ
+
.
∂Z
∂r
r ∂θ
(2.34)
With these relations (2.14) transforms into
1 ∂2
sin θ ∂
1∂ ∂
1
∂
r
+ 2 2 ψ−
−
cos θ
ψ
r ∂r ∂r
r ∂θ
R0 + r cos θ
∂r
r ∂θ
2
= −µ0 R0 + r cos θ p (ψ) − F (ψ)F (ψ) .
(2.35)
From (2.34),
Br = BR cos θ + BZ sin θ, Bθ = −BR sin θ + BZ cos θ ,
and (see (2.8))
BR =
1 ∂ψ
,
R ∂Z
BZ = −
1 ∂ψ
,
R ∂R
1 ∂ψ
,
Rr ∂θ
Bθ = −
1 ∂ψ
.
R ∂r
we obtain
Br =
(2.36)
2.2.2 Solution of the Grad–Shafranov equation
An approximate solution of (2.35) can be found as follows. Let ψ0 (r) denote the value that ψ
would take in the limit as the aspect ratio tends to zero — the cylindrical limit — and write
ψ = ψ0 (r) + ψ1 (r, θ), where ψ1 is a small modification added to ψ0 to allow for toroidal
effects, the expansion parameter being
ε ≡ r/R0 .
(2.37)
Then substituting ψ = ψ0 + ψ1 into (2.35) and splitting the result into O(1) and O(ε) terms,
we obtain
1d
dψ0
(2.38)
r
= −µ0 R02 p (ψ0 ) − F (ψ0 )F (ψ0 ),
r dr
dr
and
1∂ ∂
1 ∂2
r
+ 2 2
r ∂r ∂r
r ∂θ
=−
ψ1 −
cos θ dψ0
R0 dr
dr
d µ0 R02 p (ψ0 ) + F (ψ0 )F (ψ0 )
ψ1 − 2µ0 R0 r cos θ p (ψ0 ).
dr
dψ0
(2.39)
36
2 Tokamak magnetic fields
To solve (2.38) for ψ0 we need to specify the zero-order pressure, p0 = p(ψ0 ) and by
(2.7)2 the zero-order toroidal field, Bϕ0 = F (ψ0 )/R0 , which defines F (ψ0 ). From (2.36)
1 ∂ψ1
1 ∂ ,
Bθ = −
ψ0 + ψ1 ,
Rr ∂θ
R ∂r
which in the cylindrical limit yield
Br =
(2.40)
1 dψ0
.
(2.41)
R0 dr
We now assume that the magnetic surfaces are circular and that the center of the circle, specified by ψ = const., is displaced outwards from the major axis to R = R0 + ∆(r)
(see Fig. 2.3). It will be shown that ∆(r) is O(ε) and second-order terms in ∆ will be ignored;
thus by (2.34)
Br0 = 0,
Bθ0 = −
ψ ≈ ψ0 − ∆(r)
∂ψ0
dψ0
= ψ0 − ∆(r) cos θ
,
∂R
dr
whence
dψ0
.
(2.42)
dr
Substituting this value into (2.39), we find that the result can be rearranged in the form
2
dψ0 d∆
d
d
1d
1 dψ0
dψ0
1 dr
−∆
r
−
r
−
dr r dr
dr
r dψ0 dr
dr
dr
R0 dr
ψ1 = −∆(r) cos θ
d µ0 R02 p (ψ0 ) + F (ψ0 )F (ψ0 ) − 2µ0 R0 rp (ψ0 ) ,
dr
whence by (2.38) and (2.41),
dp0
d
r
2 d∆
2
− Bθ0 .
2µ0 r
rBθ0
=
dr
dr
R0
dr
=∆
(2.43)
To complete the solution of the Grad–Shafranov equation it remains to solve (2.43)
for given functions p0 (r) and Bθ0 (r) and with the boundary conditions, ∆(a) = 0 and
d∆/dr = 0 at r = 0. First we obtain
d∆
= −ε β̂p + 12 li ,
(2.44)
dr
where
r
2µ0
β̂p (r) ≡ − 2 2
r2 p (r1 ) dr1 ,
(2.45)
r Bθ0 (r) 0 1 0
r
and
2
2
r1 Bθ0
(r1 ) dr1 ;
(2.46)
li (r) ≡ 2 2
r Bθ0 (r) 0
thence
1
∆(r) =
R0
a
r
r β̂p (r ) + 12 li (r ) dr ,
(2.47)
where β̂p (a) = βp is the ‘poloidal beta’ (see (1.6)) and li (a) is the internal inductance. Finally
the Shafranov shift is ∆s = ∆(0).
2.2 Equilibrium in a circular torus
37
2.2.3 Magnetic fields and electric currents
It follows from (2.7)2, (2.36) and (2.42) that
1
1 F (ψ0 ) + ψ1 F (ψ0 )
Bϕ = F ψ0 + ψ1 ≈
R R
R0
d F (ψ0 )
=
,
Bϕ0 − ∆(r) cos θ
R
dr
R0
i.e.
Bϕ0 − ∆(r)Bϕ0
cos θ
,
1 + ε cos θ
Bϕ =
(2.48)
where the dash now denotes derivatives with respect to r.
To calculate Bθ we use (2.40)1, (2.41) and (2.42):
1 ∂ dψ0 R0 ψ0 − ∆(r) cos θ
=
Bθ0 − ∆Bθ0 cos θ
Bθ = −
R ∂r
dr
R
R0 Bθ0 − ∆Bθ0 cos θ ,
=
R
i.e.
Bθ0 − ∆Bθ0 cos θ
.
Bθ =
1 + ε cos θ
(2.49)
Similarly
Br = −
∆(r)
Bθ0 sin θ .
r
(2.50)
Expressions for the electric current densities now follow from (2.6), (2.34), and
jr = jR cos θ + jZ sin θ, jθ = −jR sin θ + jZ cos θ. Thus the relations
µ0 jR =
∂Bϕ
,
∂Z
µ0 jZ = −
1 ∂
(RBϕ ),
R ∂R
µ0 jϕ =
∂BR
∂BZ
−
,
∂R
∂Z
are transformed into
µ0 jr =
1 ∂ RBϕ ,
Rr ∂θ
µ0 jϕ =
1 ∂Br
1 ∂ rBθ −
.
r ∂r
r ∂θ
µ0 jθ = −
1 ∂ RBϕ ,
R ∂r
(2.51)
and
(2.52)
On substituting (2.48) to (2.50) into these equations, we obtain self-consistent currents for
Shafranov geometry.
In the cylindrical limit we get
µ0 jr0 = 0,
µ0 jθ0 = −
∂Bϕ0
,
∂r
µ0 jϕ0 =
1 ∂
(rBθ0 ) .
r ∂r
(2.53)
38
2 Tokamak magnetic fields
Suppose that there is a vacuum region just outside the limiter, a < r < b, where there is
neither current or pressure, then (2.53)3 shows that Bθ0 ∝ 1/r. In this case (2.45) and (2.46)
give
βp (r) = βp (a),
li (r) = li (a) + 2 ln(r/a),
and it follows from (2.47) that the presence of the vacuum region results in an additional
displacement
a
b2
a2 1
1
βp (a) + 2 li (a) − 2 + ln
∆v =
1− 2
.
2R0
b
b
As such a shift in the position of the discharge would result in an increased loss of plasma to
the limiter, it is usual to cancel it by an inwards magnetic force provided by a vertical magnetic
field in the vacuum region (see Wesson 2004, p. 121).
2.3 Particle trapping in magnetic fields
2.3.1 Magnetic bottles
Because of their rotational motion each charged particle moving with peculiar velocity c⊥
perpendicular to the magnetic field is a dipole with a constant magnetic moment −M b, where
M = mc2⊥ /(2B) (see (A.54)). And since in the absence of collisions their kinetic energy
remains constant, the particles move under the constraints
mc2⊥
= const.,
E = 12 mc2 = const. ,
2B
which apply only in a frame convected with the fluid. As a consequence, when a particle P
approaches a region of magnetic field of increasing strength, the increase required in c⊥ to
balance the increase in B can be found only at the expense of c ; thus P’s guiding center
G (defined in Section A.4) has a reducing value of c and eventually it may stop, reverse its
motion and move away from the strong magnetic field region. For this reason the term ‘mirror’
is an apt description of the region where B has its maximum value. Two mirrors, as depicted in
Fig. 2.4, make a ‘magnetic bottle’ and some of the particles within the bottle will be trapped
with their guiding centers oscillating between the mirrors.
It follows that c and B are related by
M=
E = 12 mc2 + MB = E(sin2 α + MB/E),
sin2 α = c2 /(c2⊥ + c2 ) ,
(2.54)
where α is the pitch angle of P’s trajectory measured from a plane perpendicular to the magnetic field. Thus as P approaches the mirror field, B may increase sufficiently to reduce P’s
value of c to zero, at which point P is reflected from the mirror.
Let Bmax be the maximum field strength of a mirror, then particles with E < MBmax will
be reflected and hence in a symmetrical field as illustrated in Fig. 2.4, they will be trapped. It
follows that those particles in a region where the field strength is B0 and having pitch angles
satisfying
1
α0 ≡ sin−1 (1 − B0 /Bmax ) 2 ,
(2.55)
α0 < α < π/2
2.3 Particle trapping in magnetic fields
39
Figure 2.4: Magnetic bottle
will escape through the magnetic ‘throat’ at the mirror. The boundary between captured
(0 < α < α0 ) and passing (α0 < α < π/2) particles is determined by the pitch angle α0 .
2.3.2 Fraction of trapped particles
Consider the magnetic bottle of Fig. 2.4. Let αc be the critical pitch angle at the point of minimum magnetic field, Bmin , then by (2.55) the bottle holds all particles for which α satisfies
−αc < α < αc ,
−1
cos2 αc = Rm
(Rm ≡ Bmax /Bmin ) ,
(2.56)
where Rm is termed the ‘mirror ratio’. Collisions will steadily scatter these particles into one
of the two loss cones, αc < α < π/2, −π/2 < α < −αc , allowing the bottle to leak on the
collisional time scale, τ , the details of which are given in Section A.10.
A simple expression for the fraction of trapped particles can be deduced by assuming
that they are part of an equilibrium distribution. With a velocity-space, spherical coordinate
system (c, ϑ, ζ), oriented so that c lies along the axis from which ϑ is measured, the element
of solid angle is sin ϑ dϑ dζ, and the trapped particles lie in π/2 − αc < ϑ < π/2 + αc . Then,
averaging over a truncated Maxwellian distribution (see (A.58) and (A.60)), we find that the
fraction of trapped particles is
2π π/2+αc
∞
2
4
1
√ ν 2 e−ν dν,
dζ
sin ϑ dϑ
fT =
4π 0
π
0
π/2−αc
i.e.
−1 12
fT = sin αc = (1 − Rm
) = c /c 0 ,
(2.57)
where the second form follows from (2.54) and the subscript “0” denotes values at θ = 0 (see
Fig. 2.5). In equilibrium conditions from (A.62), c2 = 13 c2 , so sin2 αc = 1/3, αc = 35.3◦
and fT = 0.58.
That particles can be trapped by magnetic fields has important consequences for energy
confinement in toroidal magnetic fields; it will be shown in Section 2.5.3 that the trapped particles are largely responsible for the steady and substantial losses of energy from the electron
gas.
40
2 Tokamak magnetic fields
2.4 Trapping in tokamak magnetic fields
2.4.1 Tokamak mirrors
A magnetic field line is defined by the equations
dr
r dθ
R dϕ
=
=
,
Br
Bθ
Bϕ
where here and below R can now represent R0 without ambiguity. Hence, by (2.48), (2.49)
and (2.50),
dϕ =
where
q≡
and
q dθ
,
1 + ε1 cos θ
dr = −∆ sin θ dθ,
rBϕ0
,
RBθ0
(2.58)
(2.59)
ε1 = ε − ∆ ln Bθ0 + ∆ ln Bϕ0 .
(2.60)
Some idea of magnitudes can be gained by considering the case of a uniform toroidal
current for which (2.53)3 gives Bθ0 = 12 rµ0 jϕ0 . The inductance defined in (2.46) is 12 and
(ln Bθ0 ) = 1/r. With physically realistic pressure gradients βp remains finite at the origin
and it follows from (2.43) that ∆ ∼ r2 . Hence the last term in (2.60) tends to zero with r.
Since ε1 is much smaller than unity, ϕ ≈ q θ, so q is the number of rotations of a field line
about the major axis per rotation about the minor axis. For reasons to do with plasma stability
(see Section A.24) q is termed the ‘safety factor’; typical values in tokamaks lie between 0.7
and 5. The total distance travelled along a field line per 360◦ rotation about the minor axis of
the torus is approximately 2πRq; this is termed the ‘connection length’.
The magnetic field lies in the direction of the unit vector
br ≡ Br /B, bθ ≡ Bθ /B, bϕ ≡ Bϕ /B ,
(2.61)
b = br r̂ + bθ θ̂ + bϕ ϕ̂
where B is the total field strength. From (2.50) and (2.59) to sufficient accuracy,
br = −
∆(r)
sin θ bθ ,
r
bθ =
ε
q
bϕ .
(2.62)
In tokamaks q has a minimum value a little smaller than unity close to the magnetic axis. In
this case bθ is smaller than bϕ by a factor ∼ ε and br is smaller still by a factor ε2 , so we can
replace (2.61) by
(2.63)
b = bθ θ̂ + bϕ ϕ̂
1 = b2θ + b2ϕ .
From (2.48) and (2.49) the magnetic field strength may be written
B0
2 + B2
B0 ≡ Bϕ0
B=
θ0 ,
1 + ε∗ cos θ
(2.64)
2.4 Trapping in tokamak magnetic fields
where
41
∆Bϕ0
∆Bθ0
ε∗ ≡ ε + bϕ
+ bθ
.
B0
B0
(2.65)
If we assume that a particle’s parallel speed is only sufficient to take it to the points θ = ±θ0
where it is reflected, then
at θ = 0, B =
B0
= Bmin ;
1 + ε∗
at θ = ±θ0 , B =
B0
= Bmax . (2.66)
1 + ε∗ cos θ0
Thus the tokamak field presents a continuum of magnetic mirrors mainly along the inside of
the torus.
2.4.2 Trapped particles
The mirror ratio defined in (2.56) is (1 + ε∗ )/(1 + ε∗ cos θ0 ), and hence by (2.57) the fraction
of particles that are trapped is
1
1
1 + ε∗ cos θ0 2
2ε∗ 2
fT (r, θ0 ) = 1 −
=
| sin 12 θ0 |
1 + ε∗
1 + ε∗
−π ≤ θ0 ≤ π .
A typical value for ε∗ midway between the limiter and magnetic axis is 1/6, where
fT ≈ 0.53 sin 12 θ0 . Hence a large fraction of the particles in a tokamak become trapped
in its magnetic field. Because ε∗ ≈ ε
1, it is usually sufficiently accurate to take the
fraction trapped in −θ0 < θ < θ0 to be
fT (θ0 ) = G(ε) sin 12 θ0
1
1 G(ε) ≡ 2ε/(1 + ε) 2 ≈ 2ε 2 .
(2.67)
Hence, if θ0 is increased to θ0 + dθ0 , the trapped fraction increases by an amount dfT , where
dfT (θ0 ) = G(ε) 12 cos 12 θ0 dθ0
Figure 2.5: Trapped particle orbit
(0 < θ0 < π) .
(2.68)
42
2 Tokamak magnetic fields
2.4.3 Bounce time in a tokamak field
Later we shall need an expression for the time τb (θ0 ) that it takes a particle P to travel between
the reflection points at θ = ± θ0 . In Section 3.1.3 we shall show that P drifts away from the
path between the reflection points, moving outwards on one transit and inwards on the return,
the complete orbit having a closed ‘banana’ shape. However, this drift is too small to have
much effect on the value of τb (θ0 ). We shall also ignore the small effect of the Shafranov
shift.
The element of distance along a field line is
1
ds = (R dϕ)2 + (r dθ)2 2 ,
hence from (2.58), neglecting an O(ε) term,
1
ds = 1 + (Rq/r)2 2 r dθ ≈ Rq dθ ,
(2.69)
as (Rq/r) is typically about 10. The actual distance travelled by P is therefore a factor q larger
than the direct physical displacement (see Fig. 2.5).
The parallel speed is c = ds/dt ≈ Rq dθ/dt, hence from the constants of the motion, M
and E, and (2.66) we have
12
c 2 12
1+ε
⊥
2
sin ψ ,
=c 1−
c = c 1 −
c
1 + ε cos θ
where
sin2 ψ ≡
2B0 M 1
mc2 1 + ε
is the value of (c⊥ /c)2 at θ = 0. Thus
12
κ2 − sin2 12 θ
1
|c | = (2ε) 2 c
1 + ε cos θ
and
dθ
c
c (2ε) 2
=
=±
dt
Rq
Rq
1
κ2 − sin2 12 θ
1 + ε cos θ
1+ε
2
2
cos ψ ,
κ ≡
2ε
12
,
(2.70)
where ‘+’ applies to θ increasing, and ‘−’ to θ decreasing.
The reflection points are at dθ/dt = 0, i.e. at θ = ± θ0 , whence κ = sin 12 θ0 and the
required bounce time is
12
θ0 2Rq
1 + ε cos θ
τb (θ0 ) =
dθ.
1
sin2 12 θ0 − sin2 12 θ
c(2ε) 2 0
The numerator varies between (1 + ε) and (1 − ε) with an average ∼ 1. With ε small, little
error is made by replacing the numerator by unity. The integral can then be evaluated with
help of the transformation sin φ = sin 12 θ/ sin 12 θ0 (Kadomtsev and Pogutse, 1967):
√
√
2 2Rq
Rq
τb (θ0 ) = 2π √ K(sin 12 θ0 ) ,
(2.71)
τb (θ0 ) = √ K(sin 12 θ0 ) ,
c ε
C ε
2.4 Trapping in tokamak magnetic fields
43
where K is the complete elliptic integral of the first kind, τb is the value averaged over a
1
−1
2
Maxwellian
√ distribution and C = (2kB T /m) . (We have averaged frequencies, τb and used
c = 2C/ π from (A.61).)
In typical tokamaks collision times are usually much longer than a bounce time, for example for trapped electrons in the JET tokamak it follows from (A.16) and (2.71) that τe /τb ∼ 30.
This ratio is much the same for ions although they have rather longer times for being trapped
and untrapped. Thus particles will oscillate between reflection points many times before escaping to new orbits.
2.4.4 Trapped particle resistivity
The banana orbits traced out by captured particles are properly described in the convected
reference frame, for it is only in this frame that the electric force is absent and the particle
kinetic energy remains constant. Trapping is thus a phenomenon involving peculiar velocities
c and would appear to have little effect on the average or fluid velocity v of the species in
question. However, although each banana orbit is convected by the fluid, successive turning
points — the magnetic mirrors — remain fixed in tokamak geometry, so that when particles
are captured, at the first mirror reflection they experience a magnetic force rather like a 180◦
collision with another particle; in effect they are colliding with the tokamak structure.
In the following we shall assume that the ions, which have much longer escape and capture times than electrons (see Section A.10), are almost stationary and that it is the much more
mobile electrons that are captured. The directed momentum lost by a captured electron p at
its first reflection is recovered at its second mirror interaction and as p oscillates between the
mirrors, loss and recovery follow each other until p escapes from the banana orbit. Upon
escaping it will either be only slightly deflected or deflected through ∼ 180◦ so that in either
case its parallel motion will remain almost tangential to the final banana orbit. If p escapes
moving in a direction opposite to its motion before capture, then it will have lost momentum
equivalent to a 180◦ collision with an ion. In this way magnetic mirrors impede the motion of
the electron fluid and are therefore responsible for a considerable increase in the electric resistivity; the resulting value we shall call “trapped particle resistivity”. A similar modification
termed “neoclassical resistivity”, follows from neoclassical transport theory.
Typical electrons remain captured for only about 0.65ετe ≈ τe /10 s (see (A.64)), and
just after escaping they will either be ‘returning’ in the direction opposite to their path prior
to capture or will be ‘continuing’ on with the untrapped electrons. We shall assume that the
probability of escape is the same for each side of the banana orbit, i.e. the ‘returning’ and
‘continuing’ electrons are in equal numbers. Hence the obvious reckoning for the loss of
conductivity is to assume that the diverging motions of the escaping electrons cancel out the
electric current that they would have otherwise conducted.
1
By (2.67) the fraction of passing particles is g = (1 − (2ε) 2 ) and as ne is the number
density of the charge transporting particles, in the presence of particle trapping, this number
is reduced to gne , which by (A.14) yields the trapped particle resistivity,
ηt = η /g =
α0 me
,
2
e ne gτe
g = (1 − (2ε) 2 ) .
1
(2.72)
44
2 Tokamak magnetic fields
Trapping typically increases the resistivity in JET by a factor of 2.86. It should also be noted
that increased resistivity means that ohmic heating is enhanced by particle trapping, in effect
by the randomization of the motion of the ‘returning’ electrons.
Hazeltine, Hinton and Rosenbluth (1973) applied neoclassical transport theory (see Section 3.4.1), which involves a specialized form of kinetic theory of some complexity, to obtain
the expression
g = (1 − 1.95ε 2 + 0.95ε) + O(ε)3/2 ,
ε = r/R
1
and Wesson (2004, p. 174) quotes what he describes as a being more accurate form for g,
1
namely g = (1 − ε 2 )2 . Hirshman, Hawryluk and Birge et al. (1977) have extended neoclassical theory to cover a wide range of values of the collisionality (defined in Section 3.4.1) and
in the banana region (vanishingly small collisionality), their formulae reads
g = (1 − fT )(1 − 0.28fT ) ,
where fT denotes the trapped fraction.
Figure 2.6 shows the several values quoted here for the ‘conductivity suppression factor’
g as a function of r/R; UTF is g as defined in (2.72), HHR is the Hazeltine et al. form, HHB
is the Hirshman et al. formula and WES is the expression quoted by Wesson, which in fact
is scarcely distinguishable from the HHR expression. The fact that the straightforward UTF
formula lies above the neoclassical curves implies that in neoclassical theory the ‘returning’
electrons more than cancel out the ‘continuing’ electrons; no physical mechanism has been
suggested to explain this odd result. In the rest of this text, we shall adopt (2.72), which is not
very different from the quoted neoclassical expressions.
1
0.9
0.8
0.7
0.6
g
0.5
WES
UTF
0.4
0.3
HHR
0.2
HHB
0.1
0
0
0.05
0.1
0.15
0.2
0.25
r/R
0.3
0.35
0.4
Figure 2.6: Conductivity suppression factor g from various theories (For the notation see the
text.)
2.5 Diffusivity of trapped particles
45
2.5 Diffusivity of trapped particles
2.5.1 Energy sinks at magnetic mirrors
Figure 2.7 illustrates the orbits of trapped and detrapped particles extending from the midpoint
of a banana orbit at θ = 0 to a reflection point R, at θ = θ0 . By (2.69) the total distance
measured along a field line is Rqθ0 . Although the reflection process might appear to be
reversible, with the impinging particles returning back along their banana orbits, the process
is irreversible because of a strong energy flux in the region around R; we shall now describe
the physical origin of the energy sink at R for the electron fluid.
Figure 2.7: Energy diffusion at magnetic mirrors
The mechanism is depicted in Fig. 2.8. Adjacent guiding centers for electrons at G1 and
G2 are assumed to be two Larmor radii apart in a direction lying parallel to a temperature
gradient T and a sheared electron fluid motion v . 1 The electrons with G1 as their guiding
center are hotter than those gyrating about G2 , and consequently there is an imbalance of
energy flux at the point O where the orbits touch, which gives rise to a transverse heat flux,
orthogonal to both the magnetic field and the temperature gradient, a well-known phenomenon
that will be described in detail in Section 3.5.3. If in addition the flow is sheared so that the
guiding centers are passing each other as shown in the figure, the transverse heat flux vector is
deflected, giving a component down the temperature gradient. But were the temperature and
velocity gradients parallel rather than anti-parallel, the outcome would be a heat flux up the
temperature gradient. Details of this mechanism will be presented in Section 3.5.3. We can
infer that:
If the temperature gradient and electron fluid shear are antiparallel, the heat flux is
normal, whereas if they are parallel, the heat flux is abnormal, i.e. the heat flows up
the temperature gradient.
1/2
≈ 1.07 ×10−4 T̂e /B m ∼ 1/10 mm in a tokamak; tokamak ions
have a Larmor radius of ∼ 3 mm. It follows that the distinction between the electron and ion fluid motions and their
average guiding center motions is negligible.
1 The Larmor radius for the electrons is r
Le
46
2 Tokamak magnetic fields
Figure 2.8: Heat sink due to fluid shear
Of course heat flowing up temperature gradients is not itself novel. In classical thermoelectricity this is easily accomplished by having a strong enough electric field, or equivalently
a strong electric current. Observations on the DIII-D tokamak that can be interpreted to show
that heat flows inwards, and therefore up the temperature gradient, will be described in Section 4.6.2.
The mechanism is strongest where the particle energy is mainly in the perpendicular motion around the field lines and for this reason we can think of the reflection points as being
energy sinks, although the process occurs everywhere along the banana orbit, but less strongly
as c gains strength at the expense of c⊥ . Just as thermal energy is leaving the region near R,
by the same mechanism it is also arriving from neighboring regions of higher temperature at
all points along the banana orbit and then it is carried towards R by the trapped particles. The
combination of fluid shear and radial temperature gradient is the mechanism that explains the
so-called anomalous energy loss from tokamaks.
2.5.2 Physics of diffusivity
Understanding how thermal diffusivity operates in tokamak magnetic fields is the central problem of tokamak theory, a problem that very well illustrates the point about physical modelling
made towards the end of the Preface. While formally correct equations, based on presumptions about the underlying physical mechanisms inherited from other contexts, may be correct,
in some cases they also have a good chance of being wrong. Unfortunately, this is what has
happened with thermal transport in received tokamak theory.
Convection and diffusion are described in Section A.15; here we shall repeat the basic
relations and discuss their application to tokamaks. For the diffusivity of a property φ carried
by particles there are three basic formulae, depending on the domain of application:
(a) χφ = αc2p τ ,
(b) χφ = αλ2 /τ ,
(c) χφ =
j
where α and αj are constants of order unity.
αj
2
j λj
τj
,
(2.73)
2.5 Diffusivity of trapped particles
47
The cases are:
1
(a) particles moving φ along a relatively straight path for a time τ at a r.m.s. velocity c2p 2
before passing this property on,
(b) particles that transport φ through a distance λ in a time τ and
(c) particles with several distinct modes of transport identified by the subscript j.
In (a) ‘passing’ φ need not involve complete collisions, for example with the transverse
heat flux in Fig. 2.8, the particles remain in circular orbits while their thermal energy is transmitted by grazing collisions along the tangential plane linking these orbits. There is no obvious
1
mean free path, except that after a time τ or a tangential length τ c2p 2 , the process fades out.
◦
In (b) τ is usually the 90 collision interval and λ is either the mean-free-path with straight
trajectories parallel to the field, or the Larmor radius, rL , for transport orthogonal to magnetic
field lines; the case describes classical, first-order2 transport theory, the mathematical details
of which will be given in Section 3.2.2. Finally (c) provides the rule for compounding diffusivities; they are additive in the (distance)2 /(time taken) form (cf. the additivity of frequencies,
but not of collision intervals).
Referring to (2.73)(b), the expectation in the 1950s and 60s was that the reduction of λ by
strong magnetic fields from its mean-free-path value to the relatively minuscule distance rL
would result in negligible values for the orthogonal diffusivity and readily lead to thermonuclear fusion temperatures. However, all this is made much more difficult by the presence of
energy sinks (or sources) that allow the energy to flow freely across the field lines. In this
case χ⊥ can have either sign and is controlled only by the rate at which heat flows between
the sinks along the field lines; i.e. as a result it is the parallel diffusivity, χ , that determines
the rate at which heat flows either down or up the temperature gradient across the field lines.
The parallel diffusivity for the trapped particles is different from that of the passing particles because the effective “free paths” are very different. For the trapped particles λ is replaced
by a banana length λm and τ by an average bounce time, τb , as will be described in the following section. The effective “mean-free-path” for passing particles, measured parallel to the
magnetic field, depends on whether conditions are steady or unsteady. In steady conditions
λ is replaced by half the connection length, λc = 2πRq, defined in Section 2.4.1, since over
distances greatly exceeding λc plasma conditions are periodic and longer paths along which
energy can be transported are not accessible.
However, in unsteady conditions this periodicity is lost and particles are free to transport
energy over a genuine mean-free-path3. As the connection length is typically nearly two
orders of magnitude smaller than the collisional mean-free-path, unsteady conditions greatly
increase the parallel diffusivity and therefore lead to very rapid changes in temperature, in
some circumstances even causing the discharge to collapse into a major disruption. Another
way of describing the situation is that unsteady conditions greatly increase the size of the
available energy reservoir.
2 First-order in the Knudsen number expansion, see Section A.18; the theory involved in Fig. 2.8, where it is
the combination of a temperature gradient with fluid shear that produces the thermal sink, is second-order transport
theory.
3 Because of the long-range Coulomb forces, mean-free-paths and collision intervals in a plasma are not the precise
concepts that apply to neutral gases, but are due to the continuous accumulation of very small grazing collisions (see
Section A.2), it would be difficult to take this behavior into account, and the assumption is that our adopting the
language descriptive of neutral gas transport makes little difference to the validity of the model.
48
2 Tokamak magnetic fields
There remains the problem of deducing a value for χ⊥ from that found for χ . These
diffusivities are not equal, because, as shall be shown in the next chapter, the values of χ⊥ for
cross-field heat flux in the electron and ion fluids are complicated second-order expressions
involving gradients of the electron and ion fluid velocities, whereas the corresponding values
of χ are first-order expressions, independent of derivatives. In fact the only physical properties in common are the collision, or ‘time-delay’ intervals τ and τ⊥ . The physics involved
is that τ is the time it takes for the thermal energy to be transported along the field lines to
the heat sinks, which in turn deflect this energy across the field lines (see Fig. 2.8). There is
no time-delay directly associated with the deflected heat flux, but in equilibrium this process
must proceed at the same speed as the energy is supplied by the parallel transport. Hence, we
shall replace the time-delay appearing in the second-order expression for χ⊥ by τ .
To find τ we first calculate χ via the appropriate form in (2.73) and then by (2.73)(a)
1
we get τ⊥ = χ /C 2 , where C is the thermal speed, (2kB T /m) 2 . The physics involved in
this choice is that, unlike the particles, the deflected heat flux illustrated in Fig. 2.8 lies on a
straight path with an average squared velocity of c2⊥ = C 2 (Section A.9).
2.5.3 Parallel diffusivity due to trapped particles
Returning to the description given in the last paragraph of the previous section, the speed
at which thermal energy is carried towards R by the trapped particles is proportional to the
length of the banana orbit, Rqθ0 , divided by half the bounce time, τb (θ0 ), and to find a suitable
average it is necessary to sum these speeds weighted by the distances as indicated in (2.73)(b).
More precisely, it is the sum of the weighted diffusivities that yields the appropriate transport
rate, but the process is not diffusive in the usual collisional sense. The trapped particles act as
a sort of pump, carrying the thermal energy along the banana orbit until it is dispersed without
further delay laterally down — or sometimes up — the temperature gradient.
The removal of thermal energy will be described in detail in Section 3.2.3. It involves two
stages, the first of which is the flow of heat along the banana orbits between the reflection
points in order to maintain thermal equilibrium for the second stage, which is the rapid lateral loss of energy across the field lines. ‘Parallel diffusivity’ refers to the first stage and as
described in Section 2.5.2, it enables us to find an expression for an effective ‘collision’ time
to use in the second stage; as we shall explain in Section 2.5.4, particles move far enough to
collide with other particles only in unsteady conditions.
The fraction dfT of particles that are reflected from a mirror in the interval θ0 , θ0 + dθ0
are at a distance λm = Rq θ0 from the midpoint of the banana orbit and take an average
)
to the parallel diffusivity due to these
time 12 τb (θ0 ) to reach it. The resulting increment dχ(T
particles follows from an application of (2.73)(b):
)
dχ(T
= (λ2m / 12 τb ) dfT ,
(−π < θ0 < π) .
Hence the total trapped particle diffusivity is
)
χ(T
=4
π
0
(Rq θ0 )2
dfT (θ0 ) ,
τb
(2.74)
2.5 Diffusivity of trapped particles
which by (2.68), (2.71) and ε = r/R, gives
16rq C π/2 x2 cos x
(T )
χ = √
dx .
π
K(sin x)
0
The integral is approximately 0.1976, so
)
χ(T
= k1 rq C
k1 ≈ 1.78 .
49
(2.75)
(2.76)
Next consider the fraction fU = 1 − fT of passing or “detrapped” particles. Their orbits
repeat after one connection length, λc = 2πRq, and hence their average displacement between
collisions is half this length. With τ the collision interval, their contribution to the parallel
2
)
diffusivity is χ(U
= fU πRq /τ . Hence from (2.76) the total parallel diffusivity is
2
1
C ≡ 2kB T 2 /m .
(2.77)
χ = k1 rqC + fU πRq /τ
To compare the terms in (2.77), consider the case of the electron gas. By the expression
for τe in (1.14) and that for Ce in (A.59), with the typical tokamak value ln Λ = 17 we get
Ce τe = 1.20 ×103 T̂e2 /(Zeff n19 ). Using this equation in (2.77) we find that, except very close
to the minor axis (r = 0), the contribution to the parallel diffusivity by the passing particles
is typically only about 5% of the trapped particle contribution and since the product Cτ is
independent of the particle mass, a similar conclusion holds for the ion gas. Furthermore, the
transverse particle energy is rather less in the passing particles than in the trapped ones, which
by the discussion in Section 2.5.2 further reduces their importance for thermal transport. And
in the light of the other approximations concerning temperature profile, we shall ignore the
detrapped particles, taking the parallel diffusivity to be χ = k1 rq C.
Perpendicular diffusivity, χ⊥ , required in the second stage of the cross-field transport of
energy, is a rather more complicated phenomenon, to be described in Section 3.5, but in local
equilibrium its time-scale will be the same as χ . Hence, by the argument given at the end of
the previous section, we express χ in the form C 2 τ⊥ , where
k1 rq
(C = 2kB T /m)1/2 , k1 = 1.78).
τ⊥ =
(2.78)
C
This is the time interval that will be applied to second-order thermal transport.
2.5.4 Thermal pumping
It is clear from Fig. 1.9 that diffusivity in tokamaks has time-dependent terms that are activated
in the neighborhood of minor disruptions. The very rapid oscillations shown in Fig. 1.9(a)
require an oscillatory diffusivity of amplitude much larger than the steady state value given
by (2.77). In Section 6.1 and Section 6.2 we shall show that while the diffusivity in (2.77)
proves sufficient to account for the relatively long ramp phase of the sawtooth oscillation, it
has no chance of providing an explanation of the rapid collapse phase of the sawtooth. Also,
the periodic increases in temperature imply that an apparently negative diffusivity operates
momentarily, reversing the expected direction of the heat flux and defecting it up the steady
state temperature gradient.
A mechanism that accounts for oscillatory diffusivity is illustrated in Fig. 2.9, which shows
radial distributions of the electron temperature and the electron fluid velocity, written as −ve
50
2 Tokamak magnetic fields
Figure 2.9: Normal and abnormal heat flux generating thermal waves
to make it parallel to the electric current density, jϕ ≈ −ene ve . The component figures show
the circumstances in which the heat flux is normal, i.e. down the temperature gradient, and
abnormal, i.e. up the temperature gradient. Let there be a time delay, τ∗ say, between the
evolution of Te and −ve , which are causally related with Te in the lead. Let Fig. 2.9(a) represent the initial conditions at time t0 and suppose that the Te profile evolves into that shown in
Fig. 2.9(d), then the −ve profile will follow (not shown in Fig. 2.9(d)), so that at t = t0 +τ∗ the
heat flux is abnormal as indicated in Fig. 2.9(d). The reduction in temperature increases the
−3/2
law (see (A.16) and (A.19)) and hence modifies the current
resistivity according to the Te
density, changing the profile of −ve into that shown in Fig. 2.9(c), which is attained at time
t = t0 + 2τ∗ . At this stage the heat flux is normal and the temperature profile evolves into that
arbitrary units
1
0.5
0
−0.5
−1
minor disruption
0
0.5
1
time variable
1.5
2
Figure 2.10: Thermal pumping with two time-scales
2.5 Diffusivity of trapped particles
51
Figure 2.11: Fishbone instability
shown in Fig. 2.9(b); the −ve profile follows and we arrive back at Fig. 2.9(a) at t = t0 + 4τ∗ .
Thus the profile of −ve follows that of Te , lagging an interval τ∗ behind, which accounts for
the temperature waves. We could have equally well made Te follow −ve ∝ jϕ by adopting
the path (a) → (b) → (c) → (d) → (a), which would be appropriate were the driving ‘force’
fluctuations in the loop voltage. The complete circuit is a kind of thermal pumping, which is
the name we shall attached to the process.
As shall be explained in Section 6.2.2, there are two types of thermal pumping, with values
of τ∗ separated by a factor of ∼ 20. Compare Figs 1.9(a) and 1.9(b); superimposed they
would yield a temperature wave packet of the type shown in Fig. 2.10. There are two ways for
these overstable oscillations to terminate; in one, after the final temperature fall, the recovery
fails resulting in a minor disruption (see Section 6.2.3), and in the other the wave packet is
completed and another commences; the so-called fishbone instability shown in Fig. 2.11 (from
McGuire, et al. (1985)) may be an example this latter phenomenon; Wesson (2004) offers a
rather different explanation in terms of resonance between an MHD wave and the toroidal
drift of trapped particles.
There remains the problem of the energy source for the large swings in temperature; the
flow rate of the trapped particle thermal energy is constrained by the relatively short time-scale
given in (2.78) and from (2.77) we find that the time scale for the detrapped particles is about
1/20 of that for the trapped particles. For the electrons typical values are τe ∼ 5 ×10−5 s,
τ⊥ ∼ 2 ×10−7 s, (πRq/C)2 /τe ∼ 10−8 s. However, in unsteady flow the connection length
2πRq no longer limits the range from which the energy can be drained or restored; the periodicity of the field structure is disturbed, which allows electrons to travel a full mean free
path before their energy is diverted by the shearing mechanism illustrated in Fig. 2.8. Thus in
unsteady flow the untrapped electrons dominate the parallel diffusivity and (2.78) is replaced
by
(unsteady state)
τe
(2.79)
τ⊥ = k1 rq/C ±
0
(steady state) .
The ‘steady’ state in this equation is strictly quasi-static, e.g. the ramp stage of the sawtooth oscillation is treated (in Section 6.1.3) as being slow enough for the leading term in τ⊥
to be accurate, but in the collapse phase the second term is dominant.
52
References
References
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Hirshmann, S.P., Hawryluk, R.J. & Birge, B. (1977). Nuclear Fusion, 17(3), 611.
Kadomtsev, B.B. & Pogutse, O.P. (1979). Plasma physics and controlled fusion research,
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3 Energy transport in Tokamaks
This chapter describes some of the mechanisms believed to be responsible for the loss of energy from tokamaks. The escape of thermal energy orders of magnitude more rapidly than predicted by classical or even neoclassical theory has been something of a mystery for tokamak
physicists for several decades, with turbulence as the most commonly accepted explanation.
However the consistent empirical scaling laws introduced in Section 1.5 imply the existence
of a laminar process common to all tokamaks. There is also the problem that the level of
turbulence required to explain energy transport would impede toroidal currents to negligible
values (see Section A.20). A successful mechanism is the ‘second-order’ process described in
Section 2.5, but to apply it, it is first necessary to understand the nature of the orbits traced by
particles in tokamak fields.
A convenient method of simplifying the description is to describe the motions of guiding
centers rather than the more complicated motions of the particles gyrating about these centers.
In effect we average out the gyrations and concentrate on the motion of the centers. This
works because the radius of gyration, known as the Larmor radius, rL = C/ωc, is very small,
typically about 3 mm for ions and only about 80 µm for electrons. So there is very little distinction between fluid motions and guiding center motions. However, a distinction of central
importance is that between the ‘bulk’ or fluid velocity v measured in the laboratory frame
L and the ‘peculiar’ velocity c = w − v, where w is the velocity of a bunch of like particles measured in L. The peculiar velocity — Maxwell’s ‘velocity of agitation’— is a frame
indifferent velocity measured in the convected frame. In much of the tokamak literature this
distinction appears to have been overlooked and is responsible for some significant errors.
3.1 Banana orbits
3.1.1 Drifts due to variations in the magnetic field
Guiding center drifts due to applied body forces like gravitation are discussed in Section A.14,
where it is shown that a force F causes a drift u⊥ , and conversely a drift u⊥ requires the
presence of a force F , where
F =
QB
m
b × u⊥ ,
u⊥ =
m
QB
F × b ,
(3.1)
where Q is the particle charge and m is its mass.
In addition to these guiding center drifts, there are drifts due to magnetic field inhomogeneities that depend on the particle’s peculiar velocity. The physical reason why a gradient
Theory of Tokamak Transport: New Aspects for Nuclear Fusion Reactor Design. Leslie Colin Woods
Copyright © 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
ISBN: 3-527-40625-5
54
3 Energy transport in Tokamaks
B (weaker)
electrons
e
i
B
ions
B (stronger)
Figure 3.1: Grad B drifts in P0
in the magnetic field strength B can cause a charged particle p to drift becomes apparent on
tracing the orbit of p in a plane orthogonal to the field. Assuming that in a frame P0 the electric
field is zero and p moves with constant speed, we see that the reduction of the Larmor radius
in the stronger field and its subsequence increase in the weaker field, periodically repeated,
drives the guiding centers in a direction orthogonal to both b and ∇B (see Fig. 3.1). From this
description it appears that the ions and electrons drift in opposite directions and could therefore generate an electric current in addition to that required by Ampère’s experimental law,
which, neglecting Maxwell’s displacement current, has the differential form µ0 j = ∇ × B.
However, by its definition j must also equal ene (vi − ve ), hence for the geometry of Fig. 3.1,
µ0 j = µ0 ene vi − ve = ∇ × B = −b × ∇B ,
(b ≡ B/|B|) ,
and the ion and electron fluid velocities must satisfy this basic relation.
Therefore, in a frame Pci convected with the ion fluid (vi = 0),
ve =
1
m Fem × b
b × ∇B =
ene µ0
QB
Fem ≡ −
QB
eme ne µ0
∇B .
(3.2)
When the average Larmor radius is negligible compared with macroscopic length scales
like |∇ ln B|−1 , |∇ ln n|−1 and |∇ ln T |−1 , the cross-field fluid velocity and the average
guiding center velocity are indistinguishable. Hence comparing (3.2) with (3.1), we conclude
that in the convected frame Pci there is an electromagnetic force Fem acting on the electrons
and no comparable force acting on the ions. Thus, in determining the additional influence of
magnetic field inhomogeneities on particle drifts, it is essential first to remove the electromagnetic force by choosing an origin convected with Pc .
Let Pc be a frame convected across the magnetic field at the (ion or electron) fluid velocity
and suppose that c̄⊥ denotes the average orbital speed for a particle that has no drift relative to
Pc , then particles with c⊥ > c̄⊥ will have cycloid-like motions of radius c⊥ /|ωc | > c̄⊥ /|ωc |
3.1 Banana orbits
55
Figure 3.2: Relative ion drifts in the convected frame
and hence will be more affected by the gradient in the magnetic field than the particles for
which c⊥ < c̄⊥ , and which are confined to smaller orbits. Relative to Pc the two groups
will therefore drift in opposite directions, as depicted in Fig. 3.2 for ions. Remember that
this description applies only to peculiar velocities and by definition the average taken over all
speeds and orientations of c is zero.
3.1.2 Gyro-averages
Referring to Fig. 3.3 we have for the value of B at the particle p,
B = BG + a · ∇B + O(kL2 ) ,
where BG is its value at the guiding center and kL is the small number defined by
kL ≡ |a · ∇ ln B| 1 .
Figure 3.3: Motion of a particle about its guiding center
(3.3)
56
3 Energy transport in Tokamaks
As |a| is the Larmor radius, this is the constraint mentioned in the previous section.
Thus (see (A.77))
ċ⊥ =
Q
m
c×B =
Q
m
c × BG −
Q
m
a · ∇B × c + O(kL2 ) .
(3.4)
To solve (3.4) we shall adopt the method of ‘variation of parameters’, that is we start with
the solution for the case B = const., which may be written (cf. (A.86))
Q
c=
(3.5)
x − X × BG + c + d⊥ ,
m
where X and d⊥ are constants for B uniform, but will depend on (x, t) otherwise.
In a frame following the particle,
Ḃ = DB + c · ∇B = −∇ × E + c · ∇B ≈ c · ∇B,
if |E | c|B| ,
where E is the electric field at Pc . This approximation, known as the ‘weak electric field’
constraint, requires that in a gyro-time ωc−1 the change in the particle velocity due to the
electric field E is small compared with the peculiar speed c, i.e. (QE /m)ωc−1 = E /B c.
Differentiating (3.5) and adopting this constraint we obtain
Q
Q
Q
ċ = c × B − Ẋ × B + a × c · ∇B + ċ + ḋ⊥ + O(kL2 ) ,
(3.6)
m
m
m
where the location subscript G is no longer essential. Subtracting (3.4) from (3.6) we find
Q
Q
c · ∇B × a − a · ∇B × c − ċ − ḋ⊥ + O(kL2 ) ,
F = B × Ẋ =
m
m
where by (3.1)1, F is the force on the particle due to the inhomogeneity. By (3.1)2 the corresponding drift is
Ẋ⊥ =
m
1
b × (ċ + ḋ⊥ ) + O(kL2 ) ,
c · ∇B × a − a · ∇B × c × b +
B
QB
(3.7)
of which the average over one complete orbit, i.e. over all values of θ̂, will give the average
drift velocity due to the magnetic field gradient.
The Larmor vector, a = x − X, is given by (3.5),
a=
m
m
b × (c − d⊥ ) ≈
b × c + O(kL2 ) ,
QB
QB
(3.8)
where the approximation holds if |d⊥ | |c⊥ |, which is equivalent to the constraint in (3.3).
Let r̂ be unit vector parallel to a and hence orthogonal to θ̂ and b (see Fig. 3.3), then for
present purposes it is sufficiently accurate to assume that r̂ and θ̂ are distributed isotropically,
i.e. that their values averaged over one complete gyration are zero. We shall denote gyroaverages by · · · and averages over all speeds by a bar.
From the average of 11 · A = r̂r̂ + θ̂θ̂ + bb · A , where A is any second-order tensor, it
follows that
(3.9)
r̂r̂ · A = θ̂ θ̂ · A = 12 A − bb · A = 12 11 − bb · A = 12 A⊥ .
3.1 Banana orbits
57
By (3.8), a = (c⊥ /ωc )r̂, and as r̂ · ∇b = 0 and θ̂ · ∇b = 0, we have
a · ∇B × c =
c2
c⊥
r̂ · ∇B × (c⊥ θ̂ + c b) = ⊥ r̂ · ∇B × (r̂ × b)
ωc
ωc
=
c2⊥ r̂r̂ · ∇B − r̂r̂ ·· ∇B b
ωc
=
c2⊥ ∇⊥ B − ∇⊥ · B b ,
2ωc
and since ∇⊥ · B b = −∇ · B b = −∇ B, we get
a · ∇B × c =
c2⊥
∇B .
2ωc
(3.10)
A similar calculation yields c · ∇B × a = 0.
We also require the gyro-average of ċ × b. Adopting the weak electric field constraint,
we have
ċ × b = c ḃ × b = c Db × b + c · ∇b × b ≈ c c · ∇b × b .
Hence, since c⊥ = 0, we find
c ḃ × b = c c⊥ + c b · ∇b × b = c2 b · ∇ × b .
(3.11)
Let u denote the guiding center drift velocity, then in a uniform magnetic field
u⊥ = v⊥ ,
u = v + c b,
u = v + c .
(3.12)
It follows from (3.10) and (3.11) that the gyro-average of (3.7) is the increment δu to the drift
velocity:
c2
1
c2
b · ∇b × b +
b × ḋ⊥ + O(kL2 ) .
(3.13)
δu ≡ Ẋ⊥ = ⊥ b × ∇ ln B −
2ωc
ωc
ωc
Adding this to (3.12) we get u + δu = v + c , and because the guiding centers are virtual
labels for their associated particles and are never more than a Larmor radius away from them,
the average of u + δu over all speeds must equal the fluid velocity v. Thus to O(kL2 ) accuracy
the average of δu over all speeds is approximately zero, which condition enables us to evaluate
the constant ḋ⊥ in (3.13). For this purpose we introduce the kinetic definitions,
u⊥ ≡ 12 c2⊥ ,
u ≡ 12 c2 ,
(3.14)
then (3.13) becomes
δu =
m
QB
1 2
2
2 c⊥ − u⊥ b × ∇ ln B − c − 2u b · ∇b × b ,
which applies to both the ion and electron components.
(3.15)
58
3 Energy transport in Tokamaks
Unlike (3.15) the drift equations to be found in the literature are often not frame indifferent, that is the sum of δu over all the particles at a given point does not vanish, which means
that particles manage to shed their guiding centers and move separately. For example, see Braginskii (1965), where in his section Certain Paradoxes, he claims this to be a consequence of
boundary conditions, an idea that seems to have started with Spitzer (1962, p. 26). This leads
Braginskii to introduce a magnetization current, which if consistently included in the theory,
would falsify the universally adopted MHD relation µ0 j = ∇ × B. The matter is discussed
in pp. 212 to 215 of Woods (1987), in pp. 37 to 38 of Woods (2004), and in Section A.25 there
is another way of viewing the problem.
3.1.3 Banana width
Consider a trapped particle p moving periodically between reflection points R at θ = θ0 and
R at θ = −θ0 . The line of magnetic force passing through the reflection points lies on the
magnetic surface, which is approximated by r = const., and as p moves from R to R, it will
drift away from this surface, reaching a maximum radial displacement at θ = 0. According
to Fig. 3.2 whether the drift is inwards or outwards depends on the sign of (c⊥ − c̄⊥ ). On
the return from R to R the radial drift is in the opposite direction, with the net result that
the guiding center traces out a ‘banana’-shaped trajectory as shown in Fig 3.4. To verify this
description we need the radial component of (3.15), but first we shall transform it in a more
convenient form.
The term containing ∇ ln B is known as the ‘grad B’ drift and the other term is called the
‘field curvature’ drift. The field line curvature κ is given by
κ = b · ∇b = −b × ∇ × b = −B −1 b × ∇ × B + B −1 b × ∇B × b ,
= ∇⊥ ln B + B −2 µ0 j × B ,
Figure 3.4: Banana orbit
3.1 Banana orbits
59
hence (3.15) can be written as
m 1 2
µ0 m 2
c − 2u j⊥ .
δu =
c⊥ + c2 − (u⊥ + 2u) b × ∇ ln B +
2
QB
QB
(3.16)
By conservation of charge there can be no current in the radial direction, so the radial component of (3.16) is
r̂ · δu =
m 1 2
dr
2
=
2 c⊥ + c ) − (u⊥ + 2u ) r̂ · b × ∇ ln B .
dt
QB
(3.17)
Averaging this over all particles we get
r̂ · δu = 0 .
(3.18)
Ignoring the small Shafranov shift, we find from (2.48) and (2.49) that
B=
B0
,
1 + ε cos θ
ε = r/R,
2
2 1/2
B0 = Bϕ0
+ Bθ0
,
(3.19)
and therefore by (2.62)2 and (2.63), neglecting terms O(ε2 ),
∂
∂ r̂ · b × ∇ ln B = −r̂ × ϕ̂ + sθ̂ · r̂
+ θ̂
ln 1 + ε cos θ
∂r
r∂θ
= sin θ/R .
Hence (3.17) becomes
c
q 1 2
dr
=
( 2 c⊥ + c2 ) − (u⊥ + 2u ) sin θ .
dθ
ωc
(3.20)
To sufficient accuracy, by (2.70) and the equation preceding it we get
c
dθ
=
,
dt
Rq
c2 = c2 F (θ) ,
F (θ) ≡ 2ε sin2 12 θ0 − sin2 12 θ .
(3.21)
From c2 = c2⊥ + c2 , u⊥ + u = u = 12 c2 , we find
( 12 c2⊥ + c2 ) − (u⊥ + 2u ) = 12 c2 − u + 12 c2 − u =
1 2
2 c − u 1 + F (θ)
and therefore (3.20) becomes
c
1
dr
q 1 2
1
=
c − u F 2 + F − 2 sin θ .
2
dθ
ωc
(3.22)
Omitting an O(ε2 ) term, we obtain the integral
c ∆r = ±
1
2q 1 2
c − u cos θ − cos θ0 2
1
2
ωc ε 2
(−θ0 ≤ θ ≤ θ0 ) ,
(3.23)
where ∆r is the radial displacement and the constant of integration is selected to make ∆r = 0
at the reflection points. The sign of (3.23) is positive for θ increasing and negative for θ
60
3 Energy transport in Tokamaks
decreasing and as ωc = QB/m is positive for ions and negative for electrons, ions with more
than average energy drift outwards as θ increases and inwards after reflection, as shown in
Fig. 3.4; electrons behave in the opposite manner.
Another formula for ∆r follows from (3.21)2 and (3.23),
c ∆r = ±
2q 1 2
c − u cos θ − cos θ0 .
ωc 2
(3.24)
Notice that by (3.14) the values of c ∆r and c ∆r averaged over all particle speeds are
O(kL2 ) ≈ 0:
c ∆r = 0 ,
c ∆r = 0 ,
(3.25)
the second of which means that there is no net mass flux from one end of the banana to the
other, otherwise a steady state would not exist (see Section A.25).
3.1.4 Neoclassical diffusivity
For transport via the banana orbits and particle collisions we need to calculate the orthogonal
)
diffusivity, χ(T
⊥ , of the trapped particles. Let τes be the average time that it takes a trapped
particle to escape from its banana orbit (see Section A.10), then the required diffusivity for
the electrons follows from (cf. Section 2.5.3)
)
χ(T
=
⊥
(∆r)2
dfT ,
τes
(τes = 0.65| sin 12 θ0 |ετe ) ,
(3.26)
where dfT and ∆r are defined in (2.68) and (3.23), and here · · · denotes the average taken
over all possible banana orbits. The banana orbit shown in Fig. 3.5 is an accurately drawn
example of equation (3.23) with the major axis on the left-hand side.
Hence
π 2
(2ε) 2
∆r cot 12 θ0 dθ0 ,
0.65ετe −π
1
)
χ(T
=
⊥
(3.27)
where from (3.23)
2 2
4q 2 ∆r = 2 12 c − u/c cos θ − cos θ0 ,
ωc ε
−θ0 ≤ θ ≤ θ0 .
(3.28)
The average of (cos θ − cos θ0 ) over −θ0 ≤ θ ≤ θ0 is (sin θ0 /θ0 − cos θ0 ) and therefore
χ
(T )
⊥
√
4 2q 2
=
0.65ωc2 τe ε1/2
π
0
2
(sin θ0 /θ0 − cos θ0 ) cot 12 θ0 dθ0 12 c − u/c .
(3.29)
3.2 Thermal conductivity
61
90
120
60
P
150
o..
30
180
0
Q
o
330
210
300
240
270
Figure 3.5: Banana orbit and diffusion
2
The integral in (3.29) is approximately 1.255, and (see Section A.9) 12 c − u/c = 3 C 2 /8.
Therefore the thermal diffusivity and radial neoclassical heat flux for electrons are
2
pe q 2 r2
q 2 rLe
)
= 4.10 3/2Le ,
Qnc
Te
κe = 1.5kB ne χ⊥ . (3.30)
χ(T
er = −6.15 3/2
⊥e
ε τe
ε τe Te
2
The cross-field diffusivity represented by rLe
/τe is the classical value in a uniform magnetic
field (see Section 3.3.3), so the enhancement by the factor 4.10 q 2 ε−3/2 indicates that the diffusivity at a point Pc is not determined by local conditions at Pc alone. The term neoclassical
(see Hinton and Hazeltine 1976) has been adopted to describe this departure from the princi1
ple of local action. The ion thermal diffusivity is ∼ (mi Te /me Ti ) 2 times larger. With typical
)
2
tokamak values of ε ∼ 0.15, and q ∼ 3, we find χ(T
⊥e ∼ 600 rLe /τe . Thus neoclassical effects greatly increase the classical diffusivity, although there is another process involving fluid
shear, described in Section 2.5.1 and to be developed further shortly, that generates thermal
diffusivities about two orders of magnitude greater still.
3.2 Thermal conductivity
3.2.1 Neutral gas
The diffusion vector Jφ for a property φ is written (see Section A.15):
Jφ = −χφ ∇φ ,
χφ = αλ2 /τ ,
(3.31)
where χφ is the diffusivity, α is a constant of order unity and the collision interval τ and mean
free path λ are related to the property φ.
For the heat flux vector, Jφ = q, the appropriate microscopic values are identified as τ2
and λ2 = Cτ2 , and in this case φ is the internal specific energy, u = cv T . In a monatomic gas
62
3 Energy transport in Tokamaks
the specific heat at constant volume is 32 kB /m, where kB is Boltzmann’s constant and m is the
particle mass. To obtain the classical expression for the diffusivity, we choose α = 56 , giving
χ = 56 λ22 /τ2 = 56 C 2 τ2 = 53 (p/)τ2 .
(3.32)
The collision interval for momentum transport is τ1 = 2τ2 /3 (see Woods 1993), hence
Fourier’s law for heat flux reads
κ = cv χT = 52 cv µ ,
µ = pτ1 ,
(3.33)
q = −κ∇T ,
where κ is the coefficient of thermal conductivity and µ is the coefficient of viscosity. Notice
that |q| ∝ |λ2 ∇ ln T | = O(kN ), where kN is the Knudsen number (Section A.18). For a
reason to be explained shortly, we have ignored the thermoelectric contribution, q ∝ j, to the
heat flux.
An important property of q not made explicit in the above introduction, is that it depends
on the temperature gradient at time τ earlier than the present time t, as it takes this interval for
the molecular trajectories to transport energy. Thus (3.33) is really an approximation to
q(r, t) = −κ∇T (r − τ v, t − τ ) ,
(3.34)
2
obtained by neglecting terms O(kN ).
3.2.2 Magnetoplasma
The thermoelectric contribution to the heat flux vector is proportional to j, and hence can be
ignored as it has no radial component to contribute to the energy losses. In the presence of a
magnetic field (3.33) is generalized to
2
),
q(r, t) = −κ · ∇T (r, t) + O(kN
where from Section A.7
(3.35)
κ = κ bb + κ∧ b × 11 + κ⊥ 11 − bb .
(3.36)
Expressions for κ , κ∧ and κ can be determined as follows. We write (3.35) as
κ−1 · q(r, t) = −∇T (r, t) ,
(3.37)
and evaluate the left-hand side from the fact that it takes a collision time τ for q to respond
to the thermodynamic force −∇T (r, t). It follows from (A.34), i.e. ṙ = (−ωc b) × (r − X),
that the charged particles spin about the field lines with an angular velocity −ωc b. Hence in
a frame F spinning at this rate, the effect of the magnetic field on the particles vanishes, so
that after a time τ , the instant at which collisions complete the transfer of energy, the left-hand
side of (3.37) is reduced to the vector form κ−1 q (r + τ v, t + τ ), where q is the value of
the heat flux vector as observed in the frame F at time (t + τ ). Hence (3.37) is equivalent to
q (r + τ v, t + τ ) = −κ∇T (r, t) .
(3.38)
2
If the vector R in (A.65), viz. v − v = R · ∇v + O(R ) , is convected with the fluid, its
rate of change in the laboratory frame is dR/dt = v − v, and therefore by (A.67),
, D ≡ ∂ + v · ∇ . (3.39)
DR ≡ DR − Ω × R = R · e
e ≡ 12 ∇v + ∇v
∂t
3.2 Thermal conductivity
63
The left-hand side of this equation is the rate of change of R in a frame that is both convected
and spinning with the fluid element. Similarly, the rate of change of a vector q in the frame F
spinning with the gyrating particles with angular velocity −ωc b is
(3.40)
Dq + ωc b × q = Dq + ωc b − Ω × q ,
where Dq is the usual convective time derivative. Therefore, we can adopt the expansion
q (r + τ v, t + τ ) = q(r, t) +
b × q(r, t) + τ Dq(r, t) + 12 τ 2 D2 q(r, t) + . . . .
We now impose the constraint |τ n Dn q| |q|, i.e. (τ /T )n 1, where T is the macroscopic
time scale, which by (A.106) requires that the Knudsen number kN be much less than unity.
2
), hence correct to O(kN ) (3.38) reduces to
The vector q is O(kN ) and therefore τ Dq is O(kN
q+
b × q = −κ∇T
(
= ωc τ ) .
(3.41)
By (3.39) the Knudsen number constraint can be expressed as τ |q · e| |q| or
τ |∇v| 1. In the tokamak application for detrapped particles τ is the time particles take to
travel a connection length, i.e. τ = 2πRq/C, while for trapped particles the appropriate value
for τ is the weighted bounce time τ⊥ defined in (2.78). It will be shown in Section 3.5.5 that
even in the case of detrapped electrons, the constraint kN 1 is satisfied.
The solution of (3.41) follows from the theorem in Section A.21, the parameter α of which
takes the values α = 1, α∧ = , α⊥ = 1. Thus we arrive at (3.36) with
κ = κ =
γ0 kB
pτ,
m
κ∧ =
− κ
,
1+ 2
κ⊥ =
κ
1+
2
Figure 3.6: The ratio X = κ⊥ /κ for the electron fluid
Br = Braginskii (1965); C & C = Chapman & Cowling (1970);
F & K = Ferziger & Kaper (1972)
γ0 = 3α .
(3.42)
64
3 Energy transport in Tokamaks
Figure 3.7: (a) Heat flux components and (b) the effect of fluid shear
Alternatively (3.35) may be written
q = −κ k · ∇T
k ≡ bb −
1+
b × 11 +
2
1
1+
1
1
−
bb
.
2
(3.43)
In Fig. 3.6 the ratio X = κ⊥ /κ for energy transport in the electron gas is plotted for
three different theories; the curve labeled ‘F & K’ is from Ferziger and Kaper (1972, p. 454),
the curve ‘C & C’ is from Chapman and Cowling (1970), which gives the same values as
(3.42), while the ‘Br’ curve is from Braginskii’s (1965) treatment. In addition there is a
dashed curve, which shows the conductivity ratio for the plasma (electrons plus ions) obtained
from Braginskii’s theory. These results show that there is some uncertainty in magnetoplasma
transport theory. The simplest theory, i.e. the one leading to (3.43), is physically lucid, and
gives values lying between those from the other two theories, so we shall adopt it. The three
heat flux components are illustrated in Fig. 3.7(a).
In ‘strong’ magnetic fields, defined by the condition
1, it follows from (3.43) that
q⊥ = −
κ
2
κ
∇T ,
2
=
γ0 kB mp
,
e2 B 2 τ
(3.44)
where (see Section A.2)
3/2
τe =
2.75 ×105 Te
,
ln Λ ne Zeff
1
τi =
3/2
1.67 ×107 A 2 Ti
.
ln Λ
Z 3 ne
(3.45)
The original concept of using very strong magnetic fields to contain the energy of hot plasmas
was based on (3.44).
3.2 Thermal conductivity
65
Hot
T
q
v
T
+B
2rL
q
q
electrons
transverse
heat flux
q
q
q2
Minor axis
Sheared
flow
+B
v
q
electron fluid velocity
Cool
Figure 3.8: Transverse heat flux in a strong magnetic field
We note that for a hydrogen plasma,
(κ/
2
)i ≈ 30(κ/
2
)e ,
(3.46)
i.e. in classical cross-field transport the ion thermal conductivity is dominant. Notice from
(3.32) and (3.44) that the classical thermal diffusivity for untrapped particles in strong magnetic fields is
χ⊥ =
2
5C 2 τ2
5 rL
=
6τ
6 2
2
(rL = C/ωc) ,
(3.47)
)
= 4.10q 2 ε−3/2 r2L /τ2 . Equation (3.47)
whereas with trapped particles (3.30) reads χ(T
⊥
allows the interpretation that rL /τ2 is the speed at which energy is transmitted across the
distance rL , a speed that is greatly increased with trapped particles.
6
Typical tokamak values for
= ωc τ are: e ∼ 3 ×107 ,
i ∼ 10 . Hence for both
the ions and the electrons, |q∧ |
|q⊥ |, e.g. for electrons the heat circulating around the
minor cross-section is ∼ 107 times larger than that flowing out radially. This has serious
consequences for the ability of tokamak fields to retain the plasma energy, for as illustrated
in Fig. 3.7(b), electron fluid shear can deflect q∧ into q∗∧ , which has the radial component
q∧ 2 , and even if the deflection angle is only one tenth of a degree, the resulting radial heat
flux will be ∼ 104 times larger than the classical value of q⊥ and ∼ 102 times larger than the
neoclassical value. There is also the possibility that the deflection could be inwards, in which
case a thermal instability will result. These remarks will be amplified in later sections.
Because of the importance of the transverse heat flux in tokamak theory, it is useful to
have a clear physical picture of its origin. Figure 3.8 shows that this results from the difference between the energy transported in opposite directions around adjacent orbits. The
electrons remain in their orbits, so the average velocity of all the particles passing through an
infinitesimal volume is zero, i.e. the fluid velocity is zero (see Section A.25). The assumption
that the particles trace circular orbits automatically implies zero fluid velocity, since in all
other frames the orbits are cycloidal. The radial heat flux labeled q∧ 2 in the figure is due to
the fluid shear; this flux is the key to understanding tokamak transport. It will be discussed in
the following section and given a physical interpretation in Section 3.5.3.
66
3 Energy transport in Tokamaks
Figure 3.9: Fluid shear generating a second-order heat flux, q2
3.2.3 Fluid shear and transport
2
We shall start by illustrating the mechanism of O(kN
) heat flux with the simplest possible
example. Let x̂ and ŷ denote unit vectors in the OX and OY directions, and assume that the
temperature T increases in the direction ŷ at the rate dT /dy = T as indicated in Fig. 3.9.
Then the O(kN ) or first-order heat flux is q1 = −κT ŷ, where κ is proportional to the mean
free path, λ. If in addition the plasma is flowing with a sheared velocity v = v (y)x̂, where
v = dv/dy, there will be a secondary heat flux q2 along x̂ proportional to the product T v due to the difference between the heat carried to the right in the top half of the figure and that
carried to the left in the bottom half; its magnitude is usually quite small compared with the
primary heat flux q1 . It is readily deduced from the figure that
q2 = − 12 τ v q1 x̂ = 12 τ v κT x̂ ,
(3.48)
where the dimensionless number τ v is the Knudsen number kN , and since κT is also pro2
portional to kN , q2 = O(kN
).
To deal with the general case we start from the heat flux law in a magnetic field, which
from (3.34) and (3.41) is
3
),
b × q(r, t) = −κ∇T (r − τ v, t − τ ) + O(kN
τ Dq(r, t) + q(r, t) +
(3.49)
2
where we have retained the O(kN ) term τ Dq (see derivation of (3.41)).
To evaluate the right-hand side of (3.49) we use the fact that on collisional time-scales
temperature gradients are embedded in the fluid, that is they are convected with the fluid and
spin with any vorticity present. The spinning of the charged particles about the magnetic field
lines is on much too small a scale to influence the spatial temperature distribution during a
collision interval. Therefore by an application of (3.39),
3
),
∇T (r − τ v, t − τ ) = ∇T (r, t) − τ D∇T (r, t) + O(kN
(3.50)
D∇T ≡ D∇T − Ω × ∇T = e · ∇T ,
(3.51)
where
in which Ω is the fluid spin and e is the rate of strain tensor. Hence (3.49) becomes
τ Dq + q +
b × q = −κ∇T + τ κe · ∇T .
(3.52)
3.2 Thermal conductivity
67
In Section 3.2.2 we obtained the solution of
q+
b × q = −κ∇T ,
(3.53)
which by (3.43) is the first-order heat flux
q1 = −κ k · ∇T ,
(3.54)
with
k ≡ bb −
1+
2
b × 11 +
1
1+
2
11 − bb .
(3.55)
Similarly, it follows that the solution of (3.52) can be written
q = −κ k · ∇T + τ κ k · e · ∇T − τ k · Dq .
(3.56)
Our interest is in cross-field gradients so we shall ignore parallel gradients, which in any
case are relatively small since q is unimpeded by the magnetic field. Also with strong magnetic fields we may omit terms O( −2 ), in which case (3.55) becomes
k=−
1
b × 11 ,
(3.57)
and the solution in (3.56) reduces to
q=
κ
b × ∇T −
τκ
b × e · ∇T .
(3.58)
3.2.4 Heat flux, second-order in Knudsen number
For κ∧ the coefficient γ0 in (3.42) is exactly 2.5, so in (3.58), κ/ = 5kB p/(2QB), where
Q is the particle electric charge. Also e can be replaced by its deviator (see Section A.16)
because the term omitted is parallel to the leading term in (3.58) and an order O(kN ) smaller.
Hence the second-order heat flux takes the form,
q2 = −
◦
5kB p
τ b × ∇v · ∇T .
2QB
(3.59)
It applies to both the ion and electron gases and was found first by a direct mean-free-path
argument (Woods 1983); its derivation from a modified kinetic equation is given in Woods
(1993) — but it cannot be derived from Boltzmann’s kinetic equation.
In strong magnetic fields from (3.44) and (3.58),
κ
κ
b × (τ e) · ∇T ,
q1 ⊥ = − 2 ∇⊥ T ,
q2 ⊥ = −
(3.60)
⊥
where kN = τ ||e|| is the Knudsen number. Hence
|q2 ⊥ /q1 ⊥ | =
kN ,
(3.61)
68
3 Energy transport in Tokamaks
which is an important result. For example the electron gas in a typical tokamak has e ∼ 107
and kN ∼ 10−2 , making the term about 105 times larger than the first-order term. This may
cast doubt on the convergence of the kN power series, but there are in effect two independent
series, one associated with −2 and the other with −1 , so the condition |q2 ⊥ /q1 ⊥ |
1
does not affect convergence. What is particularly important is that q2 ⊥ may either be down
the temperature gradient (as to be expected) or, depending on the structure of the fluid flow
determining e, the heat can even flow up the temperature gradient1 In Section 6.1.1, developing the introduction given in Section 2.5, we shall show that this phenomenon is responsible
for ‘sawtooth’ instabilities in tokamaks.
The solar corona provides another notable example of heat flowing up a temperature gradient. The corona is typically at a temperature of 2 000 000 K, while the only possible source
of this energy, the photosphere, is less than 6 000 K. It may be possible to explain this mystery
with the help of plasma loops and an application of (3.59).
From the discussions in Section 2.5.3 and Section 2.5.4 for trapped particles, the collision
interval τ is replaced by the time τ⊥ defined in (2.78), which for electrons is
k1 rq
τ⊥ =
(k1 = 1.78) .
(3.62)
Ce
The interval τ⊥ is the time it takes for thermal energy to be swept along the banana orbits to
the reflection points, where it is removed tangentially by the mechanism illustrated in Fig. 2.8
and to be described in detail below in Section 3.5.3.
3.3 Classical treatment of particle transport
3.3.1 Equilibrium currents
When the collision time τ is much less than the trapped particle bounce time τb , classical
transport dominates. The rate at which plasma escapes from a tokamak in these conditions was
first calculated by Pfirsch and Schlüter (1968) and although this situation does not apply in
tokamaks, the theory is valuable for reference purposes; a modified version of their treatment
follows.
If viscosity and fluid acceleration can be neglected, the equation of motion for each of the
electron and ion fluids can be written (see (A.12) and (A.13))
1
0 = Q E + v × B − ∇p − QZη · j
n
Ze = 1 , Zi = ne /ni ,
(3.63)
where Q is the particle charge, we have omitted the subscripts e and i on the dependent variables and η has the form given in (A.46). It will be assumed that the gradients are parallel to
the radial unit vector r̂. Solving for the radial component of the fluid velocity and using the
plasma equilibrium condition,
j × B = ∇p,
(p = pi + pe ) ,
(3.64)
1 This result apparently defies Clausius’ form of the second law of thermodynamics, but that electric currents can
cause heat to flow up temperature gradients is a well-known thermo-electric phenomenon; this case is similar, but the
electric currents now appear in the thermal conductivity. The second law is not involved.
3.3 Classical treatment of particle transport
69
we find
r̂ · v = b × r̂ · E /B − Zη⊥ r̂ · ∇p/B .
(3.65)
In tokamaks the electric field term in (3.63) is dominant; its value is found as follows.
An appropriate expression for the electric field is
E=−
∂φ
∂φ
r̂ −
θ̂ + Eϕ ϕ̂ ,
∂r
r∂θ
(3.66)
where φ is the electric potential. The unit vector b parallel to B is given by
b=
1
ϕ̂ + sθ̂
1
2
(1 + s ) 2
(s = Bθ /Bϕ ) ,
(3.67)
so its scalar product with (3.63) written for the electron fluid is
η (jϕ + sjθ ) = Eϕ −
s ∂φ
.
r ∂θ
(3.68)
The radial component of (3.64) can be expressed
jθ − sjϕ = p /Bϕ
p = dp/dr ,
(3.69)
it being assumed that p depends only on r.
3.3.2 Pfirsch–Schlüter current
We shall denote poloidal averages by a bar and make use of the definite integral
1
2π
2π
0
1
dθ
=
1 .
1 + ε cos θ
(1 − ε2 ) 2
(3.70)
By (2.48), (2.49) and the definition F (r, θ) = 1/(1 + ε cos θ),
Bθ (r, θ) = F Bθ0 (r) , Bϕ (r, θ) = F Bϕ0 (r) , jθ (r, θ) = F jθ0 (r) .
Setting A in (A.103) equal to E, we find that Eϕ (r, θ) = F E0 , where E0 is a constant. Hence
the poloidal averages (i.e. averages over θ) of (3.68) and (3.69) are
j̄ϕ +
E0
s
1
,
1 jθ0 =
η (1 − ε2 ) 12
(1 − ε2 ) 2
− j̄ϕ +
1
p
.
1 jθ0 =
2
sBϕ0
s(1 − ε ) 2
Solving for jθ0 , writing jθ = F jθ0 , and substituting in (3.69), we obtain the current densities
1
sEϕ
p (1 − ε2 ) 2
,
+
2
(1 + s )η
Bϕ0 1 + s2 1 + ε cos θ
1
jθ =
(3.71)
70
3 Energy transport in Tokamaks
and
jϕ =
Eϕ
p
−
2
(1 + s )η
sBϕ0
1
1
(1 − ε2 ) 2
1 + ε cos θ −
,
1 + s2 1 + ε cos θ
(3.72)
the second of which is known as the Pfirsch–Schlüter current. These equations can be combined to give
1 η p
(1 − ε2 ) 2
1 + ε cos θ −
η jϕ + sjθ − Eϕ = −
.
(3.73)
sBϕ0
1 + ε cos θ
From (3.66), (3.68) and (3.73) we get
1 ∂φ
b × r̂ · E
(3.74)
=−
+ sEϕ
B
Bϕ r∂θ
1
η p 1
=−
sEϕ (1 + ε cos θ) + 2
(1 + ε cos θ)2 − (1 − ε2 ) 2 ,
Bϕ0
s Bϕ0
where, since s = Bθ /Bϕ is relatively small in tokamaks, we have neglected s2 compared with
unity.
3.3.3 Mass diffusivity
Because mass transport is usually associated with convection, which is quite different in nature
from diffusion, the term “mass diffusivity” appears to be incongruous. However, its principal
component has the hall-mark of diffusivity in being frame-indifferent (see Section A.15).
It is readily verified from (3.65), which applies to both the electron and ion fluids, that the
mass diffusion in the radial direction is ambipolar, i.e. r̂ · (vi − ve ) = 0, otherwise charge
accumulation would occur, which is not possible on the length sales involved here. Hence for
the average particle flux of either species we may take the poloidal average,
nvD =
n
2π
2π
0
r̂ · v(1 + ε cos θ) dθ ,
(3.75)
where the factor (1 + ε cos θ) allows for the variation in the element of surface area in a plane
orthogonal to the minor axis of the torus. When ni = ne , it follows from (1.8), (3.65), (3.74)
and (3.75) that, correct to first order in ε2 ,
p
sEϕ
vD = − η⊥ + 2η q 2
−
,
2
Bϕ0
Bϕ
that is
vD = −Dp /p − sEϕ /Bϕ
(s = Bθ /Bϕ )
(3.76)
where, since 2η ≈ η⊥ (see (A.19)),
η⊥ p
D = 1 + q2 2 ,
Bϕ0
(3.77)
3.4 Neoclassical theory and its validity
71
is the mass diffusivity. Now
2me pe
C2
r2
η⊥ p
= 2
= 2 e = Le ,
2
2
Bϕ
e ne τe Bϕ
ωce τe
τe
so that
r2
D = (1 + q 2 ) Le .
τe
2
The diffusivity has two components, the cylindrical diffusion coefficient, DC = rLe
/τe ,
FS
2 2
FS
C
and the Pfirsch–Schlüter diffusivity, D = q rLe /τe . In a typical tokamak D ∼ 9D .
3.4 Neoclassical theory and its validity
The account of neoclassical diffusion presented in Section 3.1.4 follows from simple physical
reasoning, but there is an analytical treatment based on a kinetic equation for the distribution of
guiding centres, known as the “drift kinetic equation” (Section A.13). It is from this equation
plus some constraints termed “transport ordering” that the neoclassical transport equations
are derived. However, because this theory is not supported by tokamak observations and also
yields some clearly unphysical terms, the mathematical details will not be pursued beyond
listing of the equations obtained. For these details the reader is referred to either Hazeltine
and Meiss (1992) or Helander and Sigmar (2002); Wesson (2004) also gives an account of the
theory.
3.4.1 Banana and plateau regimes
Estimates for the bounce time τb and the escape time τes follow from (2.71) and (A.64):
1
τb ∼ Rq/C ε 2 and τes ∼ ετD , where τD is the particle deflection time. Let ν ≡ Rq/CτD
define a normalized frequency, then τb < τD if ν < ε3/2 . This specifies the ‘banana’ regime
in which the trapped particles are able to complete banana orbits before escaping. From
(3.30) and the corresponding equation for the ions it follows that in this regime the diffusivity
is (changing notation)
ν < ε3/2 ,
(3.78)
DB ∼ Aq 2 ε−3/2 r2L /τD
where A is a constant of order unity. The ratio ν∗ ≡ ν/ε3/2 is called the ‘collisionality’ and
the banana regime is usually defined by the condition ν∗ 1.
On the other hand, when ν > 1, collisions prevent the particles completing banana orbits
and we have the regime in which Pfirsch–Schlüter diffusion applies,
1<ν ,
(3.79)
DFS ∼ q 2 r2L /τe
so DB exceeds DFS by the geometric factor ε−3/2 .
In the intermediate range ε3/2 < ν < 1 there are two competing processes: (i) by escaping before they have completed a banana orbit, particles are unable to achieve the full radial
72
3 Energy transport in Tokamaks
Figure 3.10: Diffusivity regimes
displacement given by (3.24), an effect that tends to reduce D with increasing ν, (ii) on the
other hand more particles can contribute to the transport process, for in addition to the trapped
particles, passing particles are now more frequently trapped, displaced radially on a banana
orbit and then detrapped, which tends to increase D. The net result of (i) and (ii) is to leave
D roughly constant in a ‘plateau’ as indicated in Fig. 3.10. As ν → 1, (i) and (ii) saturate, the
banana displacements fall to zero and all particles contribute to the transport. Pfirsch–Schlüter
diffusion sets in more strongly and finally takes over.
The mathematical evaluation of poloidal averages for the radial particle flux, Γ = nvD , and
nc
of the radial heat fluxes, Qnc
er , Qir , in the neoclassical range, 0 < ν < 1 was undertaken by
Rosenbluth, Hazeltine, and Hinton (1972). Their expressions, which are a mixture of diffusion
and convection, are as follows:
2
2.44 ε 2 nE
n
n
q 2 rLe
−1.12(Te + Ti ) + 0.43Te + 0.19Ti −
,
3
n
Bθ
ε 2 τe Te
1
Γ
=
2
q 2 rLe
n
1.75 ε 2 pe E
pe
,
1.53(Te + Ti ) − 1.81Te − 0.27Ti +
3
n
Bθ
ε 2 τe Te
m T 12 q 2 r2 p
d
i e
Le i
, q = ε/ S ,
=
−0.48
Ti
=
Qnc
3
ir
me T i
dr
ε 2 τe Ti
(3.80)
1
Qnc
er =
and
(3.81)
(3.82)
where n = ne = ni .
Valid transport equations must remain valid for limiting cases of the driving thermodynamic forces. Consider isothermal conditions, with n < 0 and E = 0. By (3.80) the
particles will diffuse outwards, whereas by (3.81) the thermal energy carried by these particles will be transported inwards, an unphysical mismatch which implies that equations (3.80)
and (3.81) can not be valid transport equations. A similar conclusion follows if we retain E
and set n equal to zero — an inward mass flux is wrongly associated with an outwards energy
flux.
In addition to the above fluxes neoclassical theory predicts a parallel electric current,
j = gσ E + jb ,
(3.83)
3.4 Neoclassical theory and its validity
where
jb = −
ε 12 kB n
Bθ
n
2.44 Te + Ti
+ 0.69Te − 0.42Ti ,
n
73
(3.84)
is known as the ‘bootstrap’ current and several approximate expressions are available for the
factor g, as has been discussed in Section 2.4.4.
The inward particle convection due to E is known as the Ware pinch (Ware 1970). The
coefficient “2.44” repeating in (3.80) and (3.84) implies a relationship between the bootstrap
current and the Ware pinch. This is explained in Section A.19 as a consequence of Onsager’s
reciprocal relations and it establishes that if either of these phenomena is absent, so must
be the other. We shall show shortly that both phenomena are non-existent. The E term in
1
(3.81) can be interpreted as being the product of a velocity (1.17ε 2 E /Bθ ) and an energy
density (1.5pe ) and since the radial flow is ambipolar and the term is independent of particle
mass, one would expect to see a similar term in Qnc
ir . In classical transport theory, diffusion
and convection are distinct, separable physical processes, diffusion being frame-indifferent
and convection frame-dependent (see Section A.15); and unlike the local electric field (E +
v × B), the component E lacks this essential property, rendering the last term in (3.81)
invalid. As explained in Section A.13, the neoclassical mix-up of diffusion and convection is
a consequence of the flawed (drift) kinetic equation adopted for the distribution function f¯ of
the guiding centres.
3.4.2 Testing neoclassical theory
There are two ways of testing neoclassical theory. First one should check that the theory is
logically consistent, regardless of its relevance, and secondly one should consider whether or
not the numerical values make physical sense, i.e. are they about the same order of magnitude
as found by observation? We have noted above that the theory is inconsistent with standard
kinetic theory; next we shall consider numerical values.
We need same ‘standard’ values — for rough estimates only the following are suitable for
the JET tokamak:
Te ∼ 6 keV , ne ∼ 5 ×1019 m−3 , pe ∼ 2.4 ×104 Pa , R/a ∼ 3 , a ∼ 1.2 m , qa ∼ 3 ,
2
Bϕ ∼ 3.5 T , Bθ ∼ Bϕ /9 , E ∼ 4 ×10−2 Vm−1 , rLe
/τe ∼ 4.2 ×10−5 m2 s−1 . (3.85)
With an energy confinement time of ∼ 0.9 s (see Table 4.5) and an effective radius of ∼ 1 m,
the (observational) value of χ⊥ in JET is χ⊥ ∼ 1 m2 s−1 .
Profiles of the number density and temperature that are found to be in fair agreement with
observations at low poloidal beta are:
αn
αt
,
Te = Te0 1 − y
,
y ≡ (r/a)2 ,
(3.86)
ne = ne0 1 − y
where the constants αn and αt fall in the range (0.7, 4). Pfeiffer and Waltz’s (1979) list of
tokamak observations have average values of 1.25 for αn and 2.5 for αt .
Consider (3.81) for Qnc
er . Using the above values we find that at r/a = 0.4 the gradient
terms, i.e. those responsible for thermal diffusion, give
× 2
Qnc
er = 1.42 10 3.06 ne /ne − 2.08 Te /Te ,
74
3 Energy transport in Tokamaks
where to simplify the test we have assumed that Te = Ti . For αn and αt we shall adopt the
averages cited above, then at r/a = 0.4, we get ne /ne ≈ −1 m−1 and Te /Te ≈ −2 m−1 .
−2
, and dividing by 3pe to obtain the theoretical value of
Hence we find Qnc
er = 156 W m
the thermal diffusivity, we find χ⊥ = 2.2 ×10−3 m2 s−1 , which is almost three orders of
magnitude smaller than observed; even the more physically realistic equation (3.30), gives
χ⊥ = 2.67 ×10−2 m2 s−1 , which is still more than an order of magnitude too small. Incidentally, the convective transport term in (3.81) is ∼ 20 times larger than the diffusive term,
whereas observations show that diffusion dominates over convection.
Turning to (3.80) and applying the above figures we find that vD = (7 ×10−3 − 0.1) m s−1 ,
that is the Ware pinch contribution of −0.1 m s−1 is more than an order of magnitude larger
than the outward flow induced by the radial gradients and in a consistent neoclassical model
this would mean that tokamak discharges were impossible, being strangled by the pinch.
Wesson (2004, p. 169), who obtains an even larger value for the pinch, justifies this state
of affairs by the following argument. Starting from the collisionless equation of motion,
m
dv
dv⊥
dv
=m
+m
= Q(E + v × B),
dt
dt
dt
(3.87)
and averaging over complete orbits between bounce points, Wesson takes it as obvious that
for the trapped particles, mdv /dt = 0. Hence from (3.87), E + v⊥ Bθ = 0; thus
1
v⊥ = −E /Bθ , and since the trapped particle fraction is ∼ ε 2 , there is an inwards mass
1
flux, Γ ∼ − ε 2 nE /Bθ . Helander and Sigmar (2002, p. 203) use an equivalent argument to
obtain the same result.
However, while it is obvious that v = 0, the acceleration average, mdv /dt, cannot
vanish because the mirror forces (see (A.55)) at opposite ends of the banana orbits must differ
sufficiently to cancel the particle acceleration due to the unidirectional electric field E , that
is
dv
m
= QE .
dt
Without this constraint the banana orbits would migrate around the torus following the magnetic field lines. Hence from (3.87), Qv × B = Qv⊥ Bθ = 0, i.e. v⊥ = 0, and there
is no Ware pinch. Although this rebuttal lacks the ‘authority’ of the kinetic theory treatment
adopted to prove the existence of the Ware pinch, it embodies the essential features (see final
paragraph of Section A.25).
3.4.3 Bootstrap current
The Tokamak Community places great store on the existence of a current with a density
jb = jb b parallel to B, which seems to be a purely ‘internal’ process, unconnected either
with a driving electric field or with changes in the ambient magnetic field. This so-called
‘bootstrap’ current is believed to be driven by radial density gradients across trapped particle
orbits, and the concept can be traced to the belief that charged particles p and their associated
guiding centers G can move independently even in a strong magnetic field (see Section A.25).
Helander and Sigmar (2002) have stated that the bootstrap current is perhaps the most
important result of neoclassical theory. It has the merit, they claim, of arising spontaneously
in response to radial gradients, and since it is in the same direction as the ohmic current, it
3.4 Neoclassical theory and its validity
75
n1
r
Q
n2
r
Figure 3.11: Metaphoric bootstrap current
helps to confine the plasma by the additional poloidal field that it generates. They speculate
that the improved confinement may cause the pressure gradient to steepen further, leading to
still larger values of jb and that if this process continues, this current could even constitute
most of the total plasma current. We shall discuss the existence of what are termed ‘noninductive’ currents below in Section 5.3.2; such currents have certainly been observed, but
their origin is not the so-called bootstrap phenomenon.
The existence of jb is believed to follow from the following heuristic argument. In
Fig. 3.11 the number densities in the adjacent banana orbits are different and therefore at the
point Q this difference in the counter-streaming electrons (and ions) generates a particle flux:
1
1
c (2ε) 2 (n1 − n2 ) ≈ (2ε) 2 c ∆r dn/dr, where ∆r is the banana width (see Section 3.1.3)
1
and (2ε) 2 is the fraction of trapped particles. Averaging over the particle speeds, we obtain
the net flux,
1 dn
F = c ∆r(2ε) 2
.
(3.88)
dr
While the influence of the trapped particles cannot extend beyond the ends of the banana
orbits, it is maintained that friction transfers momentum from the trapped to the passing particles, and as the transfer rate is different for ions and electrons, the net effect is to generate an
electric current |jb | ∝ F in the passing particles.
A fatal flaw with jb is that because the width ∆r in (3.88) varies in the b-direction, ∇ · jb
is not zero (see Section A.1). More directly, the neoclassical form for jb given in (3.84) yields
the same conclusion, because from (2.49) and (2.69), dBθ /ds = (dBθ /dθ)/Rq is not zero,
i.e. ∂jb /∂s = 0. The Ampère current density, j = ∇ × B/µ0 , does satisfy ∇ · j = 0, but
the sum j + jb does not. Thus jb lacks two of the essential properties of a current density in
MHD; it runs counter to well-established principles in plasma physics, and should be set equal
to zero. Also, the derivation of (3.88) ignores the existence of the mirror forces described in
Section 3.4.2, which are required to prevent the convection of the banana orbits along with the
flux F .
Alternatively, in a convected frame with steady state conditions, (3.25)2 reads c ∆r = 0,
which result is independent of |∆r|, so even on the finest scale there is no net mass flux, and
hence no fluid velocity along the banana orbit. Mass and charge travel together, so there is no
current flowing along the banana orbits. It follows from (3.88) that F = 0, whence
jb = 0 .
(3.89)
76
3 Energy transport in Tokamaks
Then from Onsager’s reciprocal relations (Section A.19), the constraint jb = 0 eliminates the
Ware pinch, which is just as well, otherwise as noted in Section 3.4.2, tokamak discharges
would be impossible. The argument can be inverted: to be consistent with neoclassical
theory, the very existence of tokamak discharges implies that the Ware pinch is negligible and
hence that jb = 0.
3.5 Second-order transport
3.5.1 Electron thermal diffusivity
The experimental evidence is that high thermal conductivity in the electron fluid is the main
cause for the energy losses from tokamaks; conduction in the ion fluid makes a relatively
small contribution (see Section 4.4.2). Radiation is important, but mainly from the plasma
periphery, where energy is delivered from the central regions by a combination of conduction
and convection. Although plasma flow losses are high, they are usually insufficient to convect
energy from tokamaks at more than a small fraction of the observed rates.
Consider a cylindrical magnetoplasma with a strong magnetic field, B = Bz ẑ + Bθ θ̂,
where (r̂, θ̂, ẑ) is the triad of unit vectors. It will be assumed that conditions are independent
of the axial and azimuthal variables, so that ∇T = r̂ T , where the dash denotes the radial
derivative. In steady conditions the radial component of the second-order heat flux follows
from (3.59) and (3.62):
Qr ≡ r̂ · q2 =
5kB p
τ⊥ HT ,
2QB
τ⊥ =
k1 rq
C
(k1 = 1.78) ,
(3.90)
where
◦
H ≡ b×r̂ · e · r̂ .
(3.91)
For the unit vector parallel to B we have
bz = Bz /B, bθ = Bθ /B ,
b = bz ẑ + bθ θ̂
(3.92)
where B is the field strength.
The radial velocity vr of either the ion or electron fluids is suppressed by the strong field
to values much less than either the azimuthal component, vθ , or the axial component vz . With
axial symmetry and uniform conditions along the axis we obtain
∇v = vθ r̂θ̂ −
vθ
θ̂r̂ + vz r̂ẑ ,
r
(3.93)
and therefore
◦
∇v = a r̂θ̂ + θ̂r̂ + c ẑr̂ + r̂ẑ ,
(3.94)
where
a ≡ 12 (vθ − vθ /r),
c ≡ 12 vz .
(3.95)
3.5 Second-order transport
77
Whence
H = abz − cbθ .
(3.96)
From (3.90) applied to the electron gas (Q = −e) the radial heat flux is
Qer = −
5kB pe
τ⊥ He Te
2eB
He = 12 r veθ /r bz − 12 vez
bθ .
(3.97)
The assumption that the electron and ion fluids have roughly the same momentum, i.e. that
v ∝ m−1 , and therefore that the current is carried predominantly by electrons, allows us to
simplify the equation j = ene (vi − ve ) to j = −ene ve , and write
j jθ ϕ
He = − 21 r
bz −
bθ .
rene
ene
We shall assume that, at least in the present application, the viscosity is negligible and that
the plasma is in equilibrium. Let pt = pi + pe denote the total plasma pressure, then
∇pt = j × B = (jθ Bz − jz Bθ )r̂,
so that
jθ = jz Bθ /Bz + pt /Bz .
(3.98)
Eliminating jθ from He , we get
He = − 21 rbz
jz pt ,
ν +
ene
rene Bz
(3.99)
where ν is inversely proportional to the safety factor defined in (1.8):
ν≡
Bθ
1
.
=
rBz
Rq
(3.100)
The equation of a field line is r dθ/Bθ = dz/Bz . Integrating this over 0 < θ ≤ 2π, we
obtain the pitch ℘ = 2πrBz /Bθ of the helical magnetic field. Hence
ν = 2π/℘,
ν = −
2π q
℘
=
−
.
℘2
Rq 2
(3.101)
Thus, if either the pitch decreases with increasing radius or equivalently the safety factor gradient is negative, ν is positive, and if in addition, the pressure is negligible, He will be negative,
in which case it follows from (3.97) that the heat will flow up the temperature gradient and the
magnetoplasma will be unstable. Radial gradient of pitch is called magnetic shear; we shall
apply this property later in Section 4.5.5.
78
3 Energy transport in Tokamaks
3.5.2 Cylindrical coordinates
For our purpose it is sufficiently accurate to evaluate He in cylindrical coordinates — the
toroidal geometry enters through q. Thus, from bz ≈ 1, bθ ≈ S , (3.99), (3.100) and (1.9), viz.
S ≡
Bθ
µ0
=
Bϕ
Bϕ r
we get
−1
He =
2ene
r
0
jϕ r
jϕ (r )r dr ,
S r
+ rne
(3.102)
p
rne Bϕ
.
By (3.102) this can be written
1
µ0 jϕ 2 r
p
∗
∗
∗
He =
{jϕ (r ) − jϕ (r)}r dr − rne
.
2ene Bϕ r2 0
rne Bϕ
(3.103)
If the electron fluid remains steady so that from (3.90) τ⊥ = k1 rq/Ce , then by (3.97) the
thermal conductivity is is
κe =
5k1 kB pe
rqHe
2eBϕ Ce
(k1 = 1.78) .
(3.104)
By (2.48), ignoring Bϕ0
, viz. Bϕ = Bϕ0 /(1 + r cos θ/R), we can write the electron thermal
3
diffusivity, χe = κe /( 2 kB ne ), in the form
5k1 me rqCe
χe =
12e2 ne Bϕ2
2
µ0 jϕ 2
r
r
0
∗
∗
∗
{jϕ (r ) − jϕ (r)}r dr − rne
p
rne
p cos θ
−
R
,
(3.105)
where now Bϕ is the average value of the toroidal field across the minor cross-section.
In the following we shall need the averages:
n̄e ≡
1
a
and
Te ≡
a
0
ne (r) dr,
2
ne a2
a
0
ne ≡
2
a2
ne (r)Te (r)r dr ,
a
0
ne (r)r dr,
µ0 jϕ =
2Bϕ
.
Rqa
(3.106)
(3.107)
The appearance of (3.105) can be simplified by introducing the relative magnitudes
J = jϕ /jϕ ,
P = p/p,
N = ne /ne ,
(3.108)
averaging over the poloidal coordinate θ, changing the independent variable to
y = r2 /a2 ,
(3.109)
3.5 Second-order transport
79
and using a dot to denote d/dy:
5k1 me a Ce q 1
y2
χe =
3µ0 e2 ne R2 qa qa
J
y
y
0
[J(y ) − J(y)] dy
− 12 βp y
P̈ − ṖṄ /N
, (3.110)
where from (1.6),
βp =
2µ0 p
8π 2 a2 p Rqa 2
=
=
βt .
2
Bθa
µ0 Ip2
a
(3.111)
In order to calculate the electron energy confinement time τEe , we shall adopt empirical
profiles for P (y) and N (y) and deduce the profile for J(y). The integral in (3.110) is insensitive to the choice of profile, whereas the term containing βp is not. It could be accurately
included only in a self-consistent theory in which the profile shapes are deduced rather than
assumed. However, this would not be an easy task since the ubiquitous sawtooth instability
described in Section 1.5.4 implies that no genuine equilibrium state exists. In the application
of the theory to be presented in Chapter 4, it will be assumed that the empirical profiles relate
to a quasi-static equilibrium and in Section 6.1 the evolution of these profiles will be studied.
The low density tokamaks that we shall consider first have βp < 12 , so that the second term
within the brackets in (3.110) can be omitted in a first approximation. We shall consider the
quite remarkable effects of the term containing the derivatives in Section 4.2.
3.5.3 Physical mechanism for heat flux
In strong magnetic fields it follows from (3.42), γ0 = 2.5 and (3.58) that the first-order transverse heat flux is
q∧ =
5kB p
b × ∇T .
2QB
(3.112)
A physical picture of the origin of q∧ can be obtained by taking the particles in pairs, each
member of which passes near Qc in opposite directions. In Fig. 3.12 (drawn for the electrons)
Figure 3.12: Origin of the transverse heat flux
80
3 Energy transport in Tokamaks
the guiding centers at A and D are placed a distance 2arL apart, where a is a constant of order
unity. Let â be unit vector along AD and θ the angle between â and ∇T , then there will be
an energy difference of amount
2arL cos θ|∇(cv T )| =
3aCkB
cos θ |∇T |
|Q|B
per unit mass between the two groups of particles crossing AD in opposite directions and with
A and D as their guiding centers. Since C is the average speed normal to AD and nm is the
particle mass per unit volume, it follows that the net energy flux normal to AD is ξ(θ)â × b,
where b is unit vector parallel to B and
ξ(θ) = ± 3aC 2
kB p
nmkB
cos θ|∇T | = ± 6a
cos θ|∇T | ,
|Q|B
QB
(3.113)
the sign choice being + for ions and − for electrons.
The components of energy flux normal and parallel to ∇T are ξ(θ) cos θ and −ξ(θ) sin θ.
To include all the contributions from all the particles gyrating about guiding centers a distance
arL from the point Qc at which the heat flux is being estimated, we average these components
over −π/2 < θ < π/2; the result is zero flux along ∇T and a flux 3akB p|∇T |/(QB) parallel
to b × ∇T , and with a = 56 we arrive at the formula in (3.112). Of course a complete
calculation along these lines would require our averaging over a Maxwellian distribution as
well. Our objective is merely to give an explanation of why it is possible for a temperature
gradient in a magnetic field to generate a heat flux normal to itself.
Incidentally, the argument just given cannot be applied to obtain a mass flux parallel to
b × ∇T because the theory is necessarily developed in the convected frame, i.e. the mass
flux is automatically zero (see Section A.25).
Next consider the second-order term in (3.59), viz.
q2 = −
◦
5kB p
τ2 b × ∇v · ∇T ,
2QB
(3.114)
where τ2 is the collision interval for energy transport. The importance of this flux is that unlike
the transverse term q∧ , of which it is a modification due to fluid strain, it has a component
parallel to the temperature gradient. To illustrate its physical origin, we shall apply it to a
simple flow problem.
Let ŝ be unit vector parallel to ∇T , then ∇T = ŝT , where the dash denotes the spatial
rate of change in a direction along ŝ. Equation (3.114) gives
ŝ · q2 = −κT ,
(3.115)
where
κ=−
5kB p
τ2 H
2QB
◦
H ≡ b × ŝ · ∇v · ŝ .
3.5 Second-order transport
81
Figure 3.13: Second-order heat flux in sheared flow
The simplest example of the theory is provided by a transverse planar flow, sheared in a
direction parallel to the temperature gradient, assumed to be orthogonal to B. We find that
H = 12 v and
ŝ · q2 =
5kB p τ2 v T .
4QB
(3.116)
Notice that if v /Q is positive, the heat flows up the temperature gradient and the flow is
thermally unstable.
The sheared flow just described is illustrated in Fig. 3.13. The x-axis lies in stationary
fluid and the guiding centers D and A are a distance rL cos θ on either side of the axis, moving
parallel to it with speeds ± v rL cos θ, where v = dv/dy. We may treat the whole of the
x-axis as being the fluid particle Qc in which the heat flow parallel to the temperature gradient along OY is to be calculated. All the guiding centers with particle orbits crossing the
x-axis, can be paired in the way indicated in the figure. Of course the orbits shown are actually projections of the three-dimensional particle orbits on to the XY -plane, but this is not
important since we are interested only in heat transport in planes orthogonal to the magnetic
field.
The separation of the paired orbits does not alter the fact that the sum of the energies
transferred across the x-axis at an angle θ to it at points QD and QA , is the flux
ξ(θ) = ±
5kB p
cos θ|∇T |
QB
(3.117)
calculated in (3.113). The plasma at QA is moving along the x-axis towards the origin with the
speed v rL cos θ. This has components v rL cos θ sin θ parallel to AQA and v rL cos2 θ along
the tangent to the particle orbit about A. The latter velocity is swamped in magnitude by
the peculiar velocity (which lies in the same direction), and its contribution to the flux vector
ξ(θ)θ̂ is negligible. On the other hand, the velocity along AQA deflects ξ(θ)θ̂ in a manner
82
3 Energy transport in Tokamaks
similar to that described in Section 3.2.3 for sheared flow in a neutral gas. Of course, in the
latter case the primary or first-order heat flux is parallel to ∇T and the fluid shear creates a
secondary heat flux at right angles to it (see Fig. 3.9). In the magnetoplasma case directions
are interchanged—the primary flux is normal to ∇T and the fluid shear induces a heat flux
parallel to it.
The angle of deflection of the primary heat flux is equal to the ratio of the normal convection speed remaining after removal of the vorticity, namely 12 λv , to the speed of energy
transfer in the primary flux, λ/τ2 . This gave the angle 12 τ2 v shown in Fig. 3.9. We shall use
the same principle for the magnetoplasma. The convection speed normal to the primary flux
ξ(θ)θ̂ is v rL cos θ sin θ and in a remark following equation (3.47) we showed that the speed
of energy transfer of the primary flux across a Larmor orbit is rL /τ2 . With two distinct Larmor
orbits, this rate is doubled. It follows that the required deflection angle is 2τ2 v cos θ sin θ.
Applying this to the primary flux, we conclude that the fluid shear generates a heat flux
2τ2 v cos θ sin θ ξ(θ) along AQA . Resolved along OY , this gives
2τ2 v cos θ sin2 θ ξ(θ) = 2τ2 v cos2 θ sin2 θ(5kB p/QB)|∇T |
by (3.117). It remains to average this flux over − 12 π < θ ≤ 12 π, to obtain the final result,
namely the expression given in (3.116), and thus to complete our physical explanation of the
second-order heat flux vector.
3.5.4 Role of turbulence
Turbulence generated by micro-instabilities is the accepted origin of the very large plasma
mass and electron energy diffusivities. Such turbulence must possess the singular property of
increasing the cross-field thermal transport by several orders of magnitude, and yet leaving
the parallel electrical resistivity unchanged, it being observed (see Table 4) that the electric
current flows around the torus as if Spitzer resistivity (modified only by the particle trapping
factor) were the only impediment.
Assuming the classical form of the transport laws, the effect of the turbulence can be
represented by replacing the classical collision frequencies by ‘anomalous’ values. Since
q⊥ e ∝ (νee )⊥ and j ∝ A(νei ) + B(νee ) , where A and B are independent of the frequencies (see Section A.20), if electron thermal diffusivity is to be explained via an anomalous
∗
∗
∗
)⊥ , we require that (νee
)⊥
(νee )⊥ and (νee
) ∼ (νee ) . These
collision frequency (νee
are impossible constraints to meet with electric field turbulence, since such turbulence would
decrease anisotropy, not enormously increase it.
This problem is considered in some detail in Section A.20, where it is shown that
∗
/νee ∼ 5000 is required to ‘explain’ the observed values of the electron therwhile νee
mal conductivity, such a high value would almost completely suppress the toroidal electric
current. Kikuchi et al. (1990) expressed their suspicions following some observations on
JT-60: “However, it is surprising to observe such a classical behavior of the ‘diffusion-driven’
current when other transport coefficients are anomalous.” That such recognition is very rare
is the surprise.
Liewer (1985) has written an extensive review paper on measurements of micro-turbulence
in tokamaks and comparisons with theories of turbulence and anomalous transport. She concludes that no convincing correlation has been found between the low-frequency, broad-band
3.5 Second-order transport
83
microscopic fluctuations — observed in all tokamaks — and the rate at which they lose mass
and energy. No theoretical model of turbulence transport has passed the test of giving reliable
predictions over a wide range of parameters. And even the basic question as to whether electric
or magnetic fluctuations are the more likely to produce the observed “anomalous” transport is
not settled. A recent survey of the conjectured role of turbulence in tokamak transport is to be
found in Chapter 2 of ITER team (1999).
Because of its dominant role in energy losses, electron thermal conduction has attracted
most attention. From observations of the evolution of a heat pulse, it has been established that
a diffusive-type process is certainly responsible, which rules out convective cells and other
large scale flows. To verify that this process is due to turbulence, it is necessary to measure the
correlation between the electric field and pressure fluctuations. The fluctuation level δn/n is
found to vary from 0.001 or so in the center to 0.1 or more at the plasma edge, which appears
to support the turbulence-transport hypothesis, since χe is known to increase rapidly with
radius (see Fig. 4.3). However, at constant temperature the ratio (δn/n)2 , which is a measure
of the energy in the fluctuations and which we would therefore expect to be correlated with
χe , increases by two orders of magnitude over the radial distance, whereas χe changes by less
than a factor of 10. Also if fluctuations are responsible for the enhanced energy transport,
there should be a correlation between (τE )−1 and (δn/n)2 and while such a correlation has
been observed in several tokamaks (Wesson 2004, p. 201), it is not a sufficient condition, it
being necessary to establish that the fluctuations are the cause and not the consequence of
another transport mechanism2. That the observed fluctuations may be due to the presence of
a (laminar) transport process is supported by the near-disappearance of fluctuations observed
when the confinement time, τE , changes from its L-mode value to its H-mode value (see
Wesson 2004, p. 202).
Radial magnetic field fluctuations have been the basis of several speculative models of
anomalous mass and energy transport. The very large value of χ (in typical tokamaks
χ ∼ 1014 χ⊥ according to classical, first-order theory) means that quite small magnetic fluctuations are sufficient to explain anomalous heat flux perpendicular to the average field. This
model replaces χ⊥ by χ∗⊥ = α2 |δB/B|2 χ , where α depends on the phase shift (Kadomtsev
and Pogutse 1979). Experimental values of δB/B ∼ 10−3 have been observed (McGuire and
Robinson 1980), so a factor of α ∼ 10−2 suffices to give χ∗⊥ ∼ 104 χ⊥ in rough agreement
with many observations. Unfortunately the theory of magnetic turbulence is not equal to the
task of determining α free of phenomenological input.
That turbulence plays a role in transport is likely, but a minor role, probably confined to
the edge plasma, where turbulence is strongest. Perhaps the most telling observation against
the general turbulence hypothesis is the way that the great variety of tokamaks now operating,
closely follow the same empirical laws, despite the observed variability in the fluctuation
levels from machine to machine.
2 A stream flowing under gravity is often very turbulent in appearance, but it is gravity and not turbulence that
drives the water to the coast.
84
3 Energy transport in Tokamaks
3.5.5 Knudsen number constraint
It is sometimes assumed that because the electron mean free path, λe , is much larger than
the distance 2πR around the torus, a fluid description of a tokamak plasma is unlikely to be
valid — the geometric Knudsen number kN = λe /(2πR) is much greater than unity and the
particle motions are best described as being “ballistic”. However, the correct way to test the
validity of fluid theory is to use the basic equations involved in the theory and time-scales are
usually more appropriate than length-scales for this purpose.
For a valid fluid model the constraint given in (A.108), namely kN = τ /T 1, must
be satisfied. Some relevant preliminary remarks have already been made following equation (3.41). In the tokamak application we need to distinguish three cases: (i) trapped particles, (ii) detrapped, steady state particles and (iii) detrapped, unsteady state particles. We
shall first consider the electron gas for which the relevant microscopic time intervals are:
(i) τ⊥ = k1 rq/C, (ii) τg = πRq/C, and (iii) τe (see (2.79). To obtain representative values for the JET tokamak, we shall take the case T̂e = 6 keV and q = 3, when the values of the microscopic time intervals are (i) τ⊥ ∼ 6 ×10−8 s, (ii) τg ∼ 7 ×10−7 s, and
(iii) τe = 2 ×10−4 s.
By (3.97) in the steady state the macroscopic time-scale is T ∼ He = |∇ve |−1 , which
form also follows from (3.59). A value for |ve | can be estimated from
|ve | ≈ |jϕ /ene | = Ip /(πa2 ene ) ≈ 2 ×105 Îp /(a2 n19 )
which for JET at Îp = 4 MA and n19 = 4 is ∼ 2 ×105 m s−1 . A typical gradient would be
obtained by division by 2a ≈ 2 say, then T = 10−5 s. It follows from the microscopic times
given above that in cases (i) and (ii), kN 1, so that in steady conditions the fluid model for
the electrons is safe by a large margin. For the ions, since Ci = (me /mi )1/2 Ce and the fluid
velocities are similarly related, the value of kN is much the same for both species, i.e. so far
as ion heat transfer is concerned, the Knudsen number constraint is also satisfied.
For the unsteady fluctuations described in Section 2.5.4 and illustrated in Fig. 2.10, the
appropriate macroscopic time scale is half a wave length, which from the given description of
the mechanism is 2τ ∗ . The most extreme case is the fishbone oscillation shown in Fig. 2.11,
in which the fine scale oscillations have a wave length of about 2τe . This gives kN ∼ 0.5, so
in even in this case the Knudsen number constraint is satisfied by a small margin. Another
phenomenon in which time dependent diffusivity is central is the collapse time τc for sawtooth
oscillations, which for JET is found to lie in the range 50–200 µs. If we enlist the mechanism
described in Section 2.5.4 to explain this, i.e. set τc = ατe , where α is a constant a little larger
than unity, then kN = 1/α < 1.
References
85
References
Braginskii, S.I. (1965). Transport processes in a plasma. Reviews of plasma physics (ed. M.A.
Leontovich), Vol. 1, p. 205, Consultants Bureau, New York.
Chapman, S. & Cowling, T.G. (1970). The mathematical theory of non-uniform gases. Cambridge University press, Cambridge.
Ferziger, J.H. & Kaper, H.G. (1972). Mathematical theory of transport processes in gases.
North Holland, Amsterdam.
Hazeltine, R.D. & Meiss, J.D. (1992). Plasma confinement. Addison-Wesley Pub. Co.
Helander, P. & Sigmar, D.J. (2002). Collisional transport in magnetized plasmas. Cambridge
University press, Cambridge.
Hinton. F.L. & Hazeltine, R.D. (1976). Rev. Mod. Phys., 48, 239.
ITER team, (1999). Nuclear Fusion, 39(12), Chapter 2.
Kadomtsev, B.B. & Pogutse, O.P. (1979). Sov. J. Plasma Phys., 1, 389, .
Kikuchi, M., Azunii, M., Tanuji, S., Tani, K., & Kubo, H. (1990). Nuclear Fusion, 30(2), 341.
Liewer, P.C. (1985). Nuclear Fusion, 25(5), 543.
McGuire, K.M. & Robinson, D.C. (1979). Nuclear Fusion, 19(4), 505.
Pfeiffer, W. & Waltz, R.E. (1979). Nuclear Fusion, 19, 51.
Pfirsch, D. & Schlüter, A. (1968). Max-Planck-Institut, Report MPI/PA/7/62.
Rosenbluth, M.N., Hazeltine, R.D. & Hinton, F.L. (1972). Phys. Fluids, 15, 116.
Spitzer, L. (1962). Physics of fully ionized gases, 2nd edn. Interscience, New York.
Ware, A.A. (1970). Phys. Rev. Lett., 25, 15.
van Vleck, J.W. (1932). The theory of magnetic and electric susceptibilities, Oxford University
Press.
Wesson, J.A. (2004). Tokamaks, 3rd. edn. Oxford University Press.
Woods, L.C. (1983). J. Fluid Mech., 136, 423–433.
Woods, L.C. (1987). Principles of magnetoplasma dynamics. Oxford University Press, Oxford.
Woods, L.C. (1993). Kinetic theory of gases and magnetoplasmas. Oxford University Press,
Oxford.
Woods, L.C. (2004). Physics of plasmas. Wiley-VCH Verlag GmbH & Co., KGaA, Weinheim.
4 Energy losses from tokamaks
In this chapter second-order transport theory is tested against the observed steady-state behavior of a variety of tokamaks. A complete theory of energy losses employing realistic energy
source terms, the nonlinear transport equations for the heat flux vector, plus allowance for
radiation losses, would require extensive computation; and a more serious difficulty is that at
medium and high poloidal beta the temperature and density distributions perform sawtooth
oscillations, which means that strictly there are no steady state solutions. We shall avoid these
complications by adopting quasi-static, semi-empirical profiles for the distribution of temperature, density and electric current across the minor cross-section, which we shall assume is
circular. In this case a single parameter α serves to specify the shape of the temperature profile
and similarly for the other main variables. By allowing α to be a function of time unsteady
transport will be examined in Chapter 6.
The basic equations for determining energy losses have been derived in Chapters 1 and 3,
and are collected here for convenient reference; they are (1.6), (1.8), (1.26), and (3.110) (the
notation is defined after the List of Contents, except that the subscript “0” will be dropped
from R0 ):
Beta poloidal:
2µ0 p
8π 2 a2 p Rqa 2
=
=
βt .
2
Bθa
µ0 Ip2
a
βp =
Safety factor:
q(r) =
qa =
rBϕ
ε
= ,
RBθ
S
ε≡
2
r
,
R
S ≡
2
5a Bϕ
aBϕ
2πa Bϕ
=
=
RBθa
µ0 Ip R
Îp R
Bθ
µ0
=
Bϕ
Bϕ r
R
0
jϕ (r )r dr ,
Îp in MA .
Electron thermal diffusivity
(related to the thermal conductivity κe by χe = κe /( 32 kB ne )):
5k1 me a Ce q 1 J y
1
2
χe =
y
[J(y ) − J(y)] dy − 2 βp y P̈ − ṖṄ /N
,
3µ0 e2 ne R2 qa qa
y 0
where J = jϕ /jϕ , P = p/p, N = ne /ne , y = r2 /a2 ,
and the averages · · · are defined in the following section.
Theory of Tokamak Transport: New Aspects for Nuclear Fusion Reactor Design. Leslie Colin Woods
Copyright © 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
ISBN: 3-527-40625-5
1
Ce = (2kB Te /me ) 2 ,
88
4 Energy losses from tokamaks
Electron energy confinement time:
τEe ≡ We
We = 32 a2 pe .
r( 52 pe vD + Qer ) r=a ,
Energy losses via the ion fluid are considered in Section 4.4.
4.1 Low poloidal beta
The electron energy confinement time, τEe , proves to be a good approximation to the total
energy confinement time, τE , values of which have been deduced from observations on many
tokamaks. The low poloidal beta case applies to many of the earlier experiments and from the
theory point of view allows us to remove the awkward derivatives from the formula for χe .
4.1.1 Empirical profiles
For most cases a convenient independent variable is y = (r/a)2 , where a is the radius of the
minor cross-section. The distribution of a dependent variable X(y) will be taken to be one of:
α
or X = X0 exp(−αy) ,
(4.1)
X(y) = X0 1 − y
or
X(y) = X0 1 − αy + (α − 1)y 2
(y ≡ (r/a)2 ) ,
(4.2)
where α is an empirical constant chosen to produce profiles similar to those observed in tokamaks in the range 0 ≤ y < k < 1, where k (≈ 0.9) allows for uncertainty in the boundary
region. In Fig. 4.1 the algebraic profiles for α = 0.5, 1, 1.5, 2, and 3 are drawn and the exponential profile for α = 1.5 is shown as a dashed curve. For the profiles generated by (4.2)
see Fig. 4.4.
1
0.9
0.8
0.5
X(r/a)
0.7
1
0.5
α
1.5
0.6
2
algebraic
exponential
3
0.4
0.3
0.2
0.1
0
0
boundary layer
0.2
0.4
Figure 4.1: Standard profiles
0.6
r/a
0.8
1
1.2
4.1 Low poloidal beta
89
Observations are often expressed in terms of the profile averages defined in (3.106) and
(3.107):
1
1
1
1 a
ne dy , n̄e =
ne (r) dr , p =
p dy, jϕ =
jϕ dy , (4.3)
ne =
a 0
0
0
0
and
kB Te = pe /ne .
(4.4)
The measurable line-average density n̄e is usually preferred to the volume-averaged
value ne .
With the algebraic profiles,
ne = ne0 (1 − y)αn ,
Te = Te0 (1 − y)αt ,
jϕ = jϕ0 (1 − y)δ ,
(4.5)
the averages are related to the peak values by
ne0 = (1+αn )ne ,
ne0 = γn̄e ,
2Γ(αn + 1.5)
γ=√
,
π Γ(αn + 1)
and
pe0 = (1 + αn + αt ) pe ,
Te0 = αE Te ,
jϕ0 = (1+δ) jϕ , (4.6)
αE = 1 +
αt .
1 + αn
(4.7)
The ratio α0 ≡ ne /n̄e as a function of αn is shown in Fig. 4.2.
1.1
1
α0
0.9
0.8
0.7
0.6
0.5
0.4
0
1
2
αn
3
4
Figure 4.2: The ratio α0 = ne /n̄e
We also need the safety factor distribution,
−1
,
q = qa y 1 − (1 − y)δ+1
(4.8)
which follows from q = ε/S and the formula for S in (3.102). Expanding (4.8) near y = 0,
we obtain
qa = (1 + δ)q0 .
(4.9)
90
4 Energy losses from tokamaks
For the exponential profiles,
ne = ne0 exp(−αn y) ,
Te = Te0 exp(−αt y) ,
jϕ = jϕ0 exp(−δy) ,
(4.10)
and the averages are:
√
1
n̄e = 12 (π/αn ) 2 erf( αn ) ne0 ,
ne = F (αn )ne0 ,
jϕ = F (δ) jϕ0 ,
(4.11)
and
Te = F (αn + αt )/F (αn ) Te0 ,
(4.12)
pe = F (αn + αt ) pe0 ,
where F (x) ≡ 1 − exp(−x) /x , and erf(x) is the error function. The safety factor distribution is
q = qa y 1 − exp(−δ) / 1 − exp(−δy) .
(4.13)
3/2
From Ohm’s law, jϕ = σ Eϕ and σ ∝ ne τe ∝ Te (see Section A.2), we find that
δ = 3αt /2. A value for αn can be estimated from (3.98), viz. jθ − S jϕ = pt /Bϕ0 , plus the
assumption that the functions involved behave similarly over the tokamak radius; hence
αn = 12 αt = 13 δ ,
αn + αt = δ = qa /q0 − 1 ,
(4.14)
although we shall not always restrict the value of αn according to (4.14). An obvious constraint on αn is αn < δ, which follows from ve ∝ j/ne and the condition that the electron
fluid velocity be finite at the limiter, r = a. Pfeiffer and Waltz (1979) list the profile parameters for 118 observations with average values of αt = 2.5 and αn = 1.25, but in individual
cases the ratio αt /αn is often quite different from 2. When experimental details are sparse,
we shall sometimes adopt (4.14), but more often we shall specify αn and δ independently.
4.1.2 Radial distribution of thermal diffusivity
If βp 1, the electron thermal diffusivity reduces to (5k1 = 8.92),
1
χe =
1
8.92(2me ) 2 a (kB Te ) 2 q J
3µ0 e2
R2 qa ne qa y 12
y
0
[J(y ) − J(y)] dy .
(4.15)
Adopting the exponential distributions of (4.10) and writing αt = 2δ/3, but leaving αn independent, we obtain
χe = K1 H G1 (αn , δ) φ(y, δ),
where
(4.16)
1
8.92(2me kB ) 2
≈ 4.62 ×1017 ,
K1 ≡
3µ0 e2
G1 =
F (αn )
F (αn + 2δ/3)
1/2
1
2
π 12
αn
1
a Te 2
H≡ 2
,
R qa n̄e
√
erf( αn ) exp (αn − δ/3)y ,
(4.17)
(4.18)
4.1 Low poloidal beta
91
0.35
φ
0.3
3.5
δ
2.5
0.25
χ
x
x
x
x
3.0
2.0
e
10
x x
x
0.15
x
x
x
x
x
x
x
x
x
x
0.2
x
x
x
x
x
x
x
x
Experiment
x
x
x
0.1
x
0.05
xx
x x
x x
x x x x
0
0
0.2
x
x
x
x
x
0.4
0.6
0.8
1
r/a
Figure 4.3: Electron thermal diffusivity
and
1
φ(y, δ) =
δy 2 exp(−δy) [1 − (1 + δy) exp(−δy)]
.
(1 − exp(−δ))
1 − exp(−δy)
(4.19)
1
This function is plotted in Fig. 4.3 against y 2 = r/a for several values of δ.
For an estimate of χe we adopt the typical values for JET given in (3.85) and obtain
H ≈ 1.54 ×10−17 , then with αn = 1 and δ = 3, we get G1 = 1.32; finally, with φ = 0.14 (cf.
Fig. 4.3), we find from (4.16) that χe ∼ 1.3 m2 s−1 , which is close to the value of 1 m2 s−1
estimated from JET observations cited in Section 3.4.2. Both values are rough estimates, but
their agreement is encouraging initial support for the theory.
Ignoring the small exponential term in (4.18), we note that the curves in Fig. 4.3 are proportional to the thermal diffusivity. The crosses on the graph represent some experimental
values from the DIII-D tokamak reported by Wade et al. (1995); their measurements of the
thermal diffusivity χe are plotted as χe /10 with the same numerical scale as for φ. The tokamak was operating in what is known as the H-mode, which will be described in Section 4.3.1.
The theoretical curve with δ = 3 is closest to the observations — the steep increase with
radius followed by a sharp decline — is well modeled, although there is appreciable error in
the neighborhood of the minor axis, r = 0.
4.1.3 Electron energy confinement time
The formula for the electron energy confinement time τEe can be written
τEe = 32 a2 pe r 52 pe vD + Qer r=a .
(4.20)
However, this equation is unsuitable for evaluating τEe directly from algebraic profiles, the
reason for which is clear from Fig. 4.1, for if α > 1, this choice reduces Qer to zero at r = a.
While the exponential profile does not have this failing, it is evident from Fig. 4.3 that it makes
92
4 Energy losses from tokamaks
τEe too dependent on the value chosen for δ, which is merely a gross measure of overall shape;
this difficulty can be overcome as follows.
From (A.25) and (A.28) in steady conditions the electron energy equation is
(4.21)
∇ · e he ve + qe = ve · ∇pe + j · E + v × B − ∇pe /ene .
To determine the rate at which electron energy is transported radially, we write ve = vD r̂,
r̂ · j = 0 (ambipolarity, see Section 5.1.4), ∇ = r̂ d/dr, and j · ve × B = −vD r̂ · j × B =
−vD r̂ · ∇(pi + pe ) . Then (4.21) yields
1 ∂ 5
r 2 pe vD + Qer = G ,
G ≡ jϕ Eϕ − vD pi ,
(4.22)
r ∂r
where denotes the radial derivative and we have ignored the collisional transfer of heat between the electrons and ions.
Let
1
1 a y
dr
G(y ) dy
G(y) dy ,
(4.23)
A ≡
a 0
0
0
then it follows from (4.22) that
1
5
5
a
5
2
=
p
v
+
Q
er
2 e D
2 pe vD + Qer dy = a A r 2 pe vD + Qer 0 ,
0
(4.24)
which allows us to rewrite (4.20) as
1.5A pe a
.
τEe = 5
2 pe vD + Qer
(4.25)
It is evident that A is not particularly sensitive to the distribution of energy sources and that
τEe is now free from close dependence on the conditions at the limiter.
We shall assume that the ohmic heating term dominates G and that Eϕ is constant across
the minor radius; in this case (4.5)3 and (4.23) give
√
πΓ(δ + 2)
≈ 0.621 + 0.037 δ − 3.75
(2.5 < δ < 4.5) ,
(4.26)
A = 1 −
2Γ(δ + 2.5)
where the approximation is within 3% of the correct value over the given range. The value
3.75 is close to the average value for δ over many small tokamaks, whereas for JET δ = 2.5
is a good choice (see Tables 4.4 and 4.5).
Using the profiles defined in (4.5) to (4.7) to evaluate (4.15) we get
1
χe =
1
8.92 (2me ) 2 akB Te 2 12
αE φ2 (y, δ) ,
3
µ0 e2 ne R2 qa
where
1
2
4δ/3
φ2 (y, δ) = (1 + δ)y (1 − y)
1 − (1 + δy)(1 − y)δ
1 − (1 − y)δ+1
(4.27)
.
4.1 Low poloidal beta
93
Table 4.1: Values of the shape factor F
αn \δ
0.5
1.0
1.5
2.0
δ/3
1.0
0.56
0.55
-
1.5
0.48
0.50
0.50
0.48
2.0
0.43
0.45
0.47
0.47
0.44
2.5
0.38
0.41
0.43
0.44
0.40
3.0
0.34
0.38
0.40
0.41
0.38
3.5
0.31
0.35
0.37
0.39
0.36
4.0
0.28
0.32
0.35
0.36
0.34
4.5
0.25
0.29
0.32
0.34
0.32
The radial heat flux is
1
Qer = − 23 kB ne χe Te = 2a−1 ne αE δkB Te y 2 (1 − y)αt −1 χe ,
(4.28)
from which it follows that
1
Qer = 5.95
(2me ) 2 kB Te 3/2 3/2
αE Φ (δ) ,
µ0 e2
R2 qa
where
Φ (δ) = δ(1 + δ)
1
0
y(1 − y)2δ−1
1 − (1 + δy)(1 − y)δ
1 − (1 − y)δ+1
(4.29)
dy .
The value of Φ is very weakly dependent on δ, varying from 0.1117 at δ = 2 to 0.1032 at
δ = 5. We shall adopt Φ = 0.105 with negligible error incurred over the usual range of δ.
Omitting the convection term from (4.25) and using the relation
γ
n̄e kB Te ,
pe =
(4.30)
(1 + αn )
which follows from (4.6) and (4.7), we obtain the expression for the electron energy confinement time:
µ0 e2 n̄e aR2 qa
τEe = F
(4.31)
1
1 ,
(2me ) 2 kB Te 2
where F (δ, αn ) is the profile shape factor defined by
F (δ, αn ) =
2.40Aγ
3/2
αE (1 + αn )
,
(4.32)
values of which are given in Table 4.1. As to be expected, the flatter profiles are a little more
efficient at confining the electron energy. The last row of the Table gives the value of F , when
following (4.14) we set αn = δ/3. The variation in F is relatively small and in view of other
approximations in the theory, little error is incurred by adopting the value at δ = 3 , αn = 1,
which is F = 0.38. There are no entries for αn > δ for the reason given in the final paragraph
of Section 4.1.1.
Let Tˆe denote the temperature in keV, then with F = 0.38, (4.31) becomes
τEe = 7.18 ×10−22
n̄e aR2 qa
T̂e 1
2
T̂e in keV ,
which is our final expression for τEe for the low βp case.
(4.33)
94
4 Energy losses from tokamaks
Table 4.2: Typical JET discharge parameters at low βp
Îp
(MA)
2.3
3.0
2.3
2.1
2.1
1.7
2.7
2.4
3.1
3.0
3.2
2.1
Bϕ
(T)
2.5
2.5
2.5
2.5
2.5
2.6
2.1
2.6
2.6
3.4
3.4
2.5
10−19 n̄e
(m−3 )
1.62
1.80
1.41
2.30
2.11
0.69
2.19
1.81
1.93
2.84
3.01
2.12
A
(m2 )
4.3
4.6
4.5
4.4
4.4
7.0
6.4
6.2
6.4
5.0
5.0
4.8
T̂e (keV)
1.78
2.00
2.80
2.00
1.94
3.35
1.64
2.30
2.50
2.46
2.86
2.33
Z∗
5.1
5.0
7.2
5.1
3.2
5.8
4.2
4.0
4.2
4.1
6.0
2.5
τE (exp)
(s)
0.19
0.18
0.22
0.33
0.28
0.21
0.28
0.31
0.25
0.40
0.62
0.42
τEe (th)
(s)
0.23
0.21
0.17
0.35
0.33
0.21
0.42
0.39
0.33
0.44
0.41
0.34
4.1.4 Comparison of theory with observation
First observe that the functional dependence in (4.33) is exactly that specified by the row
marked ‘ideal’ in Table 1.2, namely
1
τEe ∝ n̄e aR2 qa /Te 2 ,
which was compounded from several empirical laws. The strong indication of rational values for the powers over a wide range of machine sizes and operating conditions implies the
existence of an underlying laminar theory, rather than one based on turbulence, which varies
considerably between machines.
Another formula for τEe follows from
qa =
5a2 Bϕ
Îp R
Îp in MA ,
(4.34)
and (4.33), namely:
τEe = 3.4 ×10−21
n̄e a3 RBϕ
1
2
T̂e Îp
T̂e in keV, Îp in MA .
(4.35)
In Table 4.2 we list some early JET measurements (Rebut et al. 1985) taken at values
of βp ∼ 0.1. The JET torus cross-section is elongated in the vertical direction, so for an
equivalent radius we have taken (A/π)1/2 , where A is the area of the plasma cross-section.
The major radius of the torus is 3m. In the last column are the confinement times calculated
from (4.35); they are as close as could be expected to the experimental values given in the
penultimate column. The average of the theory column is less than 2% different from the
average of the experiment column, so the theory is well supported by these JET observations.
4.2 High poloidal beta
95
4.2 High poloidal beta
4.2.1 Oscillatory temperature profiles
The low βp regime is stable and reproducible, but as βp is increased, the derivative terms in
8.92 me a Ce q 1 J y
1
2
χe =
y
[J(y
)
−
J(y)]
dy
−
β
y
P̈
−
Ṗ
Ṅ
/N
, (4.36)
2 p
3µ0 e2 ne R2 qa qa
y 0
become more important, and this stability is lost. A finite amplitude oscillation with a sawtooth appearance occurs in several of the dependent variables and is especially marked in the
electron temperature. As we shall see shortly, this requires high βp and low values of qa .
For simplicity it will be assumed that the time-scale τd for the relaxation of the toroidal
current (due to inductance, resistivity, etc.) is small enough to allow the adoption of a quasistatic model during the oscillation. For example in JET values of τd as low as 25 ms have been
obtained (Schueller et al. 1985), i.e. a 10% current change, if otherwise unimpeded, would
occur in less than 1 ms.
As βp increases from low to moderate values (βp ≥ 0.2), the typical electron temperature
profile changes from resembling the steep empirical profile specified in (4.5) to a broader
profile that cannot be reproduced simply by reducing the index αt . At high βp more realistic
empirical profiles belong to the family
Te = Te0 1 − αy − (1 − α)y 2 ,
Te0 = 6Te /(5 − α) ,
(4.37)
with the parameter α lying the range (−1, 3).
Although the number density also experiences sawtooth oscillations, except possibly near
the plasma boundary, its contribution to energy losses is small compared with that of the
temperature oscillations. For simplicity we shall therefore adopt the parabolic distribution
ne = ne0 1 − y .
(4.38)
For the present we shall omit ion thermal diffusivity and show later how to adjust the theory to
allow for it. The model that we are introducing here, based on a single parameter α, is a very
simplified representation of rather complex phenomenon. As we shall show, α is a function
of time with the profiles shown in Fig. 4.4 oscillating between about α = −1 and α = 3. The
profiles all have same average density-averaged temperature, ranging from the hollow profile
(sometimes observed) at α = −1 to the steep profile at α ≥ 2. Steeper profiles are sometimes
required, but with (4.37) these develop small negative regions near the limiter.
From (4.37) and (4.38) it follows that:
ne0
= 2,
ne pe0
12
=
,
pe 5−α
Te0
6
=
.
Te 5−α
By (4.36) χe is proportional to
J y
ψ=
[J(y ) − J(y)] dy + 12 βp y ṖṄ/N − P̈ .
y 0
(4.39)
(4.40)
96
4 Energy losses from tokamaks
From (4.38) and (4.39) the second term becomes
1
2 βp y
18(1 − α)
βp y(1 − y) .
ṖṄ/N − P̈ =
5−α
(4.41)
At moderate values of βp , α ∼ 2 is typical, so the effect of the second term in ψ is to reduce
χe . Hence if n̄e is increased at constant temperature and current, χe will decrease a little more
rapidly than 1/n̄e making τEe proportional to n̄es , where s > 1. This effect is slight, but it can
be discerned in some τEe (n̄e ) curves presented by Goldston (1984). It has also been observed
in JET, an index of s = 1.15 ± 0.1 being obtained at moderate densities (Cordey 1985).
The basic (quasi-static) profiles at constant βp are:
Te =
6Te 1 − αy − (1 − α)y 2 ,
5−α
ne = 32 n̄e (1 − y) ,
(4.42)
3
and from jϕ ∝ Te2
jϕ = jϕ0 1 − αy − (1 − α)y 2
1
4,
32
.
9
8 Te (4.43)
At y =
Te =
and so is independent of α; this occurs at the radius
r = rs = 0.5a, which is known as the “inversion” radius. With the oscillatory instability known as a ‘sawtooth’, about 50% of the thermal energy in the torus oscillates back and
forth across r = rs as indicated in Fig. 4.4; it does not (normally) continue to flow steadily
outwards because it is partially blocked by the rapid fall in thermal diffusivity at the torus
surface (see Fig. 4.3). The total energy flux consists of a large steady component upon which
is superimposed the sawtooth oscillatory flux; it is the steady component with which we shall
be concerned in the rest of this chapter. The model illustrated in Fig. 4.4 is a good choice for
JET in which rs ≈ 0.5a, but in small tokamaks rs is much closer to the magnetic axis.
Figure 4.4: Temperature profiles with the same average thermal energy
4.2 High poloidal beta
97
2.2
2
q
q
1.8
0
1.6
α = 2.0
1.5
1.4
1.0
0.5
1.2
0.0 −0.5
−1.0
1
0.8
0
0.2
0.4
0.6
0.8
1
( r / a )2
Figure 4.5: q/q0 as a function of y and α
4.2.2 Thermal diffusivity
From (1.13), i.e. qa /q0 = jϕ0 /jϕ , and (4.43) we find
q
=y
q0
y
0
2
1 − αy0 − (1 − α)y0
3
2
−1
dy0
,
(4.44)
which can be integrated algebraically as shown in Fig. 4.5, but the curves are accurately represented by the more convenient quadratic expression
q
≈ 1 + â(α)y + b̂(α)y 2 ,
q0
(4.45)
where the values of â(α) and b̂(α) given in Table 4.3 are less than 2% in error over the whole
range of α.
Using the above expressions we find that
1
y2
y
0
J(y ) − J(y) dy ≈
qa 1.5α − â + ĉy + q0 /qa − 1.5α + â − ĉ y 2 ,
q0
where values of ĉ(α) are given in Table 4.3. At y = 1, q = qa = (1 + â + b̂)q0 .
Notice from Table 4.3 that when qa /q0 < 2.5 , (1−α) is positive and the βp -term in (4.41)
increases χe . As q0 tends to be anchored near 0.7 by the snake instability to be described in
Section 4.6.1, the condition for the βp -term to oppose energy confinement is qa ≤ 1.75. For
such losses to be significant, we therefore need high βp and low qa (cf. (1.7)). In fact we shall
find in Section 6.2.3 that thermal stability sets a lower bound on qa of ∼ 2.3.
98
4 Energy losses from tokamaks
Substituting the expressions given above into (4.36), we obtain
1
1
1 6kB Te 2
5k1 (2me ) 2 a
1 − αy − (1 − α)y 2 2 1 + ây + b̂y 2 y 2
2
2
3 µ0 e ne R qa
5−α
3
qa 1 − αy − (1 − α)y 2 2 1.5α − â + ĉy + q0 /qa − 1.5α + â − ĉ y 2
×
q0
q0 18(1 − α)
(1 − y) .
(4.46)
+ βp
qa 5 − α
χe =
The radial heat flux is
Qer = − 32 kB ne χe Te = 3ne
1
6kB Te y 2
α + 2(1 − α)y χe ,
5−α a
(4.47)
and hence Qer has the volume average
Qer =
1
0
Qer dy =
1
3
(2me ) 2 kB Te 2
Φ1 (α) + βp Φ2 (α) ,
2
2
µ0 e
R qa
(4.48)
where values for Φ1 (α) and Φ2 (α) are given in Table 4.3.
4.2.3 Electron energy confinement time
Ignoring the convection term in (4.25), we have
τEe = 1.5A pe a/Qer ,
(4.49)
where pe = ne kB Te = 34 n̄e kB Te . From (1.13), (4.23), (4.45) and the assumption that
ohmic heating is dominant we get
y
1
1 a
dr
jϕ (y ) dy
jϕ dy ,
A =
a 0
0
0
i.e.
A =
qa
2q0
1
0
1
y 2 dy
1 + ây + b̂y 2
.
(4.50)
Values for A are given in Table 4.3.
The electron energy confinement time follows from (4.48) and (4.49):
τEe = F
µ0 e2 n̄e aR2 qa
1
1 ,
(2me ) 2 kB Te 2
where the profile shape factor is
F = 1.125A / Φ1 (α) + βp Φ2 (α) .
(4.51)
(4.52)
4.3 The L- and H-modes
99
Table 4.3: Functions of α required in the theory
α
â
b̂
ĉ
qa /q0
A
Φ1
Φ2
−1.0
−0.69
0.93
2.64
1.24
0.43
0.94
3.88
−0.5
−0.44
0.87
1.70
1.43
0.45
0.97
3.12
0
−0.15
0.85
0.80
1.70
0.46
1.04
2.21
0.5
0.18
0.87
−0.03
2.05
0.48
1.14
1.16
1.0
0.58
0.92
−0.73
2.50
0.51
1.27
0.00
1.5
1.01
1.13
−1.40
3.14
0.55
1.49
−1.24
2.0
1.55
1.45
−1.86
4.00
0.59
1.79
−2.49
As the ion temperature was omitted from the above theory in (4.52) βp represents
2µ0 pe /B 2 . Had Ti been included and its profile assumed to be similar to that for Te ,
βp would have its usual value, namely (see (1.6))
(4.53)
βp = 8π 2 a2 ne kB Te + ni kB Ti / µ0 Ip2 .
At high poloidal beta F is a sensitive function of the profile steepness, i.e. of the parameter
α, so it is essential to find an accurate method of assigning values to α. In Section 1.5.2 we
gave an account of the L-mode and H-mode confinement regimes and as τEe in the H-mode is
about twice its value in the L-mode, the parameter α must depend on the mode of operation.
4.3 The L- and H-modes
4.3.1 Role of boundary conditions
As we shall explain in Section 6.1.3, under normal operating conditions α is a periodic function of time following a sawtooth pattern and as temperature and number density are functions
of the profile shape, these variables also follow the sawtooth cycle. The origin of this remarkable instability will be discussed in Section 6.1. During the ramp phase of the oscillation
the steepness parameter α increases steadily from its minimum α0 to its maximum αm (see
Fig. 4.4) at which stage there is usually a sudden collapse. To sufficient accuracy for present
purposes we may assume that it is the median value of α, viz.
(4.54)
ᾱ = 12 α0 + αm ,
that determines the average heat loss due to electron thermal diffusivity.
It is important to distinguish the two components of heat flux in a tokamak — there is the
steady component of the heat flux vector, which is always directed outwards and the oscillatory component, which is inwards during the ramp phase of the sawtooth and then suddenly
outwards during the collapse phase of the sawtooth. The following account of boundary conditions applies to the oscillatory component.
The minimum value, α0 , can be found as follows. Referring to Fig. 4.4 we note that the
family of curves cross over or are “inverted” at the radius r = rs = 0.5a, where y = 14 . As the
profiles fall from their peak at α = αm initially the outwards radial heat flux in 0 < r < rs
100
4 Energy losses from tokamaks
increases. Then it decreases, while the outwards heat flux in rs < r < a increases. This
continues until α falls to a value such that the average heat flux in 0 < r < rs is zero,
while that in rs < r < a remains high. A further fall occurs until the temperature gradient
at the inversion point vanishes, which at least momentarily, requires an inwards heat flux in
0 < r < rs , and an outwards flux in rs < r < a. But this is possible only if there is a strong
heat source at the inversion radius, which is not so. Thus the profile α = α0 occurs at the
trough of the sawtooth oscillation.
It follows from (4.42) that
1
dTe
2y 2
=
−α − 2y(1 − α) ,
dr
a
which vanishes at y = 14 if α = −1. Therefore α0 = −1 is the minimum value of α. The
temperature profile for the case α = −1 is shown dotted in Fig. 4.4.
The maximum value αm is determined by conditions in the boundary region B, say
rE < r < a, where rE is determined by the boundary layer equations. The detailed theory of plasma-surface interactions is a complex, specialized subject beyond this text (e.g. see
Wesson (2004), Chapter 9, and ITER team (1999), p. 2391). The outer region of B consists
of plasma moving inwards with a fluid velocity V under a pinch action (see Section 5.2.1),
and if r = a is a limiter there will also be neutral impurity ions emitted from the surface,
often penetrating the plasma well beyond B before being ionized. Thermal speeds of the ions
and electrons will allow many particles to impinge on the wall and give up their energy to
the wall molecules. For present purposes the only properties we need for B is that it is either
a perfect thermal conductor or it is a perfect heat reservoir, or perhaps some combination of
these ideals.
Within B the electric current is relatively small, so the βp term in (4.36) dominates the
radial heat flux. From (4.41) and (4.47) we find that Qer ∝ βp (1 − y)(1 − α)(2 − α). If B
is a perfect thermal conductor, then the heat always flows into it, never back into the tokamak
plasma; hence α ≤ 1, the region α ≥ 2 being unattainable. We conclude that in this case
αm = 1 and therefore (4.54) gives ᾱ = 0. Because of the energy losses through B, this case
is called the low or ‘L-mode’. Thus the L-mode results from a close thermal contact between
the plasma and its boundary.
Now suppose that there is either a divertor at the boundary or that the pinch velocity
V is large enough to widen B. If the plasma is initially in an L-mode of operation and an
overshoot at the top of the sawtooth oscillation carries the boundary condition beyond α = 1
into the region of reversed heat flux at the boundary, then it is possible for an ‘H-mode’ to be
switched on. In this mode the thermal contact between the plasma and the boundary is weak
on the short time-scale of a sawtooth oscillation, which allows the boundary region to act as
a reservoir, receiving heat at low α and returning it at high α, a process described as being an
edge transport ‘barrier’. The interval 1 ≤ α ≤ 2 of inwards heat flux at the boundary is now
accessible and the maximum value of α is determined by the ideal reservoir condition, viz.
that the oscillatory component of the heat flux is received and returned completely each cycle.
It follows from Qer ∝ βp (1 − y)(1 − α)(2 − α) that the average of Qer is zero if ᾱ = 2.
Because of its importance in tokamak physics, we shall return to the H-mode phenomenon in
Section 6.4.1. In Section 4.5.5 an account of what are termed internal transport barriers will
4.3 The L- and H-modes
101
be given; these occur well within the plasma, sometimes at about half the minor radius and
have in common with edge transport barriers, that the heat tends to flow inwards, although for
a different reason.
4.3.2 Energy confinement in the L- and H-modes
We now have ᾱ = 0 in the L-mode and ᾱ = 1 in the H-mode. Thus from Table 4.3,
L-mode: Φ1 + βp Φ2 = 1.04 + 2.21βp ;
H-mode: Φ1 + βp Φ2 = 1.27 .
(4.55)
From (4.52) we find that for these average values the L- and H-mode coefficients are:
kL
FL =
kL = 0.50 , a0 = 2.13 ;
FH = kH (kH = 0.45) .
(4.56)
1 + a 0 βp
Notice that with βp ∼ 0.6 the confinement time in the H-mode is about twice that in the Lmode. Also at small βp it follows from Table 4.1 that (4.56) gives values in agreement with
(4.31) at αn = 1, δ = 1.5, a discrepancy resulting from the rather special temperature profile
that was adopted to model the existence of a temperature inversion radius. Observations show
that small tokamaks have inversion radii much smaller than a/2. Also these equations depend
on the boundaries being either perfect thermal conductors (FL ) or perfect insulators (FH ) on
the sawtooth time scale. Actual tokamak conditions may lie in a range of values between these
limits, so we can generalize (4.51) to
τEe = (1 − ξ)FL + ξFH
µ0 e2 n̄e aR2 qa
1
1 ,
(2me ) 2 kB Te 2
(4.57)
where 0 ≤ ξ ≤ 1, so that ξ = 0 is a pure L-mode and ξ = 1 is a pure H-mode.
As βp ∝ n̄e , it follows from (4.57) that in the pure L-mode τEe is independent of n̄e if
βp
1/a0 = 0.47, which is the case described in Section 1.5.1 (see Fig. 1.8).
Let
(4.58)
R ≡ n̄e Te + n̄i Ti n̄e Te ,
then we can express (4.53) in the form
βp = 6π 2 a2 n̄e RkB Te µ0 Ip2 .
(4.59)
For the case of relatively large βp it follows from the L-mode form of (4.57) that
τEe =
Ip2
µ 2 e2 R2 qa
3.53 ×10−3
0 1
3 .
R 1 + 0.47/βp (2me ) 2 a kB Te 2
(4.60)
The H-mode is subject to an edge instability known as an “Edge Localized Mode” (ELM),
in which the thermal barrier at the boundary is broken and a discharge of particles and thermal
energy occurs. In a sustained H-mode discharge this instability is followed by a recovery until
the edge pressure gradient increases enough to trigger another discharge; thus the H-mode
is continuously switched on and off which reduces the H-mode energy confinement time by
about 15%. Some H-mode discharges are spared this phenomenon and are described as being
“ELM free”. The cyclic process in which the ELM instability repeats regularly is described
as being an ELMy H-mode. We shall return to this topic in Section 6.4.2.
102
4 Energy losses from tokamaks
4.4 Thermal transport in the ion fluid
4.4.1 Thermal diffusivity
The collisional transfer of thermal energy between the ions and electrons follows the classical
law (e.g. see Woods 2004),
ε = (mi /2me)τe ≈ 0.117 T̂e2 /(n19 Zeff ) ,
τei
3
(4.61)
where τe is the electron collision interval defined in (A.16). The approximate form applies
to deuterium with ln Λ = 17. Let ϕe denote the rate at which energy is transferred into the
electron fluid, then it can be shown that
ϕe = 32 kB ne
3kB ne dTe
=
Ti − Te = −ϕi ,
ε
dt
2τei
which can be written
ϕe = 2.05 ×104 n219 Zeff T̂e−3/2 T̂i − T̂e .
(4.62)
With neutral beam heating (see Section 1.4.2) it is possible that the ions will become hotter
ε ≥ τE . With the
than the electrons and for the condition Ti > Te to be sustained we need τei
ignition condition (1.1), (4.61) and Zeff = 2, this requires that T̂i ≈ T̂e ≥ 30 keV, which
from the ignition curve of Fig. 1.2 is a temperature a little larger than necessary for a fusion
reactor. It is therefore a reasonable assumption that except in disruptions, the temperatures are
approximately equal, although we shall usually maintain the distinction.
The ions can lose their energy by two routes: (a) if Ti > Te , their energy passes to the
electrons and (b) they can be cooled by direct thermal diffusion. However, if their thermal
diffusivity χi is much smaller than χe , route (a) will dominate and in effect the ions will
“borrow” the electron diffusivity. With the ‘standard’ JET values given in Section 3.4.2 and
ε ∼ 172 ms. Ion thermal diffusive losses are difficult to measure and observations
Zeff = 2, τei
of them are scarce compared with those of electron thermal losses.
The second-order transport theory of Section 3.5 also holds for the ions, but to apply it
expressions for the ion fluid velocity vi are required. With the electrons we were able to
exploit the approximation ve ≈ −j/ene , which allowed practical expressions for the electron
thermal diffusivity to be deduced.
Applying (3.90) to the ion fluid we obtain
Qir =
5kB pi k1 rq
Hi Ti
2ZeB Ci
Hi = 12 r viθ /r bz − viϕ bθ ,
(4.63)
and comparing this with the corresponding equation for the electrons, (3.97), we find
χi = −
T 12 m 12 H χ
T i C e Hi χ e
i
i
i e
=−
.
T e C i He Z
Te
m e He Z
(4.64)
4.4 Thermal transport in the ion fluid
103
4.4.2 Ambipolar constraint
To determine χi from (4.64) we need a theory for the ratio Hi /He , or equivalently for the
velocity ratios viϕ /veϕ and viθ /veθ . In Section 5.1.4 it will be shown that to satisfy the
ambipolar condition, ver = vir , the viscous forces acting on the electron and ion fluids must
satisfy
2 2
pi τ1i /ωci viϕ = pe τ1e /ωce veϕ ,
(4.65)
where τ1e and τ1i are the momentum collision intervals for the electron and ion fluids; from
this principle the relations in (5.25) and (5.28) are derived for a fully ionized hydrogen plasma;
these are
viϕ = −G0 veϕ ,
where
viθ = −G0 veθ ,
5
G0 ≡ 2.61 ×10−3 Te /Ti 4 .
(4.66)
(4.67)
It follows from (4.66) and the definitions of Hi and He that Hi = −G0 He , so that (4.64)
yields
3 D ≡ 0.112 Te /Ti 4 .
χi = D χe
(4.68)
Hence the thermal properties of the ion fluid are much the same as those of the electron fluid,
except that the diffusivity is only about one-ninth of the electron value, which implies that the
ions lose most of their thermal energy via the electrons.
When convection and the collisional transfer of energy from the ions to the electrons can
be neglected, it follows from (4.49) and the corresponding equation for the ions that the ion
and electron confinement times are inversely proportional to the diffusivities, i.e.
(4.69)
τE i = τEe D .
The relation between χi and χe is rather different from the corresponding neoclassical
relation; for hydrogen (3.81) and (3.82) give
χNC
i = 17.4
Thus
T 12
e
Ti
χNC
e .
T 14 χ
χi
e
e
×10−3
=
6.44
.
NC
NC
T
χ
χi
i
e
(4.70)
2
3
The ratio χe /χNC
e is typically in the range 10 to 10 , so in a hydrogen plasma according to
NC
the theory leading to (4.68) we would expect χi /χi to fall in the range 0.7 to 7.
Some measurements give magnitudes for χi similar to neoclassical values, especially if
the ion collisionality is in the plateau regime (see Fig. 3.10). The ratio R̂ = χEXP
/χNC
i
i lies
between 1 and 3 in many cases (Hugill 1983), but in the tokamak DOUBLET III it was found
104
4 Energy losses from tokamaks
that 2 < R̂ < 7 (Ejima et al. 1982). In TFR-400 and in JET values of R̂ near 10 have been
found, and also that R̂ increases with plasma current (Brusati and Cordey 1985).
Let τE denote the total energy confinement time, then it is related to the partial times by
1
1
1
ne Te ni Ti =
+
,
τE
ne Te + ni Ti τEe
ne Te + ni Ti τE i
which by (4.58)and (4.69) can be written
R
H≡
.
τE = HτEe
1 + D(R − 1)
(4.71)
A first approximation is to assume that the two fluids have similar density and temperature
profiles, that ion diffusivity is negligible and that the ions transfer their energy into the electron
fluid, in which case H = 1 and Qir = Qer . If this transfer is neglected then H ≈ 2.
4.5 Comparison of experiment and theory
4.5.1 Neutral beam injection
By (1.60) the energy replacement time for the total input power is
τE∗ = 3π 2 Ra2 ne kB Te + ni kB Ti /P ,
where P is the sum of the ohmic and auxiliary power inputs. From the steady state relation
τE = τE∗ , which follows from (1.24) and the assumption that the radiation loss time is relatively
long, we find that P is related to the electron temperature by
kB Te = τE P/(3Rπ 2 Ra2 n̄e ) ,
( ≡ ne /n̄e ) .
(4.72)
Using this relation to eliminate kB Te from (4.60) and adopting (4.71) we arrive at
3/5 2/5
1.24 ×10−14 R3/5 I 4/5 R7/5 a4/5 n̄e qa 3/5
s.
τE = 2/5
P 3/5 1 + 0.47/βp )2/5
1 + D(R − 1)
(4.73)
An empirical formula for τE obtained statistically from a range of observations of tokamaks operating in the ELMy H-mode and adopted by the ITER team (1999, p. 2167) is:
0.44 −0.66 1.48 0.57 0.72
τE = 0.0503 HH Î 0.91 Bϕ0.15 n19
P̂
R a κ M 0.13 s ,
(4.74)
where HH is described as a confinement multiplier that depends on “the sensitivity of results
to variations in confinement”, κ is the ‘elongation’ defined as S0 /(πa2 ), where S0 is the
plasma cross-sectional area, and M is the atomic mass of the plasma particles. Comparing the
variables in common between (4.73) and (4.74) in this order, we have:
Î 0.8 , Î 0.91 ;
0.6
0.44
n19
, n19
;
P −0.6 , P −0.65 ;
R1.4 , R1.48 ;
a0.8 , a0.57 ,
which shows reasonable agreement considering the very different origins of the two formulae.
4.5 Comparison of experiment and theory
105
Figure 4.6: Energy confinement times in a variety of tokamaks (From an MIT report by
J. Kesner entitled Woods’ Scaling in Ohmic and Auxiliary Heated Tokamaks, April 15, 1988.)
To compare (4.73) with experiments the Kaye–Goldston (1985) tokamak data base, augmented by an ALCATOR-C data base was adopted, which gave results for eight tokamaks.
The experimental points are shown in Fig. 4.6, where the straight line represents theory. Bearing in mind the several approximations adopted in applying the theory, e.g. the use of empirical
temperature and density profiles, the neglect of radiation and convective losses and of energy
transfers between the electrons and ions, the agreement is good.
4.5.2 Confinement times for L- and H-modes
Tables 4.4 and 4.5 list some JET results taken from a special collection of tokamak global
confinement data coordinated by Engelman and Kardaun (1990) (published in the thirtieth
anniversary issue of the journal Nuclear Fusion). The objective of the collection was to provide the ‘tokamak community with quantitative information on a choice of well documented
discharges that can be used as a starting point for further analysis of the physics of tokamak
plasmas’. Except for A, which is the area of the minor cross-section, the variables are standard. The density profile parameter αn was deduced from the given values of ne0 and ne by (4.6)1 and likewise δ was deduced from T̂e0 /T̂e = αE = (1 + αn + 2δ/3)/(1 + αn ).
The experimental points show some scatter, especially for the L-mode observations, but the
theory averages and observation averages are quite close, being 0.61 s (th) and 0.50 s (exp) in
the L-mode and 0.54 s (th) and 0.57 s (exp) in the H-mode.
106
4 Energy losses from tokamaks
Table 4.4: Energy confinement times for the L-mode in JET
βp
qcyl
0.14
0.16
0.22
0.26
0.19
0.21
0.17
0.28
0.14
0.14
0.19
0.19
0.22
0.12
0.11
0.13
3.42
3.32
3.33
3.77
3.34
3.67
1.91
3.25
2.33
2.23
2.68
3.97
3.73
3.65
2.45
2.91
A
(m2 )
6.5
6.5
6.5
6.9
6.5
6.8
6.1
6.4
6.8
6.8
6.8
6.6
6.7
5.6
6.8
6.6
n̄19
(m−3 )
2.60
2.53
4.16
3.47
2.89
1.80
4.14
4.26
2.56
2.89
5.02
3.50
2.69
1.51
2.76
3.04
|V |
V
0.5
0.6
0.5
0.6
0.6
0.4
0.7
0.4
0.7
0.6
0.5
0.4
0.4
0.7
0.5
0.5
Zeff
αn
δ
2.7
3.0
1.8
1.8
2.4
2.3
2.5
1.8
2.0
3.0
1.9
1.6
2.4
3.6
3.0
3.3
0.75
0.80
0.63
0.67
0.76
0.80
0.45
0.58
0.66
0.70
0.52
0.69
0.91
0.70
0.67
0.62
2.57
2.14
2.41
3.71
2.94
3.06
1.93
2.70
1.26
2.07
2.21
3.07
2.70
2.91
3.52
4.48
T̂e0
(keV)
3.60
3.2
3.0
3.3
4.1
3.20
3.40
4.6
3.6
6.00
5.2
4.20
5.30
3.30
7.00
7.74
T̂e (keV)
1.82
1.73
1.51
1.33
1.94
1.50
1.80
2.15
2.39
3.31
2.64
1.90
2.73
1.56
2.91
2.72
τE (exp) τEe (th) |Îp |
(s)
(s)
(MA)
0.57
0.60
2.96
0.39
0.56
3.01
0.93
0.80
2.98
0.45
0.93
2.50
0.44
0.58
3.01
0.51
0.45
2.04
0.39
0.49
2.05
0.45
0.69
4.02
0.43
0.36
2.99
0.57
0.40
3.98
0.49
0.72
5.00
0.73
0.85
5.02
0.47
0.49
2.99
0.14
0.40
2.03
0.56
0.38
5.01
0.50
0.50
4.08
There are other moderate βp phenomena supporting second-order transport theory. One of
these is the improvement in confinement of both energy and mass by negative shear, a state
that can be generated in the central region of the plasma by injecting neutral beams during the
current ramp-up (Wesson 2004, p. 603). On a magnetic surface the average magnetic shear is
defined by
Rq r dq
=−
ν
S=
ν = Bθ /(rBϕ ) ,
q dr
r
and by (3.99) a positive value for ν reduces He and results in a smaller radial heat flux in
the region of negative shear. It may even be sufficient to give a local reversal of the heat flux,
although the thermal instability triggered by this could persist for a very short time only.
A related phenomenon arises in what is termed the ‘supershot’ regime, which is formed
by the injection of a high power neutral beam into a low density target plasma confined by
a limiter. The wall conditions need to be controlled carefully to achieve very peaked number density and temperature profiles; with central values of ne0 ∼ 1020 m−3 , temperatures
T̂e0 ∼ 12 keV and T̂i0 ∼ 35 keV, have been obtained (Wesson 2004, p. 600).
We can give a qualitative account of the supershot phenomenon using the L-mode expression for the profile shape factor given in (4.56), which we found by fixing the maximum value
of the parameter α at αm = 1 (see paragraph preceding (4.55). It was argued there that because the outwards heat flux at the boundary was proportional to (1 − α)(2 − α) and because
the limiter cannot be a heat source, either α ≤ 1 or α ≥ 2. The larger values would seem to
be inaccessible, but in the presence of a neutral beam supplying energy and forcing steeper
profiles much higher values of α would be possible. Of course, the theory really applies to
ohmic heating and is dependent on a family of rather special profile shapes whose merit is
more qualitative than quantitative despite the successes shown in Fig. 4.6 and in Tables 4.4
and 4.5. However, suppose that average values for α of 1.5 or 2 could be reached, then it
follows from Table 4.3 that the shape parameter F defined in (4.52) and which is proportional
4.5 Comparison of experiment and theory
107
Table 4.5: Energy confinement times for the H-mode in JET
βp
qcyl
0.09
0.10
0.11
0.11
0.07
0.11
0.12
0.13
0.10
0.11
0.09
0.10
0.08
0.09
0.05
0.08
3.46
3.39
3.39
3.80
3.38
3.72
3.70
2.04
3.28
2.33
2.23
2.75
3.94
3.66
2.51
2.91
A
(m2 )
6.5
6.4
6.5
6.8
6.5
6.6
6.8
6.2
6.6
6.9
6.8
6.8
6.4
6.6
6.9
6.6
n̄19
(m−3 )
2.15
1.61
1.92
1.89
1.16
1.40
1.42
2.80
2.49
2.32
2.32
3.45
2.15
1.90
2.33
2.21
|V |
V
0.4
0.7
0.5
0.5
0.6
0.7
0.7
0.6
0.7
0.8
0.6
0.5
0.5
0.6
0.6
0.5
Zeff
αn
δ
2.6
3.4
1.9
2.0
3.1
2.0
2.1
2.3
1.6
1.9
3.2
1.6
1.9
1.9
3.0
3.4
0.81
0.82
0.78
0.79
0.85
0.87
0.90
0.52
0.63
0.75
0.74
0.64
0.74
0.90
0.82
0.67
2.43
2.48
2.85
3.21
3.07
2.63
2.48
1.24
2.12
1.25
1.41
2.40
3.92
2.73
2.26
2.37
T̂e0
(keV)
2.90
3.40
3.10
2.70
4.30
2.13
2.17
2.33
2.69
3.00
3.94
3.50
3.90
3.70
4.33
4.32
T̂e τE (exp)
(keV)
(s)
1.53
0.70
1.78
0.47
1.50
0.60
1.23
0.58
2.04
0.34
1.10
0.45
1.16
0.51
1.51
0.52
1.44
0.67
2.03
0.67
2.56
0.55
1.77
0.74
1.56
0.65
1.89
0.71
2.37
0.46
2.22
0.51
τEe (th) |Îp |
(s)
(MA)
0.64
2.93
0.43
2.93
0.57
2.93
0.71
2.47
0.29
2.98
0.53
2.03
0.53
2.04
0.48
3.81
0.73
2.99
0.42
3.91
0.35
4.98
0.78
5.01
0.72
2.97
0.54
3.01
0.40
5.00
0.46
4.09
to 1/(Φ1 (α) + βp Φ2 (α)), would increase and give a larger confinement time. Near α = 2 this
factor would be quite large, tending to infinity at βp = 0.72.
4.5.3 Loop voltage
The toroidal loop voltage Vs measured at the outer surface of the torus is induced by the
changing magnetic flux through the plane of the minor axis (see Fig. 1.1). It is a simple
property to measure, but not so simple to interpret and it is obtained by measuring the voltage
around a toroidal loop of wire just outside and parallel to the torus. From Vs the voltage drop
V around the minor axis, known as the axial loop voltage, can be deduced (see Fig. 5.2) and
this is the voltage recorded in Tables 4.4 and 4.5 (Christiansen 1987). As we shall show in
Section 5.3.2 V is not the total voltage Vt driving the plasma current Ip , but for the present we
shall assume that the difference between V and Vt is negligible. Certainly for the observations
in Figs. 4.4 and 4.5 this is likely to be the case.
Thus, with V the voltage drop around the minor axis of the torus and σ the parallel
conductivity, the current density is
1 jϕ = gσ V /(2πR)
(4.75)
g = 1 − (2ε) 2 ,
where the factor g is due to particle trapping and is explained in Section 2.4.4. From (A.16),
(A.19) and the tokamak approximation ln Λ = 17, in a contaminated hydrogen plasma,
3
σ = 1.98 e2ne τe /me = 9.03 ×10−4 Te2 Zeff .
By (4.5) and (4.7),
3
1
3/2
gσ = 9.03 ×10−4 αE Te 2 Zeff (1 − y)δ 1 − (2εa ) 2 y 0.25
εa ≡ a/R . (4.76)
108
4 Energy losses from tokamaks
For JET (see Table 1.1) εa = (ab) 2 /R ≈ 0.529. The contribution of the factor g to the
integral gσ varies very little with δ; for JET it is 0.33 at δ = 2, 0.37 at δ = 3 and 0.39
at δ = 4. Ignoring this slight variation and bearing in mind the range of δ shown in Tables
4.4 and 4.5, we shall replace g by the constant gt = 0.35, which means that particle trapping
increases the resistivity by a factor of 2.86. Thus
3
3/2
gσ = 9.03 ×10−4 gt αE /(1 + δ) Te 2 Zeff ,
(4.77)
1
and from (4.75),
gt σ V
Ip
= jϕ =
,
(4.78)
A
2πR
where A is the minor cross-sectional area. Therefore the loop voltage can be deduced from
2πRIp
V =
(gt = 0.35) .
(4.79)
Agt σ In Tables 4.4 and 4.5 there are sufficient observations to allow us to deduce V (th) from
(4.79), which is the ‘theoretical’ value of the loop voltage — ‘theoretical’ in the sense of not
being measured directly. The tables give the directly measured values of this voltage, albeit
only to one figure accuracy.
Table 4.6 shows that the theory of tokamak resistivity is quite well supported by the observations. The average observed values are given and can be compared with the average theoretical values for which the standard deviations are included. In the H-mode the agreement is
within 10%, which is good considering the uncertainty in the ‘observed’ values, which were
in fact deduced from measured surface loop voltages; in the L-mode the error is an acceptable
13%. Both results are well within the range of the standard deviations.
Table 4.6: Loop voltage: theory and observations
L-mode:
V (obs)
V (th)
V (obs)
V (th)
H-mode:
V (obs)
V (th)
V (obs)
V (th)
0.5
0.60
0.7
0.27
0.6
0.75
0.6
0.34
0.5
0.51
0.5
0.35
0.6
0.39
0.4
0.54
0.6
0.50
0.4
0.30
0.4
0.46
0.7
0.78
0.7
0.37
0.5
0.51
0.4
0.39
0.5
0.49
0.4
0.64
0.7
0.40
0.7
0.69
0.8
0.40
0.5
0.48
0.6
0.63
0.5
0.50
0.5
0.52
0.6
0.53
0.5
0.37
0.7
0.53
0.6
0.35
0.7
0.52
0.6
0.68
0.6
0.76
0.5
0.66
L: V = 0.54(obs), 0.47 ± 0.15(th); H: V = 0.59(obs), 0.63 ± 0.12(th).
4.5.4 Steady state with ohmic heating
As we have a theory for the conduction of heat and for the present we are neglecting convective
transport, one would expect that for a given distribution of heating it should be possible to
4.5 Comparison of experiment and theory
109
determine the temperature profiles without any empirical elements. However, the energy input
itself depends on both the density and temperature profiles so a rather complicated iterative
computational procedure is unavoidable and in any case because of the sawtooth instability,
‘steady states’ do not really exist in tokamak transport. In the following we shall assume
quasi-steady conditions with ohmic heating and determine what constraints are imposed by
equating the energy replacement time, τE∗ , to the energy confinement time, τE . To simplify the
account we shall adopt electron-energy properties and ignore radiation losses.
Thus from (4.31), (4.32), (4.33) and equations in Section 1.3.2 we get
A γ
n̄e aR2 qa
τEe = 4.53 ×10−21 3/2
1
αE (1 + αn ) T̂e 2
and
τE∗e ≡ We
a
0
T̂e in keV ,
jϕ Eϕ r dr ,
where by (4.4), (4.6) and (4.76),
a
a
3
3 2
pe dy = 34 a2
We =
2 kB ne Te r dr = 4 a
0
and
a
0
(4.80)
0
jϕ Eϕ r dr = 12 a2
1
Hence
τE∗e = 3.01 ×10−21
0
γ
n̄e kB Te ,
1 + αn
jϕ2 dy
(1 + δ)jϕ 2
= 1.58 ×103 a2 Zeff 3/2
.
gt σ
αE Te 3/2
3/2
n̄e T̂e 5/2
γαE
,
(1 + αn )(1 + δ) Zeff ĵϕ 2
(4.81)
where T̂e is in keV and ĵϕ is in MA m−2 .
From (4.80) and (4.81) we obtain the ratio
R ≡ τEe /τE∗e = 1.50A
(1 + δ) aR2 qa Zeff Îp2
,
α3E
A2 T̂e 3
(4.82)
where A is the cross-sectional area and
2δ/3
,
αE ≡ 1 +
1 + αn
√
πΓ(δ + 2)
.
A ≡ 1 −
2Γ(δ + 2.5)
(4.83)
Table 4.7 sets out the values of the ratio R for the L-mode figures of Table 4.4 and the Hmode figures of Table 4.5 in order of the entries in those two tables. The ratio should be unity
for equilibrium, that is the supply or replacement rate should equal the loss rate, although
there are relatively small radiation and convective losses to account for. Taking the averages
we find that in the L-mode, R = 0.78 ± 0.44, while in the H-mode, R = 1.61 ± 0.44.
Thus in the H-mode the loss rate (∝ 1/τEe ) is smaller than the supply rate (∝ 1/τE∗e ) and a
factor ∼ 2 better than the L-mode, as one would expect. We should note that in the tokamak
110
4 Energy losses from tokamaks
Table 4.7: The ratio R ≡ τEe /τE∗e
L-mode:
R
”
H-mode:
R
”
1.21
0.34
1.79
0.28
1.30
0.55
1.18
0.67
0.84
0.39
0.89
1.14
0.28
0.51
0.71
0.42
2.11
1.42
1.71
1.95
1.43
1.20
2.04
1.70
0.97
1.17
2.27
0.92
2.15
1.44
2.01
1.33
observations some of the energy was supplied by NBI and RF heating, so there will be some
error in the treatment due to the different distributions of this energy input.
Bearing in mind that rather complex theories are involved in both the heating (trapped
particle dissipation) and in the thermal conduction (second-order transport) we conclude that
the averages for R being of the right order is good supporting evidence for the theories.
4.5.5 Internal transport barriers
Some tokamak plasmas are found to have improved confinement within regions bounded by
relatively sharp boundaries across which the pressure and temperature change rapidly. In Section 4.3.1 we met a similar phenomenon with H-mode plasmas near the limiter, where the
thermal contact between the plasma and the limiter was assumed to be relatively weak compared with the situation with L-mode plasmas. The existence of similar barriers to transport
within the plasmas, known as ITBs, provides a demanding test of the transport theory developed in Section 3.5. The circumstance under which an ITB could exist is illustrated in
Fig. 4.7(a), which is a copy of Fig. 2.8 except that the temperature gradient has been reversed.
In the figure the heat is now shown flowing up the temperature gradient and it is a result of
changing the relationship between the temperature gradient and the electron fluid shear. Nat-
Figure 4.7: Heat flux barrier
4.5 Comparison of experiment and theory
111
Figure 4.8: ITB in the JT-6OU tokamak
urally, when heat flows up a temperature gradient it cannot persist for long before there is a
thermal instability, however this reverse flow does present a barrier blocking the heat from
leaving the inner region of the tokamak plasma. Figure 4.7(b) illustrates an ITB at r = r∗ .
Consider the case when βp is small enough for the term involving pt in (3.99) (cf. (3.110))
to be omitted, then by (3.101),
He =
rjϕ q ,
2ene Rq 2
(4.84)
and (3.104) gives
κe =
5k1 r2 k2B Te jϕ q .
4Re2 Bϕ Ce q
(4.85)
It follows that if q(r) is a concave function such that q changes sign at r = r∗ say, then the
heat will tend to flow away from r∗ in both directions until thermal equilibrium is attained;
as a result the heat generated in the central region will be prevented from reaching the region
r > r∗ , the temperature will fall rapidly just outside r∗ and an ITB will be found at this
point. We therefore expect a temperature profile that is almost flat in 0 < r < r∗ , steep in a
narrow region r∗ < r < r∗ + δr, where δr is the width of the ITB, and relatively flat again in
r∗ + δr < r < a.
The description given in Section 4.3.1 of the mechanism determining the minimum value
of the parameter α (see Fig. 4.4) is relevant here. The minimum value occurs when the heat
flux at the inversion point r = rs is about to diverge as illustrated in Fig. 4.7(b), but the cause
of this potential divergence is that the temperature gradient goes to zero and the diffusivity
is not affected. In the case of ITBs it is the change in sign of the diffusivity that causes the
112
4 Energy losses from tokamaks
divergence. However, it is possible that a second cause of ITBs is the failure of the profile to
recover from the trough of the sawtooth oscillation, so we might expect that r∗ ≈ rs in some
cases.
For a review of the physics of internal transport barriers see Connor et al. (2004).
Figure 4.8, which shows a typical ITB with its q(r/a), Ti , Te and ne profiles in the JT6OU tokamak, is from that paper. The barrier is formed just inside the minimum value of q,
which occurs at r ≈ 12 a.
4.6 Profile instabilities
To complete our account of the diffusive transport of thermal energy in tokamaks we shall
conclude this chapter with a brief review of the basic equations and describe three types
of profile instability. In the next chapter the rather more difficult topic of convective transport is taken up and we shall find a notable difference between the two modes of transport,
namely that while convection depends on viscous, collisional forces, it appears from previous chapters that diffusive transport does not involve particle collisions at all. Yet of course
the transmission of ‘the kind of motion we call heat’1 must involve collisions. Referring to
Fig. 3.8, grazing collisions occur frequently enough in the plane in which the transverse heat
flux vectors lie to transmit thermal energy throughout this plane.
4.6.1 Safety factor
By bringing toroidal geometry into the reckoning the safety factor variable, q, plays a central
role in tokamak theory. Let q = qs be the value at the inversion radius, ys = (rs /a)2 , then
from (4.8) and (4.9),
qs 1 − (1 − ys )(δ+1) ,
qa = q0 (δ + 1) .
(4.86)
qa =
ys
The significance of the inversion radius is that it is a surface over which the temperature
gradient vanishes simultaneously at the bottom of the sawtooth oscillation, i.e. the whole of
the surface is momentarily in thermal equilibrium, which is possible only if the connection
length 2πRqs belongs to the set of values 2πR, 32 (2πR), 2(2πR), . . . allowing heat to flow
freely along magnetic fields lines to reach all of the inversion surface. The “3/2” member
is included because by symmetry half a connection length also allows the whole inversion
surface to be thermally connected. There is also the possibility that qs is independent of the
radius over a small interval δ0 , rs − 21 δ0 ≤ r ≤ rs + 12 δ0 , in which case there will exist a volume
of thermally connected plasma. Observational evidence for this will be given in Section 6.4.3
when we give an account of a related phenomenon known as a “snake”.
The functions q0 (δ) and qa (δ) for the cases qs = 1 and qs = 1.5 for JET (ys = 1/4) are
shown in Fig. 4.9. In the range 2.5 < δ < 3.5 we find that 0.65 < q0 < 0.72 with qs = 1,
and 0.98 < q0 < 1.08 with qs = 1.5. Typically, values q0 ∼ 0.7 are observed and less often
q0 ∼ 1. With small tokamaks, rs /a < 0.3, so ys is small enough to be neglected compared
1 This is the title of Stephen Brush’s excellent review of the 19th Century history of the kinetic theory of gases
(North-Holland, 1976).
4.6 Profile instabilities
113
5
4.5
safety factor
4
3.5
qa
3
2.5
2
1.5
q
0
1
0.7
0.5
0
0
1
2
δ
3
4
5
Figure 4.9: Safety factor v. profile steepness
with unity, in which case (4.86) gives qa = qs (1 + δ) and with the choice qs = 1 we obtain
q0 = 1. The fact that q0 is usually found to be appreciably less than unity in large tokamaks
renders invalid the MHD interchange instability often appealed to in models of the sawtooth
oscillation (Wesson 2004, p. 372). Sawtooth oscillations are analyzed in Section 6.1.3.
The leading role of the safety factor in the transport of electron thermal energy follows
from (2.78), (3.97), (3.00) and (3.101). If βp 1, the pt term can be omitted from He , in
which case:
5k1 k2B r2 Te jϕ q r2 jϕ q 12 Te = − 0.119
T̂e T̂e .
(4.87)
Qer = −
2
4e R Ce Bϕ q
R Bϕ q
Notice from Qer ∝ q in (4.87) that unless ohmic dissipation acts to stabilize it, a temperature profile will be thermally unstable where q < 0. We shall use the term “thermally
unstable” for the case when heat flows up the temperature gradient, although whether or not
the profile actually becomes unstable depends on the energy supply. If a local decrease in temperature increases the rate at which energy is deposited in that region, the profile will be stable;
for example ohmic heating (Section 1.4.1) is stabilizing, whereas RF heating (Section 1.4.3)
is not. It follows that with OH, temperature profiles with a temperature dip like that shown in
Fig. 4.10 can reach a stable state and yet retain the concavity (e.g. see Section 6.4.3).
From q = rBϕ /RBθ and µ0 jϕ = (1/r)∂(rBθ )/∂r,
rµ0 jϕ dq
Bϕ 2−
,
=
dr
RBθ
Bθ
so the condition for a thermal instability becomes µ0 jϕ > 2Bθ /r, i.e.
2 r
1 y
jϕ > 2
jϕ (r ) r dr =
jϕ (y ) dy ,
r 0
y 0
(4.88)
(4.89)
i.e. the value of the current density at a point p exceeds its average value between the minor
axis and p.
114
4 Energy losses from tokamaks
An example of a current density instability is shown in Fig. 4.10. Up to the point d the
value of jϕ (rc ) exceeds the average value over ad and therefore satisfies the inequality in
(4.89). Hence the heat flows up towards c from both sides; from points below d the heat flux
is normal. This transport accentuates the hollow in the temperature distribution and ultimately
destroys the distribution unless thermal energy is supplied to flatten the profile. Marginal
stability occurs when
y
jϕ (y ) dy ,
yjϕ =
0
i.e. when q, jϕ , and Te are constant over 0 ≤ r ≤ rs .
4.6.2 Thermal instability
The fact that the steeper temperature profiles retain their energy better than the flatter ones results in the sawtooth instability mentioned in Section 4.3.1 and analyzed in Section 6.1. There
is another type of thermal instability for which there is some direct observational evidence.
If βp is large enough and the electron temperature profile is sufficiently concave outside the
inversion radius, the diffusivity in the electron gas becomes negative, causing energy to flow
in the ‘wrong’ direction. It follows from (4.40) that the condition for this is
J y
ψ≡
[J(y ) − J(y)] dy − 12 βp y P̈ − ṖṄ /N ≤ 0.
(4.90)
y 0
The first term in this expression remains positive unless as explained in Section 4.6.1 the
current profile is concave. For the profiles in Fig. 4.4 only those with −1 ≤ α < −0.5 are
slightly concave about the magnetic axis.
The more interesting term in (4.90) is that containing the second derivative of the pressure.
With the distributions of (4.42) we find that
ψ = F (y; α) − 18βp y(1 − y)
α−1
,
5−α
Figure 4.10: Unstable current density distribution
(4.91)
4.6 Profile instabilities
115
where F (y; α) can be integrated algebraically. If ψ is negative we have a thermal instability.
In Fig. 4.11 the values of βp are plotted against the radius at which ψ is zero. With the
steepest profiles quite small values of βp are sufficient to induce a thermal instability outside
the inversion radius. By (1.6) the key parameter is
βp =
8π 2 a2 ne kB Te .
µ0
Ip2
(4.92)
It follows that if for a given plasma current the number density is sufficiently large and the
temperature profile is sufficiently concave for this property to be transferred to the pressure
distribution, then a thermal instability will result.
Observations on the DIII-D tokamak that give qualitative support to this conclusion have
been reported by Petty and Luce (1994). With localized off-axis electron cyclotron heating
they find that the heat flux in the electron gas is both outwards and inwards from the site of
the heating and that increasing the number density increases the magnitude of the inwards
flux. Their experimental results are summarized in Figs. 4.12 and 4.13. Figure 4.12 shows the
ohmic (QOH ) and electron cyclotron (QECH ) heating distributions and the resulting electron
and ion temperature distributions. Notice that the electron temperature distribution is strongly
concave — and therefore destabilizing — outside the sawtooth inversion radius. Figure 4.13
shows the dependence of the heat flux qe on the electron number density ne ; negative values
correspond to heat flux up the temperature gradient. The parameter βp is proportional to ne ,
and varies from about 0.1 to about 0.3. The higher values are large enough to destabilize the
rather peaked electron temperature profile.
4.6.3 Review of electron thermal transport
In steady conditions there are two stages in the transport process. First, there is rapid heat
transfer along the banana orbit magnetic field lines by typical electrons moving at their thermal
speed Ce and taking a time τ⊥ proportional to the trapped particle bounce time. Secondly,
there is the unimpeded deflection of this energy (but not the electrons themselves) across the
magnetic field by the fluid shear, as illustrated in Fig. 2.8.
inversion
radius
3
βp
α=1.5
2
unstable
α=2
α=2.5
1
stable
0
0.4
r/a
0.6
0.8
1.0
Figure 4.11: Dependence of thermal instability on βp and profile steepness α
116
4 Energy losses from tokamaks
2
Sawtooth inversion
radius
1.5
T
(keV)
1
1
electron
0.5
0.5
QOH
0
Q
(W/cm3 )
ion
0
QECH
0.2
0.4
0.6
r/a
0.8
1
0
Figure 4.12: Electron temperature profile with off-axis heating
In steady conditions the diffusivity is
χe =
5τ⊥ kB Te
5τ⊥ 2
5τe
He =
C e He =
kN Ce2
3eBϕ
6ωce
6e
e ≡ ωce τe ,
(4.93)
where kN (= τ⊥ He ) is the (collisionless) Knudsen number. So we have the unexpected circumstance that while collisions are essential to transmit energy between particles, they are
not involved in the formula for the diffusivity, which explains the paradox mentioned in the
paragraph preceding Section 4.6.1. For comparison, the expression for the classical thermal
diffusivity across magnetic field line is given in (3.47):
χ(c)
e =
5τe 2
C ,
6e2 e
(4.94)
3
13
+
1.0 10 cm
1.7 10 cm -3
+
2
13
2.2 10 cm -3
+
qe
-3
13
13
2.9 10 cm -3
2
+
(W/cm )
1
0
-1
0
0.2
0.4
r/a
0.6
0.8
1
Figure 4.13: Electron heat flux for a range of densities
References
117
(c)
that is χe is kN e times larger than χe . The Knudsen number appearing in the transport
formula is based on the time interval τ⊥ , which is about two orders of magnitude smaller than
the collision interval, τe .
Unsteady conditions arise in three forms: (i) the relatively gentle ramp phase of the sawtooth instability, which we shall treat in Section 6.1 as being a quasi-static evolution, (ii) the
very rapid collapse phase, which involves particle collisions as indicated by the second righthand term in (2.79) and (iii) the rapid over-stable, oscillatory response of the temperature, as
illustrated in Fig. 2.10.
The simplified model in Section 4.3.1 describing the difference between an L-mode and
an H-mode discharge, namely that in an L-mode the boundary is a perfect conductor, while in
the H-mode it is a perfect heat reservoir, is incomplete as it does not explain how these ideal
states are achieved. An L-mode can turn abruptly into an H-mode when the energy supplied
by NBI reaches a certain fraction of the total input energy; it is found that as a consequence,
both the thermal energy and the plasma density pile up into a pedestal formation with steep
gradients at about r/a = 0.96. This is an edge transport barrier that impedes both mass and
energy transport and occurs in a region of the tokamak plasma that is dominated by “real”
physics, with neutrals, impurities and radiation involved. It is a difficult problem and the
phenomenon is not fully understood; one speculation is that the basic transport equations
(presumed to be unknown) have a solution with a bifurcation at the critical condition. Until we
have developed the equations for the convection of plasma towards the boundary, we cannot
discuss the solution of this problem. Convection is treated in the following chapter and in
Section 6.4.2 a model of the edge transport barrier is presented.
References
(See page 25 for the reference notation.)
Brusati, M. & Cordey, J.G. (1985). V, Pt 1, 34.
Christiansen, J.P. (1987). J. Comp. Phys., 73(1), 85.
Connor, J.W. et al. (2004). Nuclear Fusion, 44 (4), R1-R49.
Cordey, J.G. (1985). Proc. of Course and Workshop on Basic Physical Processes of Toroidal
Fusion Plasma. Varenna, Italy, Sept. 1985. CEC EUR 10418 EN, Vol. 1, 209.
Ejima, S. et al. (1982). Nuclear Fusion, 22(12), 1627–49.
Engelmann, J. & Kardaun, O.J.W.F. (Co-ordinators for Tokamak Confinement Data). (1990).
Nuclear Fusion. 30(9), 1951.
Goldston, R.J. (1984). Plasma Physics and Controlled Fusion, 26(1A), 87.
Hugill, J. (1983). Nuclear Fusion, 23(3), 331.
ITER team, (1999). Nuclear Fusion, 39(12), pp. 2167, 2225, 2229.
Kaye, S.M. & Goldston, R.J. (1985). Nuclear Fusion, 25, 65.
Petty, C.C. & Luce, T.C. (1994). Nuclear Fusion, 34(1), 121.
118
References
Pfeiffer, W. & Waltz, R.E. (1979). Nuclear Fusion, 19, 51.
Rebut, P.H. et al. (1985). V, Pt 1, 11.
Scheuller, C. et al. (1985). V, Pt 1, 151.
Wade, M.R. et al. (1995). Phys. Plasmas, 2, 2357.
Wesson, J.A. (2004). Tokamaks, 3rd. edn. Oxford University Press.
Woods, L.C. (2004). Physics of plasmas. Wiley-VCH Verlag GmbH & Co., KGaA, Weinheim.
5 Plasma flow and loop voltage
Plasma particles diffuse radially outwards across the tokamak magnetic field with a velocity
vD , which near the boundary changes sign and becomes an inwards or ‘pinch’ velocity (see
Section 5.2.1). This particle flow not only convects mass and energy, but being orthogonal to
Bθ θ̂, it generates an electric field and therefore drives a toroidal electric current. When the
density and current gradients are sufficiently steep to generate relatively large values of the
average radial velocity, vD , this current can be comparable to that induced by the transformer
action illustrated in Fig. 1.1. Although this additional non-inductive current is obtained at the
expense of plasma lost from confinement, it has the merit of increasing the strength of the
confining magnetic field. Thus, with continuous refueling near the minor axis, it offers an
escape from the discontinuous operation imposed by using induction as the only current drive
(see Section 1.1.3).
The dominant force driving the plasma out of the tokamak is the radial component of
∇ · p = ∇p + ∇ · , where the viscous stress tensor, , has the Knudsen number expansion
= 1 + 2 + · · · . We shall find that neither the radial pressure gradient, r̂ · ∇p, nor even the
first-order viscous force, r̂ · ∇ · 1 , is large enough to account for the observed value of vD .
Rather — like the situation with thermal diffusion — it is the second-order term r̂ · ∇ · 2 ,
that gives agreement with observation. Also it is found that the toroidal component of ∇ · 2
accounts for the transverse diffusion of momentum from the toroidal flow around the torus.
The equations for second-order collisional transport are not easy to apply for they involve
products of derivatives and second-order derivatives. This means that temperature and density profile shapes have dominant roles in determining the value of vD , so we cannot expect
the same accuracy with mass diffusion as was achieved in the previous chapter with thermal
diffusion.
5.1 Flow of plasma across strong magnetic fields
5.1.1 Plasma particle confinement
From (1.32) and the integral of (1.30) the plasma particle confinement time is given by
(5.1)
τp = Ne ne vD a = Ne / Σe − dNe /dt ,
a
a
where
ne r dr,
Σe =
Se r dr .
(5.2)
Ne =
0
0
By ambipolarity (jr = 0), τp can also be expressed in terms of Ni and Σi .
Theory of Tokamak Transport: New Aspects for Nuclear Fusion Reactor Design. Leslie Colin Woods
Copyright © 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
ISBN: 3-527-40625-5
120
5 Plasma flow and loop voltage
As remarked in Section 1.3.3, because of the difficulty in determining the electron source
term Σe , relatively little is known about τp compared with the energy confinement time τE .
It is found that at low or medium densities τp is greater than τE , but that at high densities it
reduces to a fraction of τE . It appears from figures quoted by Hugill (1983) that for OH (ohmically heated) plasmas, τp has little if any dependence on n̄e , whereas τE ∝ n̄e (see Table 1.2,
page 20). In JET τp increases sharply with n̄e , reaches a maximum and then falls abruptly
with further increase in n̄e ; this curious behavior may be a result of changes in the density
profile (Tanga et al. 1984). Values of τp ∼ 3τE are typical at intermediate densities (see, for
example, Equipe TFR 1980).
Several tokamak groups (e.g. Gentle et al. 1984) have found that the radial flux can be
adequately modeled by the empirical formula,
nvD ≡ Γ = −Dn − (r/a)nV ,
(5.3)
where D and V are constants for a given tokamak. It is found that D ∼ 0.5 − 1 m2 s−1
and V ∼ 1 − 10 m s−1 . The reverse flow term, −(r/a)nV is due to a plasma source in the
neighborhood of the limiter from which readily ionized neutrals are emitted back into the
plasma. The particle diffusivity D is usually found to have an inverse dependence on n and a
strong temperature dependence, which provides a qualitative check for our theory.
The condition that the convection of energy dominates its diffusion at a given radial distance from the axis is D|(ln ne ) | > 0.6χe |(ln Te ) |, which follows from (4.25), (4.28) and
(5.3). However, for an overall measure of the relative importance of these processes it is necessary to compare the volume weighted averages of these quantities. Although often adopted,
the ratio τp /τE is not a good guide since it depends on the values of these times evaluated
at one point only, namely the limiter radius. We shall avoid this problem. The high collision
frequency between plasma ions and impurity ions results in the latter being swept along by the
ions. It follows that a good approximation to the radial flux of the impurity ions has the same
form as (5.3) with Γ and n replaced by the corresponding values for the impurity in question. An account of the transport of impurities is given in an article by the ITER team (1999,
p. 2436).
To this stage we have derived three rather different expression for the perpendicular component, vD , of the plasma velocity. There is a general formula in (2.32), the Pfirsch–Schlüter
formula in (3.76, and the neoclassical formula for Γ = nvD in (3.80). Comparing (3.76) and
(3.80) we find that while the neoclassical formula (without the dominating Ware pinch) gives
the larger value for vD , this is only ∼ 10−2 m s−1 , a value at least an order of magnitude too
small (see Section 3.4.2). We shall show that it is the divergence of the second-order viscosity
tensor, 2 , that yields values of vD in agreement with observations. This is the last term in
2
) term 2 rather than 1 .
(2.32), but with the O(kN
We shall first derive an expression for v⊥ , the perpendicular velocity of either the electron
or ion fluids in a direction across the magnetic field. Equations (A.12) can be rearranged into
a single form as:
v⊥ =
1 Q
1
E
×b +
− η · j − ∇ · p − Dv × b
B
ωc
m
(ωc = QB/m),
(5.4)
5.1 Flow of plasma across strong magnetic fields
121
where by (A.13) and (A.20), Re = −Qnη · j. By Dv/ωc = τ Dv/ ∼ (kN /)v⊥ v⊥ ,
we may omit the acceleration term and write
v⊥ =
1
1
E − η ·j ×b −
∇·p×b.
B
ωc
(5.5)
The radial velocity is
vr = r̂ · v⊥ =
1
1
E − η · j × b · r̂ −
∇·
B
ωc
× b · r̂,
(5.6)
the last term of which follows from p = p11 + and ∇p = r̂∂p/∂r = p r̂.
In cylindrical geometry with axial symmetry Eθ is zero. With the help of the equilibrium
relation ∇pt = j × B, where pt = pe + pi , the first term in (5.5) yields
vr(1) = −
η⊥
Ez bθ
− 2 pt .
B
B
(5.7)
This expression is the same for both ions and electrons, meaning that the flow is automatically
ambipolar; but it is orders of magnitude too small to explain the observed mass flux.
The ambipolar condition also applies to the second term in (5.5), viz.
1
1
∇ · · b × r̂ = −
∇ · · bz θ̂ − bθ ẑ ,
(5.8)
vr(2) = −
ωc
ωc
3
where = 1 + 2 + · · · + O(kN
) . The first step is to show that ∇ ·
therefore cannot contribute to the right-hand side of (5.8).
1 is parallel to r̂ and
5.1.2 Viscous stress tensor in cylindrical geometry
The first order viscous stress tensor is given by (A.132),
◦ ◦
◦
e ≡ ∇v ,
1 = −2pτ1 W ·· e
(5.9)
where τ1 is the collision interval for the transport of particle momentum and the fourth-order
tensor W is defined in equations (A.137) and (A.139). In the tokamak application, 1,
which allows us to simplify W by omitting the W2 and W3 terms from (A.137). Thus
1 = − 2pτ1
W1 +
1 ◦
W4 + 2W5 ·· e
2
≡ ωc τ1 .
(5.10)
In tokamaks gradients parallel to b are negligible, in which case W1 may be omitted, leaving
1 = −
where
pτ1
W∧ ,
(5.11)
W∧ ≡ 2W4 + 4W5 = 11∧ 11 − 11 11∧ + 311∧ 11 − 311 11∧ ,
in which 11 = bb and 11∧ = b × 11. Therefore, evaluating these terms we arrive at
1 =
p
◦
◦
◦
◦
e × b − b × e + 3bb · e × b − 3b × e · bb
2ωc
.
(5.12)
122
5 Plasma flow and loop voltage
Notice that 1 is independent of τ1 , which means that it represents a reversible process and we
would not expect it to contribute to the loss of plasma. This term corresponds to the similar
reversible term in the formula (3.112) for the transverse heat flux.
In cylindrical geometry (3.94) gives
◦
(5.13)
e = a r̂θ̂ + θ̂r̂ + c ẑr̂ + r̂ẑ ,
a = 12 (vθ − vθ /r), c = 12 vz ,
from which it follows that 1 has the form
1 = A θ̂ẑ + ẑθ̂ + B r̂r̂ − θ̂ θ̂ + C(ẑẑ − θ̂ θ̂ + D(ẑẑ − r̂r̂ ,
where A, B, C and D are scalar function of a and c. The divergence of
with the help of
∇ = r̂
θ̂ ∂
∂
+
,
∂r
r ∂θ
∂r̂
= θ̂,
∂θ
1 can be calculated
∂ θ̂
= −r̂ .
∂θ
(5.14)
The first term in ∇ · 1 is zero, while the divergences of the remaining terms are each parallel
to r̂. Thus by (5.8), 1 contributes nothing to vr(2) and we are left with
vr(2) = −
1
∇·
ωc
2·
bz θ̂ − bθ ẑ .
(5.15)
5.1.3 Radial diffusion velocity
The viscosity tensor for monatomic gases has zero trace, thus (A.134) can be written
2 = 4pτi W ·· S ,
◦
◦
◦
S=e·e .
Following the analysis that gave the formula for
2 = −
(5.16)
1 , we obtain
pτ1
{S × b − b × S + 3bb · S × b − 3b × S · bb} ,
ωc
(5.17)
where from (5.13)
S = a2 (r̂r̂ + θ̂θ̂) + ca(θ̂ẑ + ẑθ̂) + c2 (ẑẑ + r̂r̂) − 23 (a2 + c2 )11 .
Thus we find
pτ1 acbz + a2 bθ r̂ẑ + ẑr̂ − c2 bz + acbθ θ̂r̂ + r̂θ̂
ωc
+ bθ bz a2 − c2 + 3ac b2z − b2θ br̂ + r̂b .
2 = −
(5.18)
For our application it will be sufficient to assume that the axial magnetic field is much
stronger than the poloidal field, i.e. that bz ≈ 1 and bθ ≈ 0, when (5.18) simplifies to
2 = −
pτ1 2
c (r̂θ̂ + θ̂r̂) − 4ca(r̂ẑ + ẑr̂) .
ωc
(5.19)
5.1 Flow of plasma across strong magnetic fields
To calculate ∇ · 2 we use the relations
∇ · f r̂θ̂ + θ̂r̂ = r−2 r2 f θ̂ ,
123
∇ · f ẑr̂ + r̂ẑ = r−1 rf ẑ ,
which are readily established with the help of (5.14).
In this section the standard cylindrical variables (r, θ, z) have been used, being more
familiar than the local coordinates (r, θ, ϕ) of toroidal geometry (see Fig. 1.4), but now the
latter system will be restored, which is just a matter of replacing z by ϕ.
Calculating the divergence of 2 , as described above, we obtain
−∇ ·
2 =
1
4r2 ω
2
2 1 r pτ1 vϕ
rpτ1 vϕ vθ − vθ /r ϕ̂ .
θ̂ −
rωc
c
(5.20)
Because βt is small, the pressure gradient term in (3.98) is much smaller than the first righthand term and therefore jθ ≈ jϕ (Bθ /Bϕ ) ≈ bθ jϕ as bθ 1 and bϕ ≈ 1. Thus jθ jϕ ,
which for the electron fluid velocities requires that vθ vϕ . It follows that when (5.20) is
substituted into (5.15) the contribution to vr from the ϕ̂ = ẑ term is O(b2θ ) and can therefore
be omitted.
We shall set vr(2) = vD since the other contributions to the radial velocity of plasma diffusion are relatively small. Thus
vD =
2
1
r2 pτ1 vϕ
2
2
4ωc r
(5.21)
is the dominant term for the removal of plasma from tokamaks, so if it correctly represents
the physics involved, it should give estimates of about vD ∼ 0.1 m s−1 or larger, as observed in practice. In applying (5.21) to the electron fluid, we may adopt the approximation
|vϕ | ≈ |jϕ |/(ene ) = Ip /(πa2 ene ), where Ip is the total tokamak current. To obtain an estimate of the gradients we adopt the profile X(y) = X0 (1 − y)α and determine its average
gradient; this is X0 /(2a/3α), then with the choices αn = 1, αt = 2, δ = 3 (see (4.14), we
get δ/αn = α = 2. It follows that the gradients can be estimated by dividing the peak values
on the magnetic axis by a/3. From (5.21) we find that
5
vD ≈ 1.1
T̂e 2 Îp2
.
3
Zeff B 2 a7 n̄19
(5.22)
For JET some typical values are T̂e = 3 keV, B = 3 T, Îp = 4 MA, n̄19 = 2.2, Zeff = 2,
and a = 1.2 m. With these values we get vD ∼ 0.4 m s−1 , which falls in the observed range.
However, small changes in these choices make big differences to the derived value of vD , so
other tests of the theory are desirable. One of those tests will be a formula for the non-inductive
current generated by the radial flow of plasma towards the limiter.
5.1.4 Ambipolar flow
Ambipolar flow, i.e. the constraint jr = ene (vD i − vD e ) = 0, is not automatically secured by
(5.21), but requires that
pe τe1 2
pi τi1 2
veϕ =
(5.23)
2 viϕ .
2
e ωce
i ωci
124
5 Plasma flow and loop voltage
The collision intervals, τe1 and τi1 , are proportional to the corresponding values in
3
2.75 ×105 Te2
s,
τe =
ln Λ ne Zeff
1
3
1.67 ×107 A 2 Ti2
τi =
s,
ln Λ
Z 4 ni
(5.24)
where A = mi /mp is the particle mass number and Z is the ionization number. The relations
τe1 = 0.73τe and τi1 = 0.96τi follow from Braginskii’s (1965) expressions for the electron
and ion viscosity, viz. µe = 0.73 pe τe , µi = 0.96 piτi , which we shall adopt.
It follows from the above equations that ambipolarity imposes the constraint,
5
2
5
2 3 3 A 2 /Z ,
0.73 × 2.75 Te2 veϕ
= 0.96 × 167 Ti2 viϕ
which yields
5 3 3 G ≡ 2.61 ×10−3 Te /Ti 4 (Z 2 /A 4 ) ,
viϕ = −Gveϕ
(5.25)
the minus sign reflecting the fact that an axial electric field drives ions and electrons in opposite
directions.
It follows from (5.20) that
∇·
e2 +
i2
· ϕ̂ =
r2 pi τi1 1 r2 pe τe1 veϕ veθ /r +
viϕ viθ /r
r
ωce
ωci
.
(5.26)
The steady state form of (1.34) is
1 ∂ rvr vϕ + ∇ ·
r ∂r
· ϕ̂ = Fb · ϕ̂ ,
which applies to the plasma as a whole. We showed above that ∇ · 1 is parallel to r̂ and it
is easily shown from (5.21) that the term containing vr (= vD ) is negligible compared with
∇ · 2 · ϕ̂. Therefore
∇·
e2 +
i2
· ϕ̂ = Fb · ϕ̂ .
(5.27)
In the absence of a beam force the right-hand side of (5.27) is zero and for a fully ionized
hydrogen plasma we can use the ambipolar condition in (5.23) to reduce it to the condition
viθ /r
veθ /r
=
.
veϕ
viϕ
From this it follows that veθ /viθ = veϕ /viϕ , and then from (5.25)
viθ = −G0 veθ
5 G0 ≡ 2.61 ×10−3 Te /Ti 4 .
(5.28)
5.2 Particle transport
125
5.2 Particle transport
5.2.1 Particle diffusivity and the pinch velocity
From (5.21) applied to the electron fluid and with the approximation veϕ = −jϕ /ene , we
obtain
j 2 0.73me
ϕ
2
r
p
τ
.
(5.29)
vD =
e
e
4ne e4 Bϕ2 r2
ne
By (1.30) vD also satisfies
1 ∂ ∂ne
+
rne vD = Se (r) ,
∂t
r ∂r
(5.30)
in which the source term is proportional to the number of bound electrons at radius r.
The expression for vD is a sensitive function of the profile shapes, which makes the employment of empirical profiles an uncertain procedure. However, at least fair estimates can be
obtained from (5.29). For the present we shall ignore the influence of sawtooth oscillations on
vD and adopt the steady state profiles of (4.5), then by (5.24)1 with ln Λ = 17,
5
106 φ(y)T̂e 2 jϕ 2
−19
ne vD =
n̄
,
T̂
in
keV
,
n̄
19 = 10
e
e
2 a3 Z B 2
n̄19
eff ϕ
2 1 5/2 φ(y) = 2.07 αE (1 + δ)(δ − αn )/γ y 2 2 − (s + 2)y (1 − y)s−1 ,
where
and
δ = 32 αt ,
s = 5.5αt − 2αn − 2 ,
(5.31)
(5.32)
αE = 1 + 23 δ/(1 + αn ) .
The function φ(y)/ne has a negative region near r = a, which corresponds to an inward
flow from the limiter. As illustrated in Fig. 5.1, the magnitude of this pinch effect depends
on the number density profile, where the numbers on the graphs are the values of αn . The
100
1
αn
Arbitrary units, φ/n
e
80
60
40
20
1.5
1.75
2
0
−20
−40
0
0.2
0.4
0.6
0.8
1
r/a
Figure 5.1: Radial flow velocity showing the effect of pinch velocity
126
5 Plasma flow and loop voltage
steepest φ(y)/ne profiles have αn ≤ 1 and correspond to the largest pinch velocities. Profile
steepness is due to the ion source created by boundary neutrals penetrating the plasma until
they are ionized; this generates the reverse flow term in (5.3):
nvD = −Dn − (r/a)nV .
(5.33)
From (5.31) it is clear that the two-term empirical formula (5.33) can not correctly represent the underlying physics, but because (5.33) is commonly accepted, we shall find estimates
for D and V . By (4.6) equations (5.31) and (5.33) may be written,
2αn 1
1
γy 2 (1−y)αn −1 D−γy 2 (1−y)αn V , (5.34)
ne vD = 1018 n̄19 φ(y)G = 1019 n̄19
a
where
5
G≡
T̂e 2 Îp 2
.
2
3 a7 Z B 2
π n̄19
eff ϕ
(5.35)
Assuming that the diffusivity is largely determined by plasma conditions in 0 ≤ r/a ≤ 0.5
and that V results from a source term in the outer region, we shall replace D and V by the
average values:
0.25
0.25
aG
1
D=
φ(y) dy
y 2 (1 − y)αn −1 dy ,
(5.36)
20αn γ 0
0
and
G
V =
10γ
1
0.25
φ(y) dy
1
0.25
1
y 2 (1 − y)αn dy .
(5.37)
The values of D/(aG) and V /G for appropriate values of an and δ (cf. Tables 4.4 and 4.5) are
given in Table 5.1.
Values of D deduced from observations show an inverse dependence on ne and a strong
dependence on Te , in agreement with the above theory (ITER team, 1999, p. 2227). Also
observations show that there is a linear relationship between G and V as shown in Table 5.1.
The powers appearing in G make it difficult to find a typical value for this variable, and perhaps
Table 5.1: Particle diffusivity and pinch velocity
δ
an = 0.5
D/(aG)
V/G
an = 1.0
D/(aG)
V/G
an = 1.5
D/(aG)
V/G
1.5
2.0
2.5
3.0
3.5
4.0
2.60
0.20
7.44
0.75
15.75
1.62
28.14
2.69
45.37
3.78
68.60
4.75
0.20
0.02
1.02
0.28
2.58
0.79
4.99
1.49
8.34
2.23
12.77
2.91
0
0
0.13
0.01
0.57
0.35
1.33
0.84
2.43
1.44
3.89
2.03
5.2 Particle transport
127
explains why there is much uncertainty in evaluating particle diffusivity. We shall take the set
of values adopted in Section 5.1.3, which yield G = 0.037 m s−1 . So that at an = 0.75, δ = 3
it follows from Table 5.1 that D = 0.58 m2 s−1 , and V = 1.5 m s−1 , which are reasonable
values considering the heuristic definitions adopted in (5.36) and (5.37).
5.2.2 Particle confinement time
The volume average of (5.31) is
1
5
T̂e 2 jϕ 2
ne vD dy = ne vD = 106 FD 2 3
,
n̄19 a Zeff Bϕ2
0
(5.38)
where values of the shape factor FD are given in Table 5.2. Notice from the Table that for a
fixed temperature profile, the peaked number density profiles (small values of an , cf. Fig. 4.1)
are poorer at confining the plasma. For example, as the ratio αn = ne0 /ne − 1 decreases
from 2 to 1 at δ = 3, there is a sixfold increase in the radial particle flux.
We can now replace the estimate of vD deduced from (5.22) by a more accurate value.
Adopting the values of the variables listed in Section 5.1.3 and choosing an = 1, δ = 3, we
get FD = 6.23, and ne vD = 2.4 ×1019 m−2 s−1 and with n̄e = 2.2 ×1019 m−3 , we arrive at
vD ≈ 1 m s−1 , a realistic value for this average.
From (1.32) the particle confinement time is defined by
(5.39)
τp = 12 ne a2 rne vD r=a .
To reduce the boundary value uncertainty, described in Section 4.1.3 and resulting from the
use of empirical profiles, we define the ratio,
1 a
Ap ≡
ne vD r dr rne vD r=a
a 0
and show from (5.30) that in steady conditions
a
r
1 a
dr
S(r ) r dr
S(r) r dr .
Ap =
a 0
0
0
(5.40)
Then (5.39) can be written
τp =
γAp n̄e a
.
(1 + αn )ne vD (5.41)
Table 5.2: Values of the shape factor FD
αn \δ
0.5
1.0
1.5
2.0
1.0
0.55
0
-
1.5
1.75
0.39
0
-
2.0
4.27
1.35
0.31
0
2.5
8.89
3.15
1.11
0.26
3.0
16.73
6.23
2.55
0.96
3.5
29.17
11.13
4.89
2.18
4.0
47.98
18.54
8.46
4.10
128
5 Plasma flow and loop voltage
5.2.3 Plasma source term
To evaluate Ap in (5.40) we need an expression for S(r), which function depends on the
number density nn (r) of the neutral component of the working gas and on the distribution
of impurity atoms. Such profiles are difficult either to calculate or to measure. The Monte
Carlo technique of determining the history of many individual particles has been applied to
the problem and in typical conditions it yields profiles for S(r) that are sharply peaked near the
boundary (Hughes and Post 1978). When n̄e a exceeds 1019 m−3 by a few times, the neutrals
are unable to penetrate far into the plasma, consequently nn is quite small in the central region.
The ionization rate depends on Te and so with Te falling rapidly while nn increases rapidly as
the limiter is approached, a bell-shaped source term is the result.
A typical neutral particle enters the plasma, moves a short distance, is ionized and is then
driven back to the boundary layer B by the viscous force. It is neutralized at the limiter, recycled into the plasma and the process is repeated. At relatively high densities the convection
of mass and energy in tokamaks is thus largely confined to a boundary region and is therefore
sensitive to a variety of complicating influences, including neon-puffing (Section 1.5.2(ii)) and
turbulence. For this reason accurate results from (5.41) can not be expected in high density
regimes.
To illustrate the theory by a simple model, we shall adopt the empirical formula
Ap = 13 λ∗ /(a + λ∗ ) ,
(5.42)
where λ∗ is the ionizing mean-free-path for a boundary neutral. This formula is justified as
follows. With a uniform source distribution (5.40) yields Ap = 1/3, which corresponds to the
case when λ∗ /a is very large. On the other hand if λ∗ /a is small and it is assumed that S is a
delta function at r = r0 , then (5.40) gives Ap = 1 − r0 /a. It is easily verified that the average
perpendicular displacement of a particle moving a mean-free-path λ∗ from an isotropic source
on a plane wall is λ∗ /3, so Ap = λ∗ /3a. Thus (5.42) contains the limiting cases and gives a
physically reasonable variation between them.
From (5.31), (5.41), (5.42) and (1.11) we find that
τp =
Fp λ∗
n̄ 3 a4 R2 qa2 Zeff T̂e −5/2 ,
10 λ∗ + a 19
(5.43)
where Fp is tabulated as a function of δ and αn in Table 5.3.
Table 5.3: Values of the shape factor Fp
αn \δ
0.5
1.0
1.5
2.0
1.5
6.37
25.40
∞
-
2.0
2.62
7.34
28.93
∞
2.5
1.26
3.13
8.04
31.56
3.0
0.98
1.58
3.50
8.58
3.5
0.67
0.89
1.82
3.77
4.0
0.23
0.53
1.06
2.00
5.2 Particle transport
129
5.2.4 Observations of particle confinement
The apparently strong density-dependence of τp is offset by three terms in (5.43). First, Zeff is
nearly inversely proportional to n̄e , for example in JET Zeff ∝ n̄e−0.9 (Cordey et al. 1985 and
ITER team, 1999, p. 2225); secondly, λ∗ ∝ 1/n̄e since λ∗ n̄19 is of order unity; and thirdly,
since the density profile tends to flatten with increasing density, Fp decreases rapidly as n̄e
increases.
When (5.43) is applied to the averages of the variables listed by Pfeiffer and Waltz (1979)
(a = 0.18 m, R = 0.94 m, λ∗ ∼ a m, T̂e = 0.4 keV, Zeff = 4.9, qa = 5, δ = 3.75,
αn = 1.25, τEe = 0.005 s), the result is τp /τEe ∼ 2.5, which is about the expected ratio
(Hugill 1983). In TFR the experimental value of this ratio is between two and three (Equipe
TFR 1980). For the ten TFR experiments quoted by Pfeiffer and Waltz, (5.43) gives an average
τp /τEe ∼ 3. The theory seems to be satisfactory for small tokamaks.
However, when (5.43) is applied to a large tokamak like JET, determining the penetration
distance λ∗ involves some rather complicated boundary region physics (ITER team, 1999,
p. 2225). To obtain rough estimate we shall assume that λ∗ n̄19 ∼ 0.5 m−2 , then for the typical
4. A better
JET values, n̄19 = 2, qa = 3, T̂e = 3, Zeff = 2, τE = 0.5, we find that τp /τE ∼
representation involves a recycling coefficient R, and τp is replaced by τp∗ = τp 1 − R ,
but the details are beyond the scope of this text (see Wesson, 2004, p. 453).
In JET, Tanga et al. (1984) obtained a peak value of 1.18 s for τp at n̄19 = 1.1, but not all the
data required for (5.43) were given. Adopting estimates from related articles we may assume
that T̂e = 2.2 keV, qa ∼ 3, Zeff ∼ 7, R = 3 m, a = 1.05 m, δ ∼ 3.75, αn ∼ 0.75 and
λ∗ n̄19 ∼ 0.5 m−2 . With these values (5.43) gives τp = 1.20 s. Above n̄19 = 1.3, τp drops
quickly, reaching a value ∼ 0.5 s at n̄19 = 2. Table 5.3 indicates that this could be explained
by a steepening (an ↓) of the density profile and comparison of this table with Table 4.1,
indicates that τp is likely to be much more variable than τEe .
Another important observation in JET concerns the temporal behavior of n̄e as Ip is increased from zero up to a particular value, held constant for some seconds, and then allowed
to decay. It is found that n̄e follows the same time pattern as Ip , that is it increases up a ramp,
is approximately constant during the current plateau and falls away during the decline of the
discharge. Morgan et al. (1985) express this phenomenon in terms of the re-cycling coefficient
R, which is assumed to have a high value during the current rise and a low value during the
current fall, but no reason is found for this variation in R.
The phenomenon might be explained as follows. From (1.10) and (4.9) we find
δ = (2πa2 Bϕ /µ0 Rq0 )Ip−1 − 1, in which q0 can be treated as being a slowly varying function.
Thus in the JET experiment δ ≈ A/Ip − 1, where A is a constant. If Ip increases, δ decreases,
i.e. the temperature profile is broadened and by (5.43) and Table 5.3, τp is increased. Hence
n̄e also increases. Similarly n̄e and Ip decrease together. As Fp changes by a factor of ten or
more between δ = 1.5 and δ = 3.5, the changes in n̄e observed in JET could be explained in
this way.
In Section 1.5.2 (ii) we noted that τp was increased by neon puffing. Lazarus et al. (1985)
found that neon atoms were able to carry their electrons more deeply into the plasma before
complete ionization than the working gas neutrals and they are thus able to refuel the central
regions. This corresponds to an increase of λ∗ in (5.43) and hence gives a larger value of τp .
130
5 Plasma flow and loop voltage
5.3 The toroidal current and voltage relationship
5.3.1 Loop (induced) voltage
Figure 5.2 shows a cross-section of the tokamak torus on the plane Z = 0 (cf. Fig. 2.3) with
an inductive current IpIN circulating within the limiter surface. The magnetic flux through this
plane is given by
R
Bz R dR = LI IN ,
ΨR = −2π
0
where L is the inductance. Two values of the induced voltage drop, V = dΨR /dt, are important in tokamak theory; these are the surface loop voltage, Vs , where Ψ is at the limiter
position and the axial loop voltage, V , obtained by subtracting the flux passing through the
surface R0 < R < R2 shown in the figure. While Vs can be determined from observations of
the rate of change of the magnetic field just outside the torus, some theory is required to deduce V from Vs . For an account of how this is achieved for the JET tokamak see Christiansen
(1987).
From Ohm’s law (see (A.21)):
j = σ · E + v×B +
1 j × B − ∇pe ,
ene
where in the tokamak application the form using the conductivity tensor σ is more convenient.
By jr = 0, Br = 0 and the equilibrium equation, j × B = ∇(pe + pi ) , we obtain
where
j = σ · E∗ ,
(5.44)
1 E∗ = Eϕ + vr Bθ ϕ̂ − vr Bϕ θ̂ +
p r̂ .
ene i
(5.45)
Figure 5.2: The surface and axial loop voltages
5.3 The toroidal current and voltage relationship
131
On the minor axis of the torus, where vr = 0 and pi = 0, the electric field is Eϕ ϕ̂. Let V be
the voltage drop around the axial toroidal loop, then in steady conditions
Eϕ =
V
,
2πR
(5.46)
where 2πR is the average length around the torus.
Now (see (A.45))
σ = σ bb + σ∧ b × 11 + σ⊥ 11 − bb
b = ϕ̂ + S θ̂, S = Bθ /Bϕ 1 .
The conductivity σ∧ is negligible compared with σ and σ⊥ , so ignoring second-order terms
in S we have σ · ϕ̂ = σ ϕ̂ + S (σ − σ⊥ )θ̂, and σ · θ̂ = σ⊥ θ̂ + S (σ − σ⊥ )ϕ̂. Hence, changing
the notation for the radial velocity from vr to vD ,
jϕ = ϕ̂ · σ · E∗ = σ Eϕ + σ⊥ vD Bθ .
In strong magnetic fields σ⊥ ≈ 12 σ (see (A.19)), thus
Eϕ = V /(2πR) ,
jϕ = σ Eϕ + 12 vr Bθ
another form of which is
jϕ =
σ Vt
2πR
Vt ≡ V + πRvD Bθ ,
(5.47)
where Vt is the total voltage. With our choice of coordinates, Bθ > 0, and the outwards radial
flow increases the toroidal current.
From (4.77), (A.16), (A.19) and the tokamak approximation ln Λ = 17,
3
σt = gt 1.98 e2ne τe /me = 5.60 ×10−4 Te2 Zeff ,
where now we have included the banana trapping factor, gt , and adopted the approximation
gt = 0.35 explained in Section 4.5.3.
By (4.5) and (4.7),
3
3/2
(5.48)
σt = 3.16 ×10−4 αE Te 2 (1 − y)δ Zeff ,
so that
3/2
3
σt = 3.16 ×10−4 αE /(1 + δ) Te 2 Zeff .
(5.49)
Hence from (5.47),
σt Vt
Ip
=
j
=
ϕ
πa2
2πR
Vt ≡ V + V L ,
(5.50)
where
V L ≡ πR σt vD Bθ /σt ,
(5.51)
defines what we shall term the Lorentz voltage, which is ignored in accepted tokamak theory. (By “voltage” we shall always mean voltage drop, which has the opposite sign from the
standard definition.)
132
5 Plasma flow and loop voltage
5.3.2 Lorentz voltage
From (1.4), (1.13) and (4.8) it follows that
1
y ≡ (r/a)2 .
Bθ = Bθa y − 2 1 − (1 − y)δ+1
The function σ vD Bθ σ appearing in equation (5.51) defining the Lorentz voltage can be
evaluated from this equation with the help of (1.5), (5.31), (5.48), (5.49), and the profile shape
factor below.
4(1 + δ) 1
1
(5.52)
φ(y)(1 − y)δ−αn y − 2 1 − (1 − y)δ+1 dy ,
Fg ≡ √
10γ 0
where φ(y) is defined in (5.32); see Table 5.4. Thus the Lorentz voltage is
5
VL =
Fg RT̂e 2 Îp3
,
3 a8 Z B 2
2πn̄19
eff ϕ
(5.53)
where by (5.49) and (5.50),
3
Îp = 5.95 Vt
Îp in MA, T̂e in keV .
3
αE2 a2 T̂e 2
(1 + δ) R Zeff
(5.54)
The plasma current has two distinct components, an induced component ÎpIN , and a
Lorentz component, ÎpLR = (V L /V )ÎpIN . The ratio (V L /V ) can reach unity or greater.
For example, with the typical values Fg = 1, T̂e = 2.5 keV, n̄19 = 2, Zeff = 2,
Bϕ = 2.4 T, V = 0.54 V (see Table 4.6) and Îp = 3 MA, we get (V L /V ) ∼ 1.2, making the Lorentz current larger than the induced current. In this case, while the radial flow of
plasma is an obvious loss to the fusion objective, it is also a gain in generating most of the
current necessary to produce the poloidal magnetic field. However, as shall be explained in
Section 5.3.4, when V L /V ≥ 0.5, an instability occurs that modifies the current distribution
and sets a limit to this advantage.
Let L be the inductance associated with the plasma current Ip , then the magnetic energy
W = 12 LIp2 can decrease only by dissipation in a time of order L/Rc , where Rc is the toroidal
circuit resistance. In tokamaks this is a long time compared with the energy confinement
time. In the absence of a disruption giving thermal quench, it follows from (5.54) that the
total voltage Vt also changes relatively slowly. On the other hand the component V L is a
sensitive function of several variables and of the profile shape (see Table 5.4), therefore in
Table 5.4: Values of the shape factor Fg
αn \ δ
0.5
1.0
1.5
2.0
1.5
0.13
0.03
0
-
2.0
0.36
0.11
0.02
0
2.5
0.87
0.28
0.10
0.02
3.0
1.90
0.63
0.24
0.09
3.5
3.79
1.28
0.52
0.22
4.0
7.08
2.41
1.01
0.46
5.3 The toroidal current and voltage relationship
133
some circumstances a rapid interchange of values between V L and V can occur provided their
sum remains approximately constant.
From (5.53) we note that (V L /V ) ∝ Îp3 , so that an increase in the current increases
(V L /V ), which in turn further increases the current, and so on; this unstable phenomenon is
rather like that attributed to the bootstrap current in Section 3.4.3 and some of the observations
enlisted to support that assumed phenomenon also support the theory just described. An
interesting example of this will be given in Section 5.3.4.
5.3.3 Loop voltage instability
From Ip /IpIN = 1 + IpLR /IpIN = 1 + V L /V we have
J ≡ Îp /ÎpIN = 1 + V L /V ,
(5.55)
which with (5.53) yields the cubic equation
J 3 − J + 1 = 0 ,
where
=
2Fj V2 T̂e 7
3 B2
(Zeff )4 a2 R2 n̄19
ϕ
(5.56)
9/2
Fj = 16.8 Fg αE /(1 + δ)3 ,
(5.57)
(see Table 5.5 for Fj ).
In the range 0 < ≤ 1/6.75 the cubic (5.57) has two positive real roots and one negative real root. Outside this range the positive roots become complex and have no physical
relevance. In the limit → 0 one real root is J = 1, i.e. Ip = IpIN , while the second real
root is +∞ and leaps to −∞ when IpIN changes sign. Figure 5.3 shows the function J() for
the positive range of J and for the range (−4.5 < J < −3.0) on the right-hand side. The
Figure 5.3: Non-dimensional current J as function of 134
5 Plasma flow and loop voltage
Table 5.5: Values of the shape factor Fj
αn \δ
0.5
1.0
1.5
2.0
1.5
1.43
0.20
0
-
2.0
3.95
0.66
0.10
0
2.5
9.87
1.69
0.38
0.06
3.0
22.5
3.76
0.90
0.23
3.5
47.5
7.66
1.86
0.53
4.0
94.2
14.6
3.55
1.08
negative roots vary from −∞ at = 0 to −4.07 at = 0.075 and finally to −3 at the limit
→
point = 1/6.75; a section of this curve is shown as fg.
First consider the case ≤ 1/6.75. The discharge is unstable if a reduction in the voltage
→
causes an increase in the current and as ∝ V2 , the profile ac shown in Fig. 5.3 is stable,
→
while the profile ce is unstable. We therefore anticipate that initially, as increases probably
with increasing temperature, the loop voltage will remain nearly constant, but once point c is
reached at ∼ 0.148 the Lorentz component of the current begins to increase. In the absence
of a temperature collapse it takes many seconds for the total current to change (see (A.98));
hence the induced component of the current starts to drop to accommodate the increasing
Lorentz current and the voltage V required to drive it falls. The instability is now switched-on,
→
falls and the upper curve ce is followed. The more drops in value, the larger J becomes,
which further reduces V and so on, until J reaches infinity. In the case that 1/6.75 the
discharge is stable.
To check that the instability is likely in typical tokamak operational conditions, we apply
the values adopted in Section 5.3.2, except for a small reduction in the average temperature
from 2.5 to 2.39 keV. With the choices δ = 3 and αn = 1 we find ≈ 0.147, very close to
the point c in Fig. 5.3, and in this case the discharge is potentially unstable. On the other hand
at T̂e = 2.5 keV, either ∼ 0.2 and the discharge is stable, or a fall in V has reduced to less than the critical value.
Let V = V0 at the start of the instability, i.e. at J ≈ 1, then as the total current cannot
change rapidly, it follows from (5.55) that at J = ∞, where V has fallen to zero, V L = V0 . At
this stage it is possible for V to change sign, which indicates that the Lorentz component has
exceeded the original total current. The function , which is independent of the sign of V ,
begins to increase again and the solution for J now follows the negative branch of the cubic,
→
fg, until it reaches the limit point g, where J = −3. From (5.55), V = (V + V L )/J, which
becomes V = − 31 (V +V L ) at g. Thus V = − 14 V L = − 14 V0 , so that the surface voltage falls
to minus a quarter of its initial value before the voltage collapse. It will then slowly increase
to zero as the radial flow driving the Lorentz current fades over a particle confinement time τp .
Finally, when the Lorentz current vanishes, the loop voltage quickly resumes its initial value.
Figure 5.4 shows an example of a voltage instability in the TFTR tokamak (Wesson, 2004,
p. 604 and Zarnstorff, et al. 1988). The features described in the previous paragraph are
evident in the experimental curve in the figure — the negative voltage is about one-quarter of
the initial value and the recovery of the voltage is as expected. Of course, our interpretation
of the total current is different from that given in the figure — to the ohmic and NBI currents,
we add the Lorentz current, not the bootstrap current. The Lorentz current will be small
5.3 The toroidal current and voltage relationship
135
Figure 5.4: Time dependence of the loop voltage in TFTR
(The solid line is the experimental result obtained from magnetic field measurements,
and the other curves are based on various assumptions: (a) induced current only,
(b) NBI-driven and induced currents, and (c) bootstrap and NBI-driven and induced currents)
near the axis where the radial velocity vD is small and it will increase with vD until close
to the limiter, where it is possible that vD will change sign (see (5.3)) and produce a local
reversed Lorentz current. Challis et al. (1993) have observed the same phenomenon in JET
where the loop voltage falls to a negative value near zero for a period of 3 s before recovery;
they conclude that careful density control is required to establish what they term “bootstrap
dominated discharges”.
The phenomenon shown in Fig. 5.4 is taken as evidence for the existence of the bootstrap
current but such a current depends on the continuous function ∂ne /∂r (see (3.88)) and it is
difficult to imagine (even if it were to exist) how it could generate the discontinuous profile
imposed on curve c in Fig. 5.4.
5.3.4 Lorentz current
From (5.55), 1/J = IpIN /Ip and therefore the non-inductive component generated by the radial
flow velocity is
IpLR = 1 − 1/J Ip ,
(5.58)
where J is the solution of the cubic equation (5.56). Figure 5.5 shows the relationship between
the parameter defined in (5.57) and the ratio of the non-inductive component of the current
to the total current. In the widely accepted theory of tokamak plasmas it is the ‘bootstrap’
current that would feature in a similar ratio and observations in which the bootstrap current is
assumed to be the cause of the phenomena are actually evidence for the presence of a Lorentz
current and relevant to the present theory. For example, since the Lorentz current is driven by
the radial flow of the plasma, one infers that currents identified as ‘bootstrap’ are more likely
to occur when there is a strong central particle source.
136
5 Plasma flow and loop voltage
1
LR
Ip
I
p
0.8
unstable
0.6
0.4
stable
0.2
0
0
0.05
0.1
∆
0.15
Figure 5.5: The non-inductive proportion of the total current
The first point to note is that according to the theory of Section 5.3.3, the upper part of
the curve in Fig. 5.5, i.e. that part for the range J > 1.5, corresponds an unstable regime and
therefore values of J greater than 1.5 are short-lived, i.e. we would not expect the ratio of
the non-inductive component of the current to exceed one third of the total current in a stable
tokamak plasma. Support for this result is given in Wesson’s (2004) text (p. 637):
“JET ELM-free H-mode plasmas normally have a bootstrap current fraction up to
∼ 30%. In a series of ICRF heated discharges at I MA/2.8 T, ELM-free H-modes
with βp values of up to 2 were produced in which the calculated bootstrap current
reached values of 700 ± 150 kA. Although the duration of the ELM-free phase was
only 2-3 s and the current profile did not reach a steady state, the broadening of the
current profile under the influence of the bootstrap current1 was sufficient to stabilize
sawtooth oscillations. ... This regime has parallels with experimental observations
from high βp -plasmas in other large tokamaks, but was the first case of a bootstrap
dominated plasma without a strong central particle source.”
It is of interest to consider how the ‘bootstrap’ current instability develops when the unstable regime is first encountered. The integrand of the expression for Fg in (5.52) is peaked near
the minor axis (cf. Fig. 5.1, where the average value for an of about 0.7 obtained from Tables
4.4 and 4.5 will produce a much more peaked profile than that shown for an = 1), which
means that the current instability will tend first to collapse the current profile in the region
of the minor axis. This will create what is termed a ‘hollow’ current profile, similar to that
illustrated in Fig. 4.10. Some observations are reported in which there is no current flowing
along the minor axis (Wesson 2004, p. 639) and which may result from the current instability.
Kamada et al. (1994) in describing observations at high βp in the JF-60U tokamak, remark
that “in the case of bootstrap dominating discharges it is difficult to maintain a peaked current
profile in the steady state — the current profile tends to be hollow”.
1 We shall use ‘bootstrap’ to mean ‘Lorentz’ when reporting observations of the related phenomena.
5.3 The toroidal current and voltage relationship
137
5.3.5 Determining Zeff from current and loop voltage
An interesting application of loop voltage theory arises in one method of determining Zeff .
From (5.50) and (5.55),
2πRjϕ = gt σ (V + V L ) = gt Jσ V ,
where we have restored the Spitzer conductivity σ in order to make the banana trapping
∗
factor gt explicit. Let Zeff
be the value of Zeff that would be determined from observations of
Ip and V if the role of the Lorentz voltage V L and the trapping factor gt were ignored. Such
∗
measurements were originally made in the belief that Zeff
∝ 1/σ would give the required
value of Zeff , whereas these measurements actually delivered the value of Zeff /(gt J), so that
∗
Zeff = gt JZeff
(1 ≤ J < 1.5) ,
(5.59)
∗
the assumed value. In (5.59) we have
in which Zeff is the ‘true’ value of Z-effective and Zeff
restricted the range of J to the stable region shown in Fig. 5.3.
As noted in Section 1.2.4, Zeff can be determined from visible bremsstrahlung, a value we
v
∗
v
. In the 1970s discrepancies found between Zeff
and Zeff
were attributed
shall denote by Zeff
to experimental error, but when experiments in JET started in 1983 they revealed a substantial
∗
v
difference between Zeff
and Zeff
, especially at high currents and low densities. In Fig. 5.6 the
∗
values of Zeff derived from Spitzer resistivity (squares) and Zeff from neoclassical resistivity
v
(asterisks) are plotted against Zeff
with each square and asterisk corresponding to a single
instant in a JET pulse. The conclusion was that the estimate of Zeff obtained by assuming
neoclassical resistivity was in reasonable agreement with the value determined from visible
bremsstrahlung. (For ‘neoclassical resistivity’ we have the rather simpler particle trapping
factor, described in Section 2.4.4 and simplified in Section 4.5.3 to gt = 0.35.) From Fig. 5.6
we find that (5.59) is represented by 5.5 ≈ 16gt J, i.e. gt J = 0.34, so the measurement of
∗
Zeff
yields the ‘correct’ value of Zeff provided gt = 0.35 and J ≈ 1. Higher temperatures will
∗
.
switch J to its maximum stable value of 1.5 and in this case (5.59) becomes Zeff = 0.525Zeff
Figure 5.6: Values of Z-effective in JET (from Christiansen 1987)
138
5 Plasma flow and loop voltage
5.4 Toroidal velocities
Neutral beam injection in a toroidal direction is a major heating process and it also has a
stabilizing effect on certain instabilities. It is therefore important to know how long the angular
momentum generated in the plasma persists relative to the other confinement times. The
toroidal and poloidal rotational speeds of the ion and electron fluids are determined by a
balance of electric and viscous forces and — with NBI — the beam frictional force. Unless
two opposing beams are employed, toroidal speeds much higher than those in OH-plasmas
will result. Poloidal rotation can be induced by angled injection above or below the magnetic
axis.
5.4.1 Role of second-order viscosity
Two methods of determining the momentum diffusion time, τϕ were described in Section
1.3.4:
(1) switch off the beam, find the slowing down time τsϕ for the toroidal speed and assume that τϕ ≈ τsϕ , or (2) calculate the deposited beam torque from a model for the slowing
down of fast ions, use Doppler shifts to determine vϕ , and then find τϕ∗ ≈ τϕ from (1.36).
The viscous force acting on unit length of the surface of a plasma cylinder of radius r
centered on the minor axis, as illustrated in Fig. 5.7, is 2πrM, where
r
∇ · 2 · ϕ̂ r dr ,
(5.60)
M=
0
since it was shown in Section 5.1.2 that ∇ · 1 is parallel to r̂ and cannot contribute to the
tangential force. The drag force reduces the angular momentum Rvϕ , a process that can be
interpreted as being due to the lateral diffusion of the momentum away from the minor axis.
Let χϕ be the angular momentum diffusivity, then the velocity of diffusion is
vm = −χϕ vϕ /(vϕ )
= d/dr .
(5.61)
Figure 5.7: Lateral diffusion of angular momentum
5.4 Toroidal velocities
139
The rate at which momentum is lost is therefore πr2 vϕ vm = −2πrM, so that
χϕ = 2M/ r vϕ
.
(5.62)
From vϕ = i viϕ + e veϕ and the ambipolar condition (5.25) we get
vϕ =
me n i mi
me
hjϕ ,
G − 1 jϕ = 3.79
e n e me
e
(5.63)
where
h≡
1.25
1
4.79 A0.25 Z 0.5 Te /Ti
−1 .
3.79
The factor h is ≈ 1 in a hydrogen plasma and ≈ 0.8 in a deuterium plasma.
Equation (5.63) relates the toroidal velocity and the toroidal current density in an OH
plasma. In a hydrogen plasma with the profiles of Section 4.1.1 it yields the average value
vϕ =
410(1 + δ) Îp
.
γ(δ − αn ) n̄19 a2
(5.64)
Much higher speeds will be reached on the minor axis, e.g. with JET operating at Îp = 5 MA
and ne = 2 ×1019 m−3 , the average toroidal speed is ≈ 1 km s−1 , with a peak value perhaps
five times larger.
5.4.2 Angular momentum diffusivity
Equation (5.23) allows us to write (5.26) as
−∇·
2 viθ /r
veθ /r
1 r2 pe τe1 veϕ
−
.
e2 + i2 · ϕ̂ =
r
|ωce |
veϕ
viϕ
(5.65)
Ignoring viscosity for the moment and assuming that η = η11, we find from (5.5) that
v⊥ =
1
1 E − ηj × b −
∇p × b ,
Bϕ
ωc
which holds for each fluid. Hence from b ≈ ϕ̂ + S θ̂, where S = Bθ /Bϕ , the θ̂-components
of each equation are:
ene Er
p
− e ,
ene veθ − S veϕ = −
Bϕ
Bϕ
(5.66)
140
and
5 Plasma flow and loop voltage
Zeni Er
p
+ i .
Zeni viθ − S viϕ = −
Bϕ
Bϕ
(5.67)
When subtracted these equations yield the familiar equilibrium condition,
jθ − S jϕ =
1 p + pi ≈ − ene veθ − S veϕ ,
Bϕ e
(5.68)
and it follows from (5.66) to (5.68) that
Er ≈ pi /ene Bϕ ,
viθ ≈ S viϕ .
From the above equations and (5.28) we find
veθ /r
viθ /r
−1 pe + pi
−
= .
veϕ
viϕ
veϕ rene Bϕ
Substituting this relation into (5.65) and writing2 τe1 = 0.73τe , by (5.60) we arrive at
pe + pi
me pe τe M = 0.73 r2 2 2 veϕ
.
(5.69)
e Bϕ
rene
From (A.16) with ln Λ = 17, the approximate relation veϕ ≈ −jϕ /ene , and equations
(4.5) and (4.58), we can write (5.69) as
7
M = −374 Φ(y)R
me T̂e 2 jϕ ,
e a2 Bϕ2 n̄19 Zeff
(5.70)
where
Φ = (aE7/2 /γ)(1 + δ)(δ − αn )(αt − 1)(αt + αn ) y 2 (1 − y)(5αt −αn −3) .
Evaluating (vϕ ) by differentiating (5.63) and substituting the result and (5.70) into (5.62),
we obtain
7
RT̂e 2
χϕ = 98.7 φ(y) 2 2
,
(5.71)
ha Bϕ n̄19 Zeff
where φ(y) = (aE7/2 /γδ)(δ − αn )(αt − 1)(αt + αn )y(1 − y)(3.5αt −αn −2) .
The average diffusivity follows upon integrating over 0 < y ≤ 1:
7
χϕ =
2Fϕ RT̂e 2
,
ha2 Bϕ2 Zeff n̄19
(5.72)
where values of the profile shape factor Fϕ are given in Table 5.6.
2 This relation is taken from Braginskii’s (1965) account of transport theory, the equations of which are collected
in Woods (2004), p. 162.
5.4 Toroidal velocities
141
Table 5.6: Values of the shape factor Fm
αn \δ
0.5
1.0
1.5
δ/3
1.5
0
0
0
0
2.0
4.87
3.31
2.12
4.22
2.5
10.45
6.62
4.60
7.56
3.0
17.39
10.33
7.10
10.33
3.5
26.09
14.66
9.81
12.66
4.0
36.90
19.78
12.85
14.63
To deduce the momentum confinement time we note from the definition of diffusivity that
the time required for the angular momentum to diffuse a distance r from the minor axis is
τ (r) = r2 /χϕ . To avoid the singularity at y = 1 we shall approximate by replacing χϕ by its
average value, then averaging τ (r) over 0 ≤ r ≤ a, obtain the momentum confinement time,
τϕ = 13 a2 /χϕ =
ha4 Bϕ2 n̄19 Zeff
7
6Fϕ RT̂e 2
.
(5.73)
5.4.3 Comparison of theory and observation
Table 5.6 reveals a difficulty that arises when inferring scaling laws for τϕ from observations.
For example, variations in δ from 2 to 3.5 at αn = 1 due to small changes in boundary
conditions, or impurity levels, or changing between modes, could alter τϕ by a factor of ∼ 4.4.
Table 5.7: Experimental and theoretical values of τϕ in TFTR
n̄19
T̂e (keV)
R
Bϕ
δ
Fϕ
Zeff
τϕ (exp)
(ms)
τϕ (th)
(ms)
3.91
3.38
4.37
4.73
3.19
3.73
2.00
1.67
1.41
1.79
1.65
2.13
1.77
1.54
1.94
1.40
1.82
1.98
1.98
2.14
1.98
2.01
2.38
3.48
3.84
3.84
3.84
4.70
4.70
3.88
3.85
3.85
3.96
2.69
3.42
3.33
4.46
4.22
4.78
5.11
14.5
8.7
12.3
11.9
16.2
15.4
17.1
18.0
3.4
3.8
3.1
2.9
4.0
3.5
5.7
6.6
134
89
130
76
97
80
65
78
152
98
97
60
81
95
28
55
A data base of 44 TFTR discharges containing nearly all the measurements3 required
to apply (5.73) was compiled by Scott et al. (1985). Two parameters not given were the
density profile parameter αn and Zeff . The ratio Te0 /Te was given, which enabled values
0.8
of the parameter δ to be found and hence by (4.14), αn ≈ δ/3. For Zeff the value 10/n̄19
3 Incomplete reporting is an unsatisfactory feature of far too many fusion research papers, which makes it very
difficult for ‘extramural’ researchers to check their theories.
142
References
was adopted — a rough estimate based on graphs for OH-plasmas given by Efthimion et al.
(1984). To compile Table 5.7 we have taken the first and every sixth entry from the given
data base, which was arranged in order of magnitude of the toroidal velocity at the end of the
beam pulse. The column τϕ (exp) was determined from (1.35) and entails assumptions about
the deposited beam torque and the profiles ne (r) and vϕ (r). The theoretical values of τϕ were
calculated from (5.73). In view of the many assumptions involved, theory and experiment are
in satisfactory agreement.
It is also found in TFTR that when the minor radius a is reduced from 0.81 m to 0.58 m,
τϕ falls from ∼ 83 ms to ∼ 30 ms; this implies that τϕ ∝ a3 if it can be assumed that all the
other variables, including profile shapes, are unchanged. But there are wide error bars in the
τϕ measurements and in view of the uncertainties, the a4 law of (5.73) is certainly not ruled
out.
References
(See page 25 for the reference notation.)
Braginskii, S.I. (1965). Transport processes in a plasma. Reviews of plasma physics (ed. M.A.
Leontovich), Vol. 1, p. 205, Consultants Bureau, New York.
Challis, C.D. et al. (1993). Nuclear Fusion, 33, 1097.
Christiansen, J.P. (1987). J. Comp. Phys., 73(1), 85.
Cordey, J.P. et al. (1985). V, Pt 1, 26.
Efthimion, P.C. et al. II, Paper A-I-2.
Equipe TFR (1980). Nuclear Fusion, 20(10), 1227–45.
Gentle, K.W. et al. (1984). Plasma Physics and Controlled Fusion, 26(12A), 1407.
Hughes, M.H. & Post, D.E. (1978). J. comp. Phys., 28, 43.
Hugill, J. (1983). Nuclear Fusion, 23(3), 331–73.
ITER team, (1999). Nuclear Fusion, 39(12), pp. 2167, 2225, 2227, 2229.
Kadomtsev, B.B. (1975). Sov. J. Plasma Phys., 1, 389.
Kamada, Y. et al. (1994). Nuclear Fusion, 34(12), 1605.
Lazarus, E.A. et al. (1985). Nuclear Fusion, 25(2), 135.
Morgan, P.D. et al. (1985). V, Pt II, 535.
Pfeiffer, W. & Waltz, R.E. (1979). Nuclear Fusion, 19, 51.
Scott, S.D. et al. (1985). Private communication, PPL, Princeton, N.J.
Tanga, A. et al. (1984), JET-P (84)09.
Wesson, J.A. (2004). Tokamaks, 3rd. edn. Oxford University Press.
Woods, L.C. (2004). Physics of plasmas. Wiley-VCH Verlag GmbH & Co., KGaA, Weinheim.
Zarnstorff, M.C. et al. (1988). Physical Review Letters, 60, 1306.
6 Thermal Instabilities
Tokamaks are beset by several macroscopic instabilities, the principal of which are sawtooth oscillations and major disruptions. Also packets of magnetic waves often occur in the
Bθ -field during the rise of the plasma current and sometimes during the main discharge; these
are termed Mirnov oscillations (Mirnov and Semenov, 1971) and a similar response known
as a fishbone oscillation (see Fig. 2.11) sometimes occurs in high energy NBI plasmas. An
important H-mode instability known as an ‘edge localized mode’ (ELM) was briefly described
at the end of Section 4.3.2. These are all essentially thermal disturbances that modulate the
electric current and their explanations follow from the complex nature of second-order heat
flux in tokamak magnetic fields.
Sawtooth events were described in Section 4.3.1. They occur so commonly that their
presence is accepted as a signal that the tokamak is operating normally. The current in the
central region of the plasma climbs slowly in a ramp phase between repeated sudden collapses
described as ‘internal’ or ‘minor disruptions’. The rise time of the sawtooth offers a good test
of the theory developed in Chapters 4 and 5. Major or ‘external’ disruptions in the temperature
lead to an abrupt loss of confinement, a sudden drop in the plasma current and to an end to the
discharge. They usually occur when for a given energy input poloidal beta exceeds a critical
value and as βp is proportional to the number density, they pose a threat to the economic
viability of tokamak reactors.
Major disruptions involve substantial energy transfers and hence involve sudden and very
large increases in the thermal diffusivity. It has not yet been accepted what mechanism causes
this increase in thermal diffusivity, but the oscillations that are observed accompanying the
instabilities indicate that a resistive MHD instability known as a tearing mode may be implicated. However, the theory based on this instability — known as the ‘reconnection model’
— does not agree with observations since the predicted growth rate is much too slow. The
explanation to be presented in Section 6.2 depends on the unstable nature of the second-order,
cross-field transport of energy. However, in some major disruptions MHD instabilities can
trigger a fast thermal quench, so in Section 6.3.2 an account is given of the dominant MHD
instability known as a ‘ballooning’ mode. The temperature collapses in a two-stage process;
in the first stage an MHD instability is switched on and flattens the temperature profile on the
tearing mode time scale, which then triggers a much faster thermal instability.
One other curious thermal phenomenon that perhaps should not be classed as an instability
is called a snake. This consists of a low temperature, relatively dense, rope-like plasma column
that follows magnetic surfaces around the torus at rational values of q, preferably on q = 1
and sometimes q = 3/2. It forms during pellet injection and remains stable for relatively
long times. The density within a snake has been observed to be as much as twice that of the
surrounding plasma.
Theory of Tokamak Transport: New Aspects for Nuclear Fusion Reactor Design. Leslie Colin Woods
Copyright © 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
ISBN: 3-527-40625-5
144
6 Thermal Instabilities
6.1 Sawtooth oscillations
6.1.1 Some observations of temperature and density sawteeth
In Section 4.3.1 we gave a brief account of a typical sawtooth oscillation and defined the
inversion radius rs at which the family of temperature profiles cross over (see Fig. 4.4). Figure
6.1 shows the temperature evolution measured in JET by Tubbing et al. (1985) at five different
positions. There are two normal sawteeth with an inversion radius at rs ∼ 0.5a. The figure
also shows a ‘partial’ sawtooth collapse at 47.45 s followed by a packet of large amplitude
oscillations. All of the dependent variables respond to the instability, although the temperature
fluctuations are the dominate feature. The sawtooth density fluctuations shown in Fig. 6.2 were
obtained during the rise time of a JET discharge (Campbell et al. 1985) and in Fig. 6.3 there is
an example of sawteeth in q0 and n0 plotted against t/τs , where τs is the sawtooth period (see
Wesson, 2004, p. 373). Notice that the percentage changes in q0 and n0 at the sudden jumps
are ∼ 4% and ∼ −9%, whereas from Fig. 6.1 the temperature changes are much larger, being
of the order of 40%.
Sawteeth usually occur in a regular pattern with a single period and amplitude, dependent
on plasma conditions and tokamak dimensions. In Fig. 6.1 the sawtooth period is 23 ms,
while in Fig. 6.2 it is ∼ 60 ms. An interesting feature sometimes visible is the presence of
sharp spikes at either or both end of the steep collapse, a phenomenon clearly revealed in a
study of internal disruptions in the TFR tokamak by Dubois et al. (1983), who remarked that
a complete theory should be able to predict it.
Why the sawtooth oscillation should occur at all has not yet been explained; but two
instabilities are required to drive the process, one to explain the abrupt collapse and the other to
explain the ramp phase. We shall assume that (4.40) and (4.41) are qualitatively correct during
the ramp phase except in the neighborhood of the peak temperature at α = αm . Consider
Figure 6.1: Temperature sawtooth oscillations in JET
6.1 Sawtooth oscillations
145
Figure 6.2: (a) Density sawteeth occurring during the rise of a JET discharge (b) The corresponding temperature profile
a temperature profile with α just larger than its minimum α0 . A small steepening of the
profile, i.e. an increase in α, reduces χe , especially in the neighborhood of y = 0.5 (r ∼
0.7a); hence the central region loses less heat by conduction and assuming that other losses
and gains of heat are unaltered, the profile is further steepened. This steepening process
continues until α reaches αm , when a different instability switches on and usually generates a
bunch of thermal waves that herald a minor disruption to complete the cycle. To analyze this
terminating collapse phase it is necessary first to understand the physics of the ramp phase.
6.1.2 Kadomtsev’s model of sawtooth oscillations
Most attempts to explain sawtooth oscillations are derived from Kadomtsev’s model, which
can be summarized as follows (Kadomtsev 1975). Assume that during the ramp phase of the
sawtooth the current gradually peaks on the magnetic axis,
then by
q0 = 2Bϕ /(µ0 jϕ0 R), q0 will fall and eventually be less than unity (see Fig. 6.3). Since
qa > 1, there will be a resonant surface (Section A.24), q = 1, in the plasma enclosing the
axis. Within this surface the exchange instability generates turbulence, which flattens the
temperature and current profiles. The resistive tearing mode instability allows a magnetic
island to form around a point on the q = 1 surface, which grows rapidly. As it does so, the
island pushes aside the original magnetic axis and its surrounding turbulent region, eventually
restoring a new axial plasma with flatter profiles and with q > 1. Ohmic heating then causes
the current to peak again on the axis and the cycle is repeated. The tearing mode magnetic
island on the q = 1 surface has poloidal and toroidal wave numbers m = 1, n = 1 and it
is believed that this island is responsible for the m = 1, n = 1 oscillatory MHD modes that
have frequently been observed as precursors to internal disruptions.
146
6 Thermal Instabilities
This model has several shortcomings, the most serious of which is that it yields a collapse
time, τc , for the disruption orders of magnitude longer than observed. An estimate for τc is
given by w2 /ξ⊥ , where w is the width of the island and ξ⊥ is the magnetic diffusivity, which
−3/2 2 −1
in tokamaks has the value (see (A.98)) ξ⊥ ≈ 4.4 ×10−2 T̂e
m s . For example with
w = 1 cm and T̂e = 3 keV we get τc ∼ 10 ms; for JET Kadomtsev’s theory gives τc ≥ 10 ms,
whereas from observations τc lies in the range 50 − 200 µs (Campbell et al. 1985).
A mechanism for greatly speeding up the island formation is essential if the MHD model
is to be reconciled with observation. Another problem is that the model gives no precise
specification for the occurrence of a disruption. Furthermore, the precursor oscillations that
ought to be concomitant with a magnetic island are sometimes absent as is the case with the
large amplitude oscillations known as ‘giant’ or ‘compound’ sawteeth found in JET in some
operating conditions (Campbell et a1. 1985). ‘Single’ or isolated sawteeth have also been
observed lacking precursor signals.
There is also the problem posed by the existence of ‘double’ sawteeth with a longer and
sometimes erratic period and a larger amplitude (Pfeiffer 1985). Figure 6.4 (from Campbell et
al. 1986) shows an example of a double sawtooth at ab with a partial collapse midway up the
ramp phase. To explain their superposition on a regular series of smaller and ‘normal’ sawteeth
it is necessary to complicate Kadomtsev’s model by requiring a hollow current profile, giving
two q = 1 surfaces. The smaller sawtooth jumps are then explained by a partial magnetic
reconnection that does not reach the axis.
Using numerical methods, Dubois et al. (1983) found two other defects in the reconnection
model, namely that it could not reproduce the sharp spikes on the disruption profile and that at
the outset of the disruption the island size was much too small. They did find good agreement
with observations of sawteeth in TFR by adopting a phenomenological model in which the
temperature flattening was due to the propagation of a turbulent region outwards from the
q = 1 surface, but they gave no physical justification for this process.
Figure 6.3: Time dependence of the averaged behavior of q0 and n0 in TEXTOR
6.1 Sawtooth oscillations
147
6.1.3 Sawtooth ramp phase
In Section 4.3.1 we assumed the existence of temperature sawtooth oscillations and proceeded
to determine the range of the ‘steepness’ parameter α (see Fig. 4.4). As described in Section 4.3.1, its minimum α0 (−1) was fixed by the condition that no heat source exists at the
inversion radius rs since for α < α0 the theory implies that such a source would be necessary.
The maximum αm depended on the nature of the thermal boundary condition at the outer
edge of the discharge — perfect thermal conductivity gave the L-mode with αm = 1 and a
perfect heat reservoir gave the H-mode with αm = 3. These results are based on a very simple
model that reproduces only the essential features of a temperature sawtooth oscillation. Associated oscillations in variables like ne , jϕ and q are assumed to depend on the temperature
oscillation, but not included in this model.
The method we shall adopt is to hold the total energy of the plasma constant and then
derive an equation for the rate of change of the energy U within the inversion radius, r = rs ,
thus yielding a relationship between α̇ = dα/dt and α. The theory therefore applies to
sawteeth within the surface r = rs ; the value of rs depends on the profile shape — here we
shall assume that it is known.
Because the collapse phase is so rapid, we may take the rise time from α = α0 to α = αm
as being equal to the sawtooth period τs . To obtain an expression for τs in OH-plasmas we
start from the integral of (1.25) taken over the range (0, rs ):
rs
d rs 3
kB ne Te r dr + rQe rs =
jϕ Eϕ r dr ,
(6.1)
dt 0 2
0
where we have ignored convection, radiation and energy transfer between the electrons and
ions.
Let
rs
ys
2
3
(6.2)
kB ne Te dy
y ≡ (r/a)2 ,
Us ≡
2 kB ne Te r dr = 0.75a
0
0
where in our model the inversion radius is at rs = 12 a, i.e. ys = 14 . Using (4.42) to evaluate
the integral we get
a0 = 0.2145, b0 = 0.0218 ;
Us (α) = 6.75a2 n̄e kB Te (a0 − b0 α)/(5 − α)
Figure 6.4: Compound sawtooth activity
148
6 Thermal Instabilities
therefore, with Te held constant,
−2
dUs
= 0.712a2n̄e kB Te 5 − α
α̇ .
dt
(6.3)
By (5.48), (5.50) and Eϕ = V /2πR, the right-hand side of (6.1) can be written
a2 Vt V
POH =
8π 2 R2
ys
0
η −1 dy ,
η −1 = σ = 9.03 ×10−4 gt Te3/2 /Zeff ,
where Vt is the sum of the loop and Lorentz voltages and gt (y) is the electron trapping factor.
In Section 4.5.3 it was found that for the range 0 ≤ y ≤ 1 the constant value gt = 0.35
(see (4.79)) is a good approximation. The trapped fraction has a smaller average over 0 ≤
y ≤ 0.25 and by the method of Section 4.5.3 we find that for this range gt = 0.44 is a good
approximation over 2 < δ < 4. Hence by (4.37), (4.39) and (4.45),
POH = 3.87 ×1030
a2 Vt V
3
kB Te 2
4π 2 R2 Zeff
6
5−α
32
ys
1 + âys + b̂ys2
.
(6.4)
By (4.46) and (4.47),
rQer rs =
where
ψ1 =
1
3
30k1 (2me ) 2 akB Te 2 ψ1 + βp ψ2 ,
2
2
2
(5 − α) µ0 e
R qa
6
5−α
12
(6.5)
(5 − α)ys2.5 (qa /q0 ) α + 2(1 − α)ys
2
× 1 − αys − (1 − α)ys2 (1 + âys + b̂ys2 )
× 1.5α − â + ĉys + (q0 /qa ) − 1.5α + â − ĉ ys2 ,
and
ψ2 = 3
6
5−α
32
(6.6)
(5 − α)ys2.5 (q0 /qa )(1 − α)(1 − ys ) α + 2(1 − α)ys
1 × 1 − αys − (1 − α)ys2 2 1 + âys + b̂ys2 .
(6.7)
Shortly we will require the function
1 −1
.
ψ0 ≡ 6(5 − α) 2 ys 1 + âys + b̂ys2
In the above equations â, b̂, ĉ, and (qa /q0 ) are the functions of α given in Table 4.3.
(6.8)
6.1 Sawtooth oscillations
149
6.1.4 Sawtooth period
Now substitute (6.3) to (6.5) into (6.1) to obtain the following equation for α̇:
τ ∗ α̇ = 75 ξψ0 − ψ1 − βp ψ2 ,
where
τ∗ ≡
µ0 e2 n̄e aR2 qa
1
1 ,
(2me ) 2 kB Te 2
(6.9)
(6.10)
and
ξ = 0.26 Vt V aqa /Zeff = 0.26 V2 Jaqa /Zeff
(J ≡ Vt /V ).
(6.11)
The function ξ is a dimensionless measure of the input power and V L is the Lorentz voltage
defined in (5.53):
5
Fg RT̂e 2 Îp3
VL =
.
3 a8 Z B 2
2πn̄19
eff ϕ
(6.12)
Notice from (4.51) that τ ∗ is proportional to the electron energy confinement time, τEe .
From (6.9) the sawtooth period is given by
τs = τ ∗ Fs (ξ, βp ) ms ,
where
Fs (ξ, βp ) = 13.3
(6.13)
αm
α0
|Fs (ξ, βp )| > 0
dα
ξψ0 − ψ1 − βp ψ2
(6.14)
This function is shown in Table 6.1 for the L-mode, i.e. for the range α0 = −1 to αm = 1;
the constraint on Fs ≥ 0 sets a lower bound for ξ, which is why values for ξ < 0.20 are not
included in the table. The sawtooth period is:
1
τs = 1.61 ×10−2 Fs (ξ, βp )n̄19 aR2 qa T̂e − 2 ms .
(6.15)
Let τs (α0 → αm ) denote the time taken for the temperature profile to climb up the indicated α-range and then take τs (−1 → 1) to be the sawtooth period for the L-mode; this
Table 6.1: The function Fs (ξ, βp )
βp \ξ
0.20
0.22
0.26
0.30
0.4
0.5
0.6
0.8
1.0
0.0
0.2
0.4
0.6
0.8
1.0
230
259
294
345
423
560
163
178
197
222
256
305
115
123
132
144
158
176
91
97
102
109
117
127
61
64
67
70
73
77
47
48
50
51
53
55
38
38
39
41
42
43
28
28
28
29
29
30
21
22
22
22
23
23
150
6 Thermal Instabilities
range is based on the assumption of perfect thermal conductivity at the plasma boundary
(Section 4.3.1). A small increase in the α range results in a very large increase in the sawtooth
period, for example with the notation Fs (ξ, βp ; α0 → αm ), we get Fs (0.22, 0.2; −1 → 1) =
178, as shown in Table 6.1, whereas Fs (0.22, 0.2; −1 → 1.134) = 363, or Fs (0.22, 0.2; 1 →
1.134) = 184. An increase of αm to 1.135 extends the range over the zero of the denominator
of the integrand in (6.15): Fs (0.22, 0.2; −1 → 1.135) = −323, which as shall be explained
in Section 6.2.2, initiates the collapse phase of the sawtooth. The flattening of the profile in
this phase increases χe and more heat is lost, α is further reduced and so on, until a stage is
reached where at least for a minor disruption, the collapse is halted because further collapse
would require a heat source to exist on the inversion surface (see Section 4.3.1). Also, by
(6.12) and J = 1 + V L /V the ratio J is larger at the temperature peak of the sawtooth cycle
than at the low temperature of the bottom of the cycle and as Vt remains fairly constant (see
Section 5.3.2), the growth in Vt2 /J can usually increase ξ sufficiently to change the sign of α̇.
If it fails to do this a major disruption is possible.
For an H-mode to occur we find that Fs (ξ, βp ; −1 → 3) is negative unless much larger
values are adopted for the power input parameter ξ, for example to obtain a positive value for
τs (−1 → 3) at βp = 0.4 it necessary to have ξ > 0.44, whereas in the L-mode ξ > 0.19 is
sufficient. That there is a lower bound on the heating required for the H-mode to operate has
been found from observations and is described in Wesson (2004, p. 187).
The theory is easily extended to NBI plasmas. Suppose that the fraction x of the total
beam power Pb is absorbed within the inversion radius, then the sawtooth period is given by
(6.15) with the modification that in calculating Fs , ξψ0 is replaced by (ξψ0 + ξb ψb ), where
ψb = 0.2(5 − α)2 and
3
ξb = 2.49 ×10−2 xP̂b Rqa {aT̂e 2 }
P̂b in MW, T̂e in keV .
(6.16)
Notice from (6.16) and Table 6.1 that the sawtooth period gets longer and longer with
increasing βp and with reducing ξ. It follows from (6.15) that sawteeth will have a long
period at a point P in the neighborhood of the line
ξ = ξ0 = ψ1 /ψ0 + βp (ψ2 /ψ0 ) ,
(6.17)
points on which correspond to τs = ∞. However, if P falls below the line into the region
where τs is negative, a major disruption is likely as shall be explained in Section 6.2.2.
When the tokamak conditions admit very large values of τs , the description “sawtooth
free” is applied to the regime and the sawtooth is said to be “stabilized”; there are several
ways in which this stabilization can be obtained (Wesson, 2004, p. 627).
6.1.5 Theory v. observation for the sawtooth period
To test the formula for τs requires a range of data not usually provided in papers dealing with
sawteeth; in particular, because Fs is a rapidly changing function of ξ at small values of ξ (see
Table 6.1) and ξ ∝ Vt V , accurate values are required for these two voltages. Some observations made in JET (Wesson and Gowers 1986) are given in Table 6.2. In applying the theory
allowance for the elongated cross-section of JET was made by replacing the minor radius a
1
by (ab) 2 and values for Zeff were calculated from the empirical formula (see Section 1.2.4)
6.2 Disruptions
151
Table 6.2: Some sawtooth periods in the JET tokamak at B = 3.4 T.
Pulse
No.
4768
4770
4774
4771
4718
4728
4786
4791
4782
a
(m)
1.17
1.16
1.14
1.14
1.14
1.14
1.21
1.21
1.21
b/a
1.59
1.51
1.33
1.28
1.33
1.34
1.51
1.51
1.52
Va
(V)
0.52
0.91
0.63
0.66
0.77
0.84
0.77
0.76
0.75
T̂e0
(keV)
2.80
2.00
3.70
2.70
2.85
4.20
4.35
3.00
3.15
n̄19
qcyl
ξ
0.79
1.55
1.24
1.86
2.43
1.42
1.78
3.01
2.82
11.7
9.9
4.9
4.6
3.3
3.3
3.2
3.2
3.2
0.50
2.07
0.24
0.26
0.32
0.24
0.26
0.40
0.38
Fs τs (exp) τs (th)
(ms) (ms)
47
36
83
10
39
34
134
51
112
115
84
150
83
90
98
134
63
76
115
99
86
61
138
94
66
173
93
0.9 0.7
Zeff = 11/(n̄19
qa ). Unfortunately there was insufficient information available to enable
the Lorentz voltage to be calculated, which means that we would expect to overestimate Fs ,
especially at low values of n̄19 .
As a random mix of ‘simple’ and ‘compound’ sawteeth is present, the values of τs (exp)
given in Table 6.2 are average values subject to errors of 20% or more (cf. the last two pulses
in similar conditions). The first entry in Table 6.2 is at the lowest density and by (6.12) should
have an appreciable Lorentz voltage. The largest stable value of J is 1.5 (see Fig. 5.3) and if
we assume that this applies to the first entry, ξ is increased to 0.75, which reduces Fs to 30
and brings τs (th) down to 53 ms, an improvement. The almost random nature of the observed
values is smothered by averaging, which provides a better test; the average value of τs (th) is
88 ms and that of τs (exp) is 86 ms. In view of the approximations involved in arriving at each
of the last two columns, a 2% discrepancy is good agreement.
6.2 Disruptions
6.2.1 Description of major disruptions
For a given value of the plasma current there is a critical average number density n̄e∗ above
which tokamak discharges are unstable. This instability develops very rapidly with a sudden
cooling of the central plasma and a flattening of the current profile, usually followed by a
slower decay of the plasma current to zero; it is termed a major disruption. Occasionally the
second phase does not occur and the discharge recovers. Major disruptions present a severe
engineering problem for tokamaks because the rapid fall in plasma current generates large
electromagnetic forces in the structure and if the released energy is unevenly deposited, a
large amount of wall material can be vaporized; in addition, by limiting the attainable plasma
density major disruptions may restrict the plasma beta to uneconomic values. The expression
minor disruption applies to the collapse phase of a sawtooth oscillation and will be considered
separately in Section 6.2.3, although it has much in common with major disruptions.
Considerable attention has been paid to the behavior of the plasma prior to the temperature collapse, since it is thought that this should reveal the cause of the instability. First
152
6 Thermal Instabilities
Figure 6.5: A typical Hugill diagram
there is a change in the plasma conditions, such as an increase in density or a change in the
plasma current, which it is conjectured moves the magnetoplasma state into an unstable domain. Perturbations are characterized by poloidal wave numbers m = 1, 2, . . . and toroidal
wave numbers n = 1, 2, . . .. If a sawtooth oscillation is present it usually stops. Growing m = 2 magnetic oscillations generally appear, but other low-m modes have also been
observed. In medium-sized tokamaks these oscillations persist for about 10 ms before the
sudden temperature collapse, which takes about 1 ms or less. In JET the total radiation from
the plasma increases, reaching ∼ 100% of the input power at the instant of the disruption
(Wesson et al. 1985).
During the collapse the current profile is flattened and the system inductance results in a
brief negative voltage pulse of between 10 to 100 times the applied voltage; this phenomenon
and the post-disruptive phase was studied by the TFR Group (1985), who found that the central
plasma is rapidly lost to the peripheral regions and that convection and conduction are the
dominant energy transport paths during the final decay of the current.
Believing that there was no satisfactory theory of disruptions, tokamak physicists adopted
empirical expressions for the density limit. For qa greater than about three, Murakami, Callen,
and Berry (1976) obtained n̄e∗ ∼ k1019 Bϕ /Rqa , where the parameter k is in the range 10–
20. Disruptions also occur if qa is too small, e.g. if qa ≤ 2. Fielding et al. (1977) incorporated these observations in a stability diagram plotted in the (1/qa , M ) plane, where
M ≡ n̄19 R/Bϕ is termed the ‘Murakami parameter’. A typical plot known as a ‘Hugill diagram’ (Hugill 1983) appears in Fig. 6.5. The stable region lies to the left of the marginal
stability curves. Notice that the operating regime is reduced for contaminated OH-plasmas
and extended beyond the Zeff = 1 boundary when auxiliary heating is employed.
It is unsatisfactory to express stability criteria in terms of a dimensional parameter, so it
is not surprising that when points where disruptions have occurred are plotted on the Hugill
6.2 Disruptions
153
diagram the scatter around the assumed marginal stability curves is considerable. There is
experimental evidence that temperature is one of the variables missing from the abscissa in
Fig. 6.5 since the Murakami density limit can be exceeded with low temperature profiles (von
Geoler 1975).
Notice that the density limit curve in Fig. 6.5 is approximated by the straight line
1/qa ≈ M/15, which by the relation (1.10), viz. qa = 5a2 Bϕ /(RÎp ), can be expressed
as n̄20 ≈ Îp /πa2 , in which form it is known as the Greenwald (1988) limit. The elongation
κ also plays a role replacing n̄20 by n̄20 /κ. The upper limit in Fig. 6.5 is qa ≈ 2, so that for
stability observations yield the conditions
n̄20 ≤ Îp /πa2 ,
qa > 2 .
(6.18)
It is widely believed that the disruptive instability is due to a tearing mode instability,
similar to that described in Section 6.1.1 for the collapse phase of the sawtooth oscillation. In
major disruptions it is assumed that a (m = 2/n = 1) magnetic island grows until it interacts,
either with islands of different helicity (e.g. m = 3/n = 2) (Carreras et al. 1979; Biskamp
and Welter 1979), or with the limiter or the cold gas region (Sykes and Wesson 1980). In the
first variant it is suggested that where the islands overlap the field lines become stochastic and
that this flattens the local temperature profile through parallel heat transport. However, unlike
direct interactions with the plasma boundary region, the mode-coupling process is unable to
provide a sufficiently rapid fall in the average temperature; and neither version of the magnetic
island mechanism attempts to explain the origin of the density threshold for the onset of the
disruption.
The density limit has also been attributed to a thermal instability in the boundary plasma
(Vershkov and Mirnov 1974; Gibson 1976; Ohyabu 1979; Ashby and Hughes 1981; Wesson
et al. 1985). Impurity radiation occurs mainly in the edge region of the plasma, where the
temperatures are ∼ 100 eV or less and it increases sharply if the temperature of the region
falls. It is therefore argued that if ohmic or NBI heating is only marginally able to balance the
radiation losses, the temperature profile is unstable and contracts inwards. This phenomenon
destabilizes the current profile and hence triggers the magnetic island growth, which leads to
a disruption as described above (Roberts 1983; Wesson et al. 1985). The density limit can
be estimated from the radiation balance condition, which in some operating conditions is in
rough agreement with experiment. Some support for radiation being the trigger for disruptions
is given by the observation that n̄e∗ is reduced as plasma impurities are increased (Axon et al.
1980).
However, there are several difficulties with the idea that radiation triggers the instability. First, even with the relatively clean plasma obtained in a gettered torus, density limits are obtained that are evidently related to the limits found in contaminated OH-plasmas
(cf. Fig. 6.5). Secondly, NBI is found to increase n̄e∗ above its OH-value more in the most
contaminated plasmas despite the fact that such plasmas radiate a higher fraction of the injected power. Thirdly, the relatively simple empirical relation between n̄e∗ and qa suggests
that a complex atomic process like radiation is not involved. Fourthly, although contractions
of the current channel sometimes occur, this is not an invariable precursor to a disruption (Engelhardt et al. 1979). Another contrary observation in the Pulsator tokamak (von Geoler et al.
1979) is that near the density limit, a.c.-modulation of the discharge can trigger a disruption.
154
6 Thermal Instabilities
6.2.2 Precursor waves
The same conditions presage the collapse phase for both minor and major disruptions, although these precursors can take quite different forms as illustrated in Fig. 6.6 for sawtooth
collapses in JET, which shows the temperature variation in three minor disruptions within
the inversion radius. Although the precursor ‘waves’ in Figs 6.6(a) and 6.6(c) look very
different, they can be attributed to the same mechanism, terminating at different stages. In
Fig. 6.6(b) the precursor waves have a much longer period, although the final collapse occurs at much the same speed as in the other two examples. It is observed that there are
two sorts of precursor waves; those with periods typically ∼ 100 µs and those with periods typically ∼ 2 ms (cf. Fig 6.6(b)). The short-period waves are of the type appearing in
Fig. 6.6(a), in which example the period of 130 µs is several times the electron collision interval, τe . There are insufficient details for an accurate calculation of τe , but with the typical
JET values, T̂e = 2.5, n̄19 = 2.5, Zeff = 2.5, we get τe = 37 µs, making the wave period
in Fig. 6.6(a) ∼ 3.5τe .
For a reason that will become clear shortly, we shall call the short period waves ‘thermal
waves’. A likely mechanism for the generation of thermal waves is as follows. Consider the
oscillating temperature profile shown in Fig. 6.7, which is controlled by the L-mode ‘perfect’
conducting boundary condition at the edge c. This does not allow heat to flow inwards, yet the
tendency for the temperature profile to steepen as described in Section 6.1.1, produces an ini-
Figure 6.6: Three types of sawtooth collapse: (a) and (c) thermal instability driven oscillations,
(b) poloidal magnetic field driven oscillations
6.2 Disruptions
155
tial steepening beginning at time t say, when the profile has reached the maximum steepness
that can be attained without an inward flux of heat from the boundary. In the simple model
of Section 4.2 this occurs at α = 1. This boundary constraint distorts the temperature profile,
which continues to steepen within the inversion radius and sets in motion the restoring mechanism described in Section 2.5.4 as thermal pumping. The response time is the time it takes to
transfer the electron thermal energy from the local plasma element to neighboring elements,
namely the electron collision interval τe . Hence, at time t + τe the temperature profile will
have peaked upwards, at which stage thermal pumping starts to return the profile to its initial
shape, in fact it overshoots reaching a minimum at about t + 3τe , and again the combination of
temperature gradient and electron fluid shear act to restore the profile to its equilibrium shape;
the wave period is therefore ∼ 4τe . The essential feature is the time lag between changes in
the temperature profile and the response of the thermal conductivity, which generates an overstable oscillation, so the amplitude of the wave continues to increase until a final plunge takes
the temperature down to a level from which immediate recovery is not energetically possible.
With a minor disruption the boundary condition described in Section 4.3.1 checks a further
fall and the temperature profile begins to climb relatively slowly up the ramp phase of the
sawtooth and the cycle is repeated. However, with a major disruptions, as will be explained
in Section 6.2.3, the rate of heating falls below what is required to balance the thermal losses
and the collapse continues on to shut-down.
The longer period waves are most often seen in observations of sawteeth collapse in OH
plasmas (e.g. see Wesson 2004, p. 352). They are called Mirnov oscillations (Mirnov and
Semenov, 1971) and they often occur during the rise of the current and sometimes during
the main discharge. They do not necessarily presage a disruption, for example in Fig. 6.1
they follow the sawtooth collapse at t = 47.45, whereas in Figs 6.2(a) and 6.6(b) they appear
to have triggered the minor disruptions. The Mirnov period in each of these discharges is
τM ≈ 3.6 ms.
Mirnov oscillations are usually attributed to tearing mode instabilities (see Section A.24)
and while there may be such instabilities present, how they could generate significant variations in the temperature profile such as shown in Fig. 6.6(b), on time-scales of a few ms
remains unexplained (see Wesson, 2004, ch. 7). Rather than MHD instabilities, substantial
and rapid fluctuations in the thermal diffusivity must be an essential element in any explana-
Figure 6.7: Precursor oscillations, cf. Fig. 2.8
156
6 Thermal Instabilities
tory theory; this would generate waves in the local temperature distribution and thence modify
the toroidal current leading to the Mirnov oscillations that are observed in Bθ (Wesson, 2004,
p. 352).
A thermal accounting of these waves is as follows. Suppose that a fluctuation in the temperature profile increases the steepness parameter δ, then from Table 5.4 and (5.53) the result is
a sharp increase in the Lorentz voltage V L , which will steepen the current density profile. We
now have the configuration represented in Fig. 6.7 (also see Fig. 2.9(a)). The increased shear
in the electron fluid now enhances the thermal conductivity and the temperature profile falls,
the dependent variables following the oscillatory pattern already described in Section 2.5.4.
The essentially new feature is the time lag τ∗ say, between changes in the temperature profile and the response of the Lorentz voltage. Once the increased voltage has accelerated the
electrons and changed the current density profile, the time lag between changes in jϕ and the
response of the thermal conductivity is τe as before. Referring to Fig. 2.9, the sequence of
changes is now in the clockwise direction:
τe
τ∗
τe
τ∗
(b) −→
(c) −→
(d) −→
(a) ,
(a) −→
(6.19)
and the wave period is 2(τ∗ + 2τe ).
The Lorentz voltage is generated by the radial velocity vD of the plasma crossing the
Bθ field, so τ∗ is determined by the time taken to change the diffusion velocity vD , which
can be traced to the time it takes to satisfy the ambipolar constraint on vD given in (5.23).
This constraint involves both τe and τi , which by (5.24) are related by τi = 57τe (Ti /Te )3/2
for a hydrogen plasma. It follows that for ambipolarity to reach equilibrium, we need
τ∗ = τi (ignoring possible temperature differences). Using the JET values adopted above
for the thermal waves, we get τi ≈ 3.5 ms, which is sufficiently close to observed values to
support the model.
Returning to the collapse phase of a minor disruption triggered by Mirnov waves, we see
that it occurs in the second stage of (6.19), i.e. it takes a time τe to return to the midpoint of the
oscillation limits and then a further time ∼ τe to collapse completely. In Fig. 6.6(a) there is
a partial recovery into a Mirnov half-wave (stage (a)−→(b)), but in Fig. 6.6(c) the half-wave
is only just visible. With major disruptions the same trigger for the collapse applies and it
remains to explain why the collapse continues on until shut-down.
6.2.3 Collapse phase
A theory of tokamak disruptions can be based on (6.11), (6.12) and (6.17):
5
ξ = 0.26 Vt (Vt − V L ) aqa /Zeff ,
and
ξ = ξ0 (α) = ψ1 /ψ0 + βp (ψ2 /ψ0 ) ,
VL =
Fg RT̂e 2 Îp3
,
3 a8 Z B 2
2πn̄19
eff ϕ
(6.20)
(6.21)
where Fg is given in Table 5.4. From the table we see that if δ increases from 2 to 3, Fg
increases at least fourfold, which by (6.20) may result in an appreciable reduction in ξ.
Equation (6.21) is the line in the (βp , ξ) plane where τs changes from +∞ to −∞ as
ξ passes from +ξ0 (α) to −ξ0 (α). Since ξ is proportional to the ohmic power supplied to
6.2 Disruptions
157
the tokamak, the interpretation follows that if ξ falls below the line ξ = ξ0 , there may be
insufficient power to restore the plasma to its pre-collapsed state, an interpretation supported
by the observed extension of the stable region obtained by the inclusion of auxiliary heating
(Fig. 6.5). We have used the conditional mood here because as the profile falls the parameter
δ (or α) may decrease sufficiently to increase ξ above ξ0 . One has to remember the delay time
τe between changes in the temperature and the thermal diffusivity. If the collapse does not
occur on a first passage of ξ across the line singularity, on a later transit there will be a larger
variation in the temperature profile shape and a major collapse is more likely and when this
happens the collapse time will be τc ≈ kτe , where k is 2 or more.
Figure 6.8 shows the lines ξ = ξ0 (α) in the (βp , ξ) plane for the values of α indicated. The
stable domain lies on the positive side of the lines and there is no doubt about the unstable
region below α = 0. There is an intermediate region (not labeled) from which a partial
recovery is possible. Consider the vertical line a → b taken to be a trajectory of a discharge
D close to a ‘qa ’ type disruption. As D crosses the line α = 1, its profile parameter α will
fall towards α = 0.75, the next line drawn in the figure. If this occurs quickly enough, D will
remain in a stable region and will be able to make a partial recovery until with increasing α
the line ξ = ξ0 is again encountered and a second collapse is triggered. A disruption from
which a temporary recovery is made is called a ‘soft’ disruption, a phenomenon that has been
observed in JET (Tubbing et al. 1985).
There are two other important qualitative conclusions from the theory that agree with
observations of JET behavior. The first is that with ξ just a little larger than ξ0 , so that a major
disruption is imminent, by (6.15) the sawtooth period τs becomes too large to be observed,
0.4
0.35
ξ
βN
Stable
α = 0
limit
0.3
a
0.25
0.50
low qa limit
0.2
0.75
α = 1.0
0.15
− 0.5
b
0.1
− 0.75
0.05
Unstable
0
−0.05
0
− 1.0
0.5
1
1.5
β
p
Figure 6.8: Tokamak stability diagram
2
2.5
3
3.5
158
6 Thermal Instabilities
so these oscillations appear to stop. Secondly, the collapse commences from the maximum
value of α, which in the L-mode corresponds to zero thermal conductivity at the boundary
(Section 4.3.1). Thus just prior to a disruption all the heat lost by the plasma is transferred by
radiation.
In the L-mode, the line α = 1 represents the most peaked temperature profile and the
section 0 < βp < 1 corresponds to the low qa limit identified in Fig. 6.5. On this line,
ξ0 = 0.19 and therefore by (6.20) the stability constraint reads 0.26V Vt aqa /Zeff > 0.19,
which yields
0.73Zeff
Vt = JV ) .
qa >
(6.22)
aV Vt
From Table 4.4, which lists values for the L-mode in JET, we find the typical values
1
Zeff = 2, V = 0.54, a = (Aπ) 2 ≈ 1.45, so in this case, giving J its maximum stable
value of 1.5, we obtain the stability condition qa > 2.3. By drawing trajectories terminating
disruptions in the Hugill diagram (see Fig. 7.8.3 of Wesson (2004)) it has been found from
experiments in JET that stability requires qa > 2.2.
Referring to Fig. 6.8 we see that except near α = 0.75, most of the line α = 0 corresponds
to the density limit. The equation of this line is βp = 10.7ξ − 0.06 and therefore the stability
condition is βp < 10.7ξ − 0.06. Adopting the normalized beta defined in (1.7), we obtain
βN < 56.3
Vt (Vt − V L ) a2
RZeff
− 0.53
a
.
Rqa
(6.23)
From the average JET values used in the previous paragraph and the choice J = 1, we obtain
the condition βN < 5.7, which is a little larger than the empirical MHD limit of 3.5 quoted
in equation (1.7). On the other hand values of βN as large as 6.7 have been obtained in the
DIII-D tokamak (see Fig. 38, p. 2309 of ITER team (1999)) However, as we shall show in
Section 6.3.3, both the MHD and thermal instabilities are involved in disruptions with the
MHD instability initiating the thermal collapse.
The observation cited in Section 6.2.1 that low temperature plasmas exceed the Murakami
limit follows from the theory because the limit is really determined by βp which is proportional to Te . Finally we note that Zeff in (6.22) and (6.23) reduces the region of stability, in
agreement with the empirical result visible in the Hugill diagram of Fig. 6.5.
6.3 MHD instabilities
6.3.1 Ideal and resistive instabilities
The disruptive instabilities with which we have been concerned so far can be explained as consequences of the second-order transport of heat in toroidal geometry. The received treatments
of MHD instabilities have failed principally because the growth rates deduced for disruptions
have been orders of magnitude too slow. However, observations indicate that MHD instabilities play some role since m = 1, n = 1 oscillatory MHD modes are often observed as
precursors to disruptions.
6.3 MHD instabilities
159
Strait (1994) in reviewing the stability problem of high beta tokamak plasmas, observes
that: “Stability at high beta . . . is an important requirement for a compact, economically attractive fusion reactor. It is also important in present large tokamak experiments, where the best
performance is now often limited by instabilities rather than by energy transport.” However,
these performance limitations are not necessarily separable.
By scaling the maximum stable βt with the normalized plasma current, Troyon et al.
(1984) found the formulae given in (1.7) for maximum beta that has been confirmed in many
observations. By (1.10) this limit can be written:
βN, max = 20βp a/Rqa = 20βt Rqa /a = β̂t Bϕ /Îp = Cβ ,
(6.24)
where β̂t is the toroidal beta expressed as a percentage and Cβ is a parameter that depends
on the instability for which the equilibria are optimized. The value Cβ = 3.5 given in (1.7)
represents a serious restriction and much research has been concerned with finding ways of
increasing this number, which is found, inter alia, to depend on the steepness of the temperature and number density profiles. There is no precise beta limit because stability depends
on profile shapes and choice of mode numbers. The limit in (6.24), while based in theory,
involves such choices and therefore is described as being ‘semi-empirical’; it is quite different
from the slightly less constraining thermal limit obtained in (6.23).
There are three MHD instabilities that are particularly relevant in the precursor stage of
thermal disruptions; these are kink instabilities with the toroidal mode number, n = 1, ballooning instabilities with large values of n and the resistive, tearing mode instability in which
the field lines “reconnect”, creating stationary loops that slowly fade away. The kink and
ballooning instabilities are based on ideal MHD equations since the inclusion of resistivity
in these equations makes relatively little difference to the value of Cβ . However, once the
MHD instability is triggered, the rate at which it modifies the temperature profile and thereby
switches on the fast thermal instability depends on the magnetic diffusivity. We therefore need
the fastest growth rate for the resistive tearing mode instability, which is (e.g. see Woods 2004,
p. 118),
1
1
γmax = 1/τA τd 2 , τA = /vA , vA = B/ µ0 2 , τd = 2 /ξ ,
where is the shear length, |B/B |, vA is the Alfvén speed and ξ = η/µ0 is the magnetic
1
diffusivity. The time scale for the evolution of the resistive tearing mode is τm = τA τd 2 .
Thus
0.25 0.75 −0.5 1.5
a ms .
(6.25)
T̂e B
τm = 1.81n̄19
The kink instability is stabilized by the presence of a conducting wall outside the plasma
boundary within the radial distance 1.5a and in most tokamaks this stabilization occurs, so the
rest of this section will be concerned with the ballooning instability.
6.3.2 Theory of the ballooning stability limit
Figure 9 in Section A.24 illustrates the basic principle involved in the ballooning instability —
where magnetic field lines are concave towards the plasma, the configuration is destabilizing
and where they are convex it is stabilizing. The major curvature of the torus is therefore
stabilizing on the side closest to the major axis (see Fig. 1.1) and destabilizing on the outer
160
6 Thermal Instabilities
Figure 6.9: Ballooning mode stability diagram (Figure 6.14.1 (p.341) from “Tokamaks 3/e” by
Wesson, J (2004))
side. At low values of βt the overall effect is stabilizing provided q ≥ 1, but with higher
pressures and therefore higher gradients, the perturbations tend to accumulate in the outer,
destabilizing regions. They liberate more energy than required to bend the field lines in the
neighborhood of a flute instability, and what is termed a “ballooning” mode results (Connor,
Hastie and Taylor 1978).
The theory is algebraically complicated, so we shall be content with an approximate account that involves the essential features. The basic variables are:
s=
r dq
,
q dr
α=−
2µ0 Rq 2 dp
,
B 2 dr
(6.26)
where s is the (stabilizing) magnetic shear and α is the (destabilizing), normalized pressure
gradient. The (α, s) plane stability diagram, obtained numerically for circular flux surfaces
and a parabola current distribution, is shown in Fig. 6.9, which is reproduced by permission
of Oxford University Press. Almost all the stable region (1) is bounded by the straight line
s = 1.8α drawn in the figure. The stable region (2) can be accessed only by adopting particular
pressure and q profiles in a tokamak of unusual cross section.
Because Fig. 6.9 applies to a parabolic current distribution (δ = 1 in (4.5)3 ) and we wish
to generalize our formula for the beta limit to other values of δ, we shall generalize the linear
relation between s and α to s = 1.8kα, and then assign k by the condition that our formula
yields the Troyon number 3.5 at δ = 1.
From (4.5)3 and (4.8):
−1
, qa = q0 (1 + δ) . (6.27)
jϕ = (1 + δ)jϕ (1 − y)δ , q = qa y 1 − (1 − y)δ+1
The straight line approximation to marginal stability shown in Fig. 6.9 yields the relation
−
Bϕ2 r dq
dp
= 0.32k
,
dr
µ0 R q 3 dr
(6.28)
6.3 MHD instabilities
161
Table 6.3: Determining βN, max
δ
G
qa
i
βN
4i
0
0
1
0.5
0
2.00
0.5
0.16
1.40
0.73
2.28
2.92
1
0.31
1.75
0.92
3.53
3.66
2
0.57
2.31
1.23
4.93
4.93
3
0.80
2.73
1.46
5.85
5.85
4
1.01
3.05
1.65
6.60
6.60
5
1.20
3.29
1.81
7.27
7.23
6
1.37
3.47
1.94
7.91
7.78
7
1.54
3.56
2.06
8.53
8.25
8
1.69
3.70
2.17
9.14
8.69
9
1.84
3.77
2.35
9.76
9.40
10
1.98
3.83
2.43
10.33
9.72
the stable region being where the left-hand side of (6.28) is smaller than its right-hand side.
The average pressure is
a
a
dp 2
1
2
r dr ,
pr dr = − 2
p = 2
a 0
a 0 dr
and evaluating this with the help of (6.27) we find the marginal pressure
pm = G(δ)
Bϕ2 a
,
2µ0 Rqa2
(6.29)
where G, adjusted to get the Troyon number 3.5 at δ = 1, is
1
y −3/2 1 − (1 + δy)(1 − y)δ 1 − (1 − y)δ+1 dy .
G = 0.595
(6.30)
0
For qa we shall adopt (4.86) with qs = 1, i.e. qa = 4 1 − 0.75δ+1 . From βt = 2µ0 p/Bϕ2 ,
(6.29) and (1.7) we arrive at the maximum value of normalized beta,
βN,max = 20G(δ)/qa ,
βp ≤ G(δ) R/a .
or
(6.31)
The function G(δ) is given in Table 6.3.
The fourth row of Table 6.3 gives the values of the internal inductance calculated from
1
1
2
1 − (1 − y)δ+1 dy ,
(6.32)
i =
y
0
which follows from (2.46) and the distribution
2
Bθ2 = Bθa
1 − (1 − y)δ+1
2
/y ,
derived from (1.4) and (6.27)1. The external inductance is much smaller than i , which can
therefore be treated as being the circuit inductance. Notice the strong dependence of i on the
steepness of the current profile.
Excepting at the ends of the range of δ in Table 6.3, there is close agreement between 4 i
and βN, max , which can be traced to the similarity between the integrals that arise in calculating
i and G (Lao et al. 1992); there is no physical significance in this coincidence and although
the use of i and sometimes ( i − 0.5), as abscissa and βN as ordinate in stability diagrams is
common (see Fig. 6.10); the parameter G has more merit.
162
6 Thermal Instabilities
Figure 6.10: Some achieved values of βN in DIII-D L-mode discharges (open circles) compared
with the ballooning mode limit (solid circles). The dashed line is 4i and the curved line is the
theoretical limit from (6.31).
6.3.3 Some observations of limiting betas
Figure 6.10 shows observations of some peak values of βN and some explicitly calculated
values of βN, max for ideal MHD ballooning limits plotted against i (Lao et al. 1990). The
theoretical limit (6.31) is not distinguishable from 4 i except at the limits of the i range.
There is good agreement between theory and observation. From (5.53) and Table 5.4 we
find that as profiles steepen, V L increases and therefore the threshold in (6.23) and (6.24) for
activating the thermal instability is reduced. Thus profile steepening makes the change to a
thermal collapse more likely.
Figure 6.11, from ITER team (1999, p. 2330), shows the two time scales involved in
disruptive instabilities. For typical JET values (6.25) gives τm ∼ 4 ms, and for the thermal
collapse the time scale is 3τe ∼ 250 µs. (There is some uncertainly with the duration of the
thermal collapse; several times τe is our estimate.) These values are close to those shown for
JET in Fig. 6.11 and the a1.5 dependence for τm in (6.25) agrees with the observations.
Disruptions involve the following sequence of events:
The disruption sequence
1. The sawtooth oscillation ceases and radiation reaches 100% of the input power;
2. the limit defined in (6.31) is reached and the ballooning instability is switched on;
3. the current profile is steadily flattened by the resulting turbulence, which by (6.25) takes
a time τm to saturate;
6.4 L ⇒ H transition, ELMS, Snakes, PEPS, and MARFES
163
4. during this time the inductance L falls (see Table 6.3) and as the magnetic energy 12 LIp2
is constant on this time scale, there is a sharp increase in Ip ;
5. by (6.20) the Lorentz voltage, V L , sharply increases and since Vt = V + V L ≈ const.,
there is a corresponding negative spike in V ;
6. by (6.23) the negative spike in V makes Vt < V L and exposes the plasma to the thermal
instability;
7. there is a rapid drop in the temperature in a time ∼ 2kτe , where k > 1.
8. the current quench occurs soon after.
Figure 6.11: Thermal quench times for various tokamaks as a function of minor radius
6.4 L ⇒ H transition, ELMS, Snakes, PEPS, and MARFES
First we shall explain the mechanism that transforms tokamak discharges from the L-mode
into the H-mode and then introduce four more thermal instabilities that have a bearing on
the efficient operation of a fusion reactor. The first instability is the Edge Localized Mode,
which reduces the H-mode confinement advantage by about 15%, and as it is believed that a
successful reactor must operate in the H-mode, it is important to understand the physics of
ELMs. A snake is a cool, high density filament that forms as a consequence of pellet injection
164
6 Thermal Instabilities
on either the q = 1 or the q = 3/2 magnetic surface. It can even reach twice the density of the
ambient plasma and a width (in JET) of ∼ 25 cm, and being remarkably long-lived, represents
a loss in reactor efficiency. Pellet injection changes the plasma properties, producing what is
described as being a Pellet Enhanced Performance mode, and for a second or two PEPS have
even better energy confinement than H-mode plasmas. A MARFE (Multifaceted Asymmetric
Radiation From the Edge) is a radiation instability that appears in a boundary region of cool
(Te ∼ 10 − 100 eV ) recombining plasma. This instability occurs when the temperature of the
region is being reduced by radiation to a value where the radiation rate further increases and
so on. The usual outcome is a loss of H-mode and a return to L-mode confinement.
6.4.1 The L ⇒ H transition
In the model of the H-mode adopted in Section 4.3.1, the improved energy confinement is
assumed to be due to the thermal insulation of the plasma from the boundary. It is observed
that both the thermal energy and the plasma density pile up into a narrow region of steep
gradients, called a pedestal (see Fig. 6.14), at about r/a = 0.96. This is an edge transport
barrier that impedes mass and energy transport and which can be simply explained by secondorder transport as follows.
By (1.7), (1.11), (3.99), and (3.101), equation (3.110) can be expressed in the form
χe =
1
2aJ aq y2q
5k1 kB Te y
−
βN P̈ − Ṗ Ṅ /N ,
2
6µ0 Re Ce ne Rqa N q
10qa
(6.33)
where k1 = 1.78, J = jϕ /jϕ , N = ne /ne , P = pe /pe , βN = 20aβp /Rqa and
y = (r/a)2 . We shall assume that the L⇒H transition is triggered by the boundary conditions in the pedestal region, where y ≈ 1. From (4.8) we find that at y = 1, q̇ = q = qa and
hence q /q = 2/a. Therefore (6.33) becomes
χe =
4aJ
βN 5k1 kB Te
−
P̈ − Ṗ Ṅ /N .
2
6µ0 Re Ce ne Rqa N
10
(6.34)
Adopting the distributions of Section 4.1.1 we obtain
P̈ − Ṅ Ṗ /N = (αt + αn )(αt − 1)(αt + αn + 1)(1 − y)αt +αn −2 ,
and
J
δ+1
=
(1 − y)δ−αn
N
αn + 1
αt = 23 δ .
It follows from (6.33) that heat flows inwards against the temperature gradient if
βN > F
a
(1 − y)αs ,
Rqa
(6.35)
where
F≡
40(δ + 1)
,
(αt + αn )(αt − 1)(αn + 1)(αt + αn + 1)
αs ≡ 13 δ + 2(1 − αn ) .
6.4 L ⇒ H transition, ELMS, Snakes, PEPS, and MARFES
165
40
30
F
an
20
0.5
H−mode
10
1
L−mode
1.5
0
2
3
δ
4
5
Figure 6.12: The profile shape function F(δ)
Let r be the width of the pedestal, so that the center of the pedestal is at x = a − 12 r.
Assuming that r/a
1, we find that 1 − y = r/a and (6.35) becomes
a
r αs
βN > F
,
(6.36)
Rqa a
which is a necessary condition for the L⇒H transition. An alternative form is
r αs
F
.
βp >
20 a
(6.37)
The function F(δ) is graphed in Fig. 6.12 for three values of αn .
We also need the expression for the velocity vD in the pedestal region. Restoring the
electron fluid velocity, veϕ = −jϕ /ene in (5.29), we get
vD =
0.73me
r2 pe τe (veϕ
)2
4ne e2 Bϕ2 r2
pe τe ∝ Te5/2 .
We shall assume that r(pe τe ) pe τe , i.e. that the temperature gradient in the pedestal region
is much larger than Te /r. In this case we have
vD =
0.73me pe τe 5Te 2
(v
)
+
2v
v
eϕ eϕ ,
4ne e2 Bϕ2
2Te eϕ
(6.38)
where from (A.16)
pe τe = 0.103T̂e5/2 /Zeff .
(6.39)
The two terms in (6.38) make quite different sorts of contribution to the plasma flow velocity vD ; the temperature gradient term drives the plasma inwards, up the pressure gradient,
whereas the term containing veϕ
generates a flow pattern that tends to overturn the veϕ distribution, as illustrated in Fig. 6.13. The initial (flattest) profile in the figure shows the electron
166
6 Thermal Instabilities
Figure 6.13: Electron fluid velocity profile steepening
fluid velocity falling to zero as the boundary wall is approached and the curvatures veϕ
act so
as to drive the plasma towards the point of inflexion, which has the effect of strengthening the
transport barrier.
The practical problem involved in this theory is that defining profile shapes by the parameters αn and δ is not likely to be very accurate in the region of the boundary layer, especially
when pedestals appear and perhaps the best we can expect is a qualitative account of the transition. A further complication is that sometimes this transport barrier appears to be spread
over a large part of the outer plasma radius (Wesson, 2004, p. 187).
6.4.2 Edge Localized Modes
As remarked in Section 4.3.2, the H-mode of operation is prone to a periodic edge instability,
known as an ‘ELM’, that switches an H-mode discharge momentarily back into the L-mode
and then returns it to the H-mode at a frequency dependent on the power supplied by the auxiliary heating; most of the time the discharge is in the H-mode. Apart from their adverse effect
on energy confinement, ELMs can also can damage the limiter surface, and understanding
their physics has developed into an important topic in fusion research.
There are several types of ELM with different amplitudes, frequencies and power dependencies. Small amplitude, high frequency edge instabilities, known as type III ELMs, occur
when the flow of power to the plasma edge is only a little above the threshold power required
to change a discharge from the L to the H-mode (see Section 4.6.2). When this power flow
substantially exceeds the threshold power, the instability becomes a high amplitude, low frequency, type I ELM. Between these frequencies there is a region in which the instability is
absent, which is described as being an ‘ELM free’ H-mode. A survey of the subject of ELMs
is given by the ITER team (1999), where many references are provided.
A theory of ELMs can be based on (6.36):
a
r αs
,
(6.40)
βN > F
Rqa a
which is the condition that the heat flows inwards, making an L⇒H transition likely.
Suppose that the transition L⇒H has occurred, but with NBI only marginally above the
threshold power required to make the transition. An immediate effect of the transformation
6.4 L ⇒ H transition, ELMS, Snakes, PEPS, and MARFES
167
Figure 6.14: Sequence of density profiles following an L⇒H transition
will be to steepen the density profile in the pedestal region and to flatten it in the central
region. This is equivalent to a reduction in the parameters αn and δ and the operating state
moves along the appropriate curve in Fig. 6.12 towards the F axis. For example, Fig. 6.14
(from Wagner et al. 1990) shows a sequence of density profiles at the times indicated on the
figure. The initial L-mode profile is ne = ne0 (1 − (r/a)2 )αn with αn ≈ 1.16, whereas the
final H-mode profile has αn ≈ 0.25; the flattening of the central region is also clear from the
figure.
After several electron collision intervals, the value of F is sufficiently increased to reverse
the inequality in (6.40) and the plasma returns to the L-mode, where the initial conditions are
repeated and drive the plasma back to the H-mode, and so on. As the changes involved are not
large, they can occur quickly and generate what are termed ‘dithering’ ELMs, which are high
frequency, type III ELMs. The frequency of these oscillations is proportional to (P − Prad ),
where P is the heating power and Prad is the radiated power in the pedestal region.
When there is enough input power to move the plasma state clear of the transition boundary, the change in the profile shapes is insufficient to return the plasma to the L-mode and
provided the value of βN is not too close to the ballooning limit defined in Section 6.3.2, there
will be no oscillatory response and an ELM-free H-mode will result.
Finally, type I ELMs occur when the input power is large enough to take the plasma
into the unstable region defined by the ballooning limit. The input power both increases
βN , which is proportional to the internal energy density 32 p and reduces F through profile
flattening. And if βN is increased up to its limit, βN, max , a type I ELM will be triggered.
These resemble minor disruptions in some respects, with precursor waves and two time scales
involved in the collapse; some steps of the sequence set out at the end of Section 6.3.3 are
relevant. When the temperature collapse occurs, the steepening process depicted in Fig. 6.13
is suddenly reversed, which accounts for the flood of particles to the walls. Figure 6.15 shows
168
6 Thermal Instabilities
Figure 6.15: H-mode operational diagram for ASDEC-Upgrade
an H-mode, (ne , Te ) ELM diagram, where the values of ne and Te are measured at the top of
the pedestal (Kaufmann et al. 1997). The description given above matches the distribution of
ELM types shown in the figure, and supports the model. The further study of H-mode ELMs
is a large and specialized subject beyond the scope of this book.
6.4.3 Snakes
Figure 6.16 (from Weller et al. 1987) shows an example of the soft-X-ray emission following
the injection of a D2 pellet in JET, which results in the formation of a rope-like filament
called a snake. This is a high density structure with typical poloidal and radial dimensions of
θ ≈ 25 cm and r ≈ 17 cm that forms on the q = 1 surface and which rotates about the
minor axis. Snakes can survive for ∼ 2 s regardless of frequent disturbances from sawtooth
oscillations, although they are vulnerable to soft disruptions. A snake therefore acts as a sort
of probe for studying of the position of the q = 1 surface during a sawtooth cycle; the theory
for this coincidence is given in Section 4.6.1, where the point is made that if q is independent
of radius for a small distance δ0 there will be volume of thermally connected plasma.
It is observed that before pellet injection the q = 1 surface coincides with the sawtooth
inversion radius, and that for a snake to be formed the pellet penetration needs to be inside the
q = 1 surface. The density and temperature of a typical snake are:
n = 3 ×1019 m−3 , nb = 6 ×1019 m−3 ,
Te = −140 eV, Tb = 1200 eV,
where
denotes snake perturbations and the subscript ‘b’ denotes background values;
Fig. 6.17 is a sketch illustrating these variations. During the formation of the snake it is
found that, following an initial large drop, the electron temperature within the snake quickly
6.4 L ⇒ H transition, ELMS, Snakes, PEPS, and MARFES
169
Figure 6.16: Contour plot showing the snake-like perturbation of soft X-ray emission following
the injection of a deuterium pellet
rises to within 10% or so of the ambient plasma temperature. The initial temperature drop can
be attributed to the energy required to ionize the pellet atoms.
The discovery that a relatively cool, high density filament could form and remain intact in
a very hot tokamak plasma for seconds presents the greatest, imaginable challenge to classical
continuum fluid dynamics. Three well established principals are set to nought — first, heat
must flow up the temperature gradients at the snake boundary, secondly, there must be a mass
flux into the snake up the density gradient and thirdly, these ‘unnatural’ transports must be
stable and persist for many sawtooth collapses following the ablation of the pellet. The only
complete account of the snake phenomenon is due to Deane (1989), who adapted equations
(6.33) and (6.38) to show that the transport of heat out of a snake and the influx of mass were
well accounted for by second-order transport. He also considered the stability of snakes and
Figure 6.17: Cool, heavy snake
170
6 Thermal Instabilities
derived expressions in agreement with observation for snake sizes and lifetimes. Snakes are
complicated phenomena of great interest for the challenge they offer to transport theory, but
we here shall restrict our attention to the physical principles involved.
Explaining why the snake should be located on the inversion surface r = rs is the easy
part; in Section 4.3.1 we showed that the bottom of the sawtooth oscillation was determined
by the condition that there was no heat source at rs , for at this point the temperature gradient
is zero and any further downward displacement of the temperature profile at rs would required
heat to diverge from rs as indicated in Fig. 4.7(b). However, this divergence will continue for
an electron collision interval or so until thermal equilibrium is restored and the sawtooth ramp
phase begins; thus for a brief period (∼ 40 µs in JET) at the bottom of the sawtooth oscillation
there will exist a cool channel, C say. For thermal equilibrium, the high parallel conductivity
along C requires a closed structure and therefore C can exist only where q has rational values;
observations show that while q = 1 is the preferred value, similar structures can occur on the
q = 3/2 surface.
The supply of cool deuterium atoms within the q = 1 surface triggers the formation of a
snake as follows. These particles are swept outwards by the radial plasma motion until reaching the region C of the q = 1 surface, where if the collapse phase of the sawtooth oscillation
has just occurred, C is already at a relatively low temperature. Ionization of the deuterium
atoms as they cross C absorbs considerable energy and results in the large temperature drop
observed at the beginning of the snake’s lifetime. As explained in Section 4.6.1, a depression
in the temperature profile may evolve into an equilibrium structure and need not completely
disappear. Equilibrium will require a nearly constant pressure and therefore initially, when C
is relatively cool, it will become appreciable denser than its surroundings.
Further progress to a fully developed snake depends on the transport of more particles
into C and the maintenance of a temperature depression as depicted in Fig. 6.17. The mass
flux is given by (6.38) and since the density perturbation is much larger than the temperature
perturbation, it is sufficiently accurate to write
5
veϕ
.
vD ∝ Te2 veϕ
Figure 6.18: Evolution of a snake
(6.41)
6.4 L ⇒ H transition, ELMS, Snakes, PEPS, and MARFES
171
From jϕ ≈ ene veϕ it follows that the density increase will be mirrored by a dip in the electron
fluid velocity, veϕ , as depicted in Fig. 6.18. From (6.41) the radial velocity vD vanishes at
inflexion points on the veϕ (r) profile, which are the points labeled p in Fig. 6.18. The radial
velocity is directed towards these points, increasing the local density and further impeding
the toroidal velocity, with the effect of distorting the veϕ (r) profile from apb towards cqb as
indicted in the figure. This process continues until the inflexion points vanish and the final
form is the almost rectangular profile cqb. The initial increase in density occurs when the
dip in the temperature profile is large, for by (6.41) the flow towards p from the side pb is
associated with a higher temperature and is therefore larger than the opposing flow.
Besides acting as a probe for determining the position of the q = 1 surface, the angular velocity ωs of the snake about the minor axis provides a diagnostic from which the
ion temperature can be estimated (Ashby 2005). From an enlarged Fig. 6.16 we find that
ωs = 3.6 ×103 radian s−1 at the snake radius rs = 0.55 m. A formula relating Ti and ωs can
be found from the principle of the conservation of total angular momentum, which we shall
assume — at least for the short initial period of Fig. 6.16 — remains constant at the value of
zero it had in the initial low-temperature stage of the discharge. In the gyration of the ions
about their guiding centers, the average angular momentum per particle is mc̄rL = mc̄ 2 /ωci
and each guiding center has an angular velocity of mr2 ωs about the minor axis. Hence
c̄ 2 /ωci + r2 ωs = 0 and from c̄ 2 = 8kB Ti /(πmi ), and ωci = eB/mi , it follows that the
ion temperature in keV at the snake position is
π
T̂i = 10−3 rs2 ωs B .
(6.42)
8
With the above values and B = 2.5 T, we get T̂i = 1.07 keV, while from the typical temperatures given earlier (2nd paragraph), T = 1.06 keV.
Incidentally, the above theory allows us to derive an expression for the radial electric
field, Er , the value of which can presently only be obtained inferentially (ITER team, 1999,
p. 2570). From the radial component of (A.12)2, Er = vϕ Bθ − vθ Bϕ + pi /ene , where values
for vϕ and vθ follow from (5.63) and vθ = rωs = −(8/π)103 T̂i /(rB).
6.4.4 Pellet enhanced performance mode (PEP)
The ablation and ionization of a pellet that reaches the central region of a tokamak plasma
will initially cool the ambient plasma, reduce the conductivity, and the local current density,
and create a peaked number density profile. The new plasma will flow outward, and because
of the increase in the density gradient, by (5.29) the radial velocity vD will be increased. By
(5.51) this will increase the Lorentz voltage between the minor axis and the limiter and hence
increase the off-center, non-inductive component of the current density. The outcome will be
a hollow current density profile like that illustrated in Fig. 4.10; the safety factor will develop
negative gradients in the central plasma and (see Figs. 4.7 and 4.8) create an ITB. As a result
of this barrier the central temperature will start to increase and with auxiliary heating switched
on at this stage, a very high temperature can be achieved. Shortly afterwards, the reduction in
the radial heat flux near the limiter will switch the boundary conditions into the H-mode.
The above sequence of events is visible in Fig. 6.19 taken from Fig. 1 of Hugon et al.
(1992). When the pellet has ablated away and the increment to vD declines, the PEP mode
ceases; its lifetime is about the same as the particle confinement time in JET, that is between
172
6 Thermal Instabilities
Figure 6.19: A typical PEP; (a) ICRH input power, (b) central temperatures, (c) central and
volume averaged electron density
1 and 2 seconds (see Section 5.2.4). The authors find that that the D-T fusion rate is much
larger in the PEP mode than in non-enhanced plasmas, and speculate that this transient mode
could be used to ignite the plasma in the next generation of tokamaks.
Figure 6.20: Electron temperature oscillations in a PEP
Figure 6.20 shows a rapid temperature oscillation in a PEP plasma reported by Hugon et al.
(1992), which is an example of the thermal pumping phenomenon described in Section 2.5.4.
The temperature scale is normalized to a temperature T0 , whose value was not quoted in the
paper. The curve ab is drawn to be tangential to the average value at each end and enables an
estimate to be made of the extent to which the oscillations are non-sinusoidal, being extended
somewhat in the higher temperatures.
By the theory of Section 2.5.4, the period of the waves should be ∼ 4τe ; the quarterperiod of the waves in Fig. 6.20 is 140 µs. From graphs given in the paper we obtain the
6.4 L ⇒ H transition, ELMS, Snakes, PEPS, and MARFES
173
estimates, T̂ = 10 keV, n19 = 2, and assuming that Zeff = 2 (cf. Table 4.4), we find that
τe = 126 µs, in reasonable agreement with theory. The non-sinusoidal character follows from
the fact that τe ∝ T̂ 3/2 , so that the profiles will be distorted in the direction of the higher
temperature. From the temperature range of the two peaks on either side of the maximum
peak, this displacement is represented by the factor ∼ (0.6/0.4)1.5 ≈ 1.8, which is about the
same magnitude as the distortion of the peaks in question.
6.4.5 MARFES
Radiation from tokamaks is an important and complex phenomenon beyond the scope of this
text, however the thermal instability known as a MARFE (Multifaceted Asymmetric Radiation
From the Edge) should be mentioned since some consider it to be the cause of L-mode density
disruptions. To ensure ignition, impurity radiation from the core plasma is reduced as much
as possible, whereas near the walls a high level of radiation is desirable to protect the divertor
plates from being overheated. The emissivity of most of the important impurities (mainly
carbon from the wall materials), reach maxima at temperatures Tm in the range 10 − 200 eV,
which means that at temperatures Te > Tm the radiation term in (1.25) has the form
∗
A/T̂eα
L = n19 n̄19
(α > 0) ,
(6.43)
∗
where n̄19
is the number density of the radiating elements and A and α are constants.
It is apparent from the form of (6.43) that a radiation instability — a MARFE —is likely
if the number densities are large enough and there exits a process that reduces the electron
temperature towards Tm , since by (6.43) reducing temperature means greater radiation which
reduces the temperature still further, and so on. Figure 6.15 shows the domain for such an
instability and also indicates that an L-mode disruption may be the consequence. On the
other hand, there could be a different cause for the disruption, with the MARFE a precursor
triggered by it.
Lipschultz et al. (1984) gave an early account of the MARFE phenomenon observed in
medium to high density Alcator C discharges. They found that a relatively small MARFE
region emitted a large fraction of the total radiated power and that this region was located on
the inside edge of the torus closest to the major axis, extending over a small poloidal arc and
the whole of the toroidal arc. The only poloidal asymmetry introduced in our transport theory
appears in (3.105), which includes the asymmetric term
5k1 me rqCe
p cos θ
·
·
·
−
.
χe =
12e2 ne Bϕ2
R
Since p < 0 near the boundary θ = π, it follows that compared with other values of θ, at the
MARFE location the thermal diffusivity is smaller. Therefore, for the temperature of magnetic
surfaces to remain more or less constant, a larger radial temperature gradient is necessary in
the MARFE region, i.e. the pre-MARFE temperature is lower than elsewhere both around
the torus in the poloidal direction and further into the main plasma. With a positive density
perturbation, due perhaps to accumulating impurities, we have initial conditions similar to
those required for the growth of a snake. In fact during a MARFE Lipschultz et al. found that
the ion density increased by a factor of up to ten at the largest value of r on θ = π, while
174
6 Thermal Instabilities
the temperature dropped by 50% or so. Our hypothesis is that the mechanism triggering a
MARFE is the same as that described in Section 6.4.3 for snakes, but of course with MARFEs
the outcome is a radiation instability.
The consequences for plasma confinement are variable (see account given by the ITER
team 1999, pp. 2409-16). It appears that the loss of H-mode confinement is not always the
outcome of a MARFE, although in the region of that instability the density profile steepens
(cf. Fig. 6.14), i.e. the coefficient αn is reduced, and the inequality in (6.14) is likely to be
reversed. As this inequality is the condition that an L-mode can transform into an H-mode, its
reversal means that the H-mode is lost and the plasma returns to an L-mode. Furthermore, if
the temperature profile is sufficiently flattened, the value of G in (6.31) will be reduced (see
Table 6.3) and βN,max will exceed its limit and a disruption will ensue.
6.5 Minimum reactor size for ignition
To bring the various elements of tokamak transport together, our final contribution will be to
apply the theory to the problem of determining the minimum size for a fusion reaction to reach
ignition. The initial aim of the International Thermonuclear Experimental Reactor (ITER) was
a reactor that would ignite and produce fusion power in the GW range by an extended burn
in deuterium-tritium plasmas. Here we shall obtain an estimate of the minimum size that a
toroidal machine with a circular cross-section would need to be in order to achieve this aim. Of
course there are many approximations and we have ignored all the real engineering problems
— the calculation provides no more than a guide. For a serious account of the problems
involved in the design of an burning reactor, the reader can consult the articles in ITER team
(1999) and the many references supplied there.
6.5.1 Stability constraints
Ignition:
30 ≤ τE n̄20 T̂ ,
Ballooning:
βp ≤ G(δ)R/a
Pressure:
p ≤ G
Thermal:
qa >
Transition:
a Bϕ2
,
Rqa2 2µ0
G in Table 6.3 ,
p = 3.21 ×103 α0 n̄19 T̂ ,
0.73Zeff
aqa V Vt
,
βp < 6.4
− 0.06 ,
aV Vt
Zeff
r αs
F
(F in Fig. 6.12) ,
βp >
20 a
µ0 e2 n̄e aR2 qa
n̄19 aR2 qa
= 1.89 ×10−2 F
,
1
1
1
(2me ) 2 kB T 2
T̂ 2
Confinement:
τE = F
Mode:
FL =
0.5
,
1 + 2.13βp
Beta:
βp =
8πp R2 qa2
R2 qa2
×10−3 α0 n̄19 T̂ =
8.06
.
107 a2 Bϕ2
a2 Bϕ2
FH = 0.45 ,
(6.44)
(6.45)
(6.46)
(6.47)
(6.48)
(6.49)
(6.50)
(6.51)
6.5 Minimum reactor size for ignition
175
Inequalities (6.44), (6.45) and (6.46) follow from (1.1), (6.31) and (6.29). The thermal
constraints are (6.22) and (6.23), the L to H-mode transition is (6.37), the confinement time is
taken from (4.56) and (4.57), we have assumed that τE ≈ τEe , and some definitions have been
included in the list. The distinction between Ti and Te has been ignored, p denotes the total
pressure and pi is assumed to have the same radial profile as pe .
6.5.2 Minimum dimensions
From (6.45) and (6.51)
n̄20 T̂ ≤ 12.4
G aBϕ2
,
α0 Rqa2
hence from (6.44)
30 ≤ τE n̄20 T̂ ≤ 12.4τE
G aBϕ2
F G n̄19 Bϕ2 2
=
0.234
a R.
α0 Rqa2
α0 T̂ 12 qa
(6.52)
There needs to be a substantial gap between the end values of this double inequality to accommodate the middle term. Let M > 1 be the margin separating these terms, then a necessary
condition for ignition is
1
Mα0 T̂ 2 qa
a3 ≥ 128 εa
,
(6.53)
FG
n̄19 Bϕ2
where εa = a/R.
To complete the theory it is necessary to include the other restrictions given above, e.g.
from (6.18) qa > 2, etc., and to make a judgment about M. We shall ignore the thermal
inequalities in (6.46), but as V = Vt − V L and the Lorentz voltage V L can be relatively large,
these inequalities could become dominant when a large non-inductive current is part of the
design. To get a rough estimate, we shall adopt the values:
α0 = 0.75, F = 0.45, (H-mode), G = 0.8, εa = 1/3, M = 2,
choices taken from Fig. 4.2, equation (6.50), Table 6.3, a typical value for a/R and the judgment that a margin smaller than M = 2 would be unsafe, whereas a larger value would
be uneconomic. A temperature of ∼ 10 keV is suggested by Fig. 1.2, while from (6.18),
Îp = 15 MA and a ∼ 2 m, values of n̄19 ∼ 7 and qa ∼ 3 should be safe. The strongest possible magnetic fields are necessary to reduce the size of the system; we shall set Bϕ = 5 T. With
these values (6.53) yields a ≥ 2.13 m. From (6.51) βp = 0.34, so from (6.50) FL = 0.29.
Hence in the L-mode the required radius is a ≥ 2.47 m.
However, the merit of (6.53) is not so much as a means of finding the dimensions of
an igniting tokamak, for there are many estimates and approximations involved, but as an
indication of the changes obtained by altering the parameters. To apply the above theory to a
tokamak with a “D” shaped cross section, one would need to know how to translate an “a × b”
cross-section (in the notation of Table 1.1) into a corresponding circular cross-section. The
extension of the theory to “D” shaped cross-sections is clearly of some importance; perhaps
this could be achieved by representing the magnetic surfaces as a family of concentric ellipses.
176
6 Thermal Instabilities
It would appear from the figures given above that the 2.0 × 3.7 tokamak High-Q ITER, to
be constructed at Cadarache, France (see Preface), should achieve its stated objectives of an
extended burn with a ratio of fusion power to auxiliary heating power of at least 10. However, designing a fusion reactor is a complex engineering task; this text deals only with the
underlying transport theory, upon which a successful design should be based. The physical
theory presented in this book should enable tokamak physicists to improve tokamak design
and operation; in ITER team (1999) the reader will find many other important considerations,
too often based on empirical equations, that play central roles in this design.
Postscript
Finally, in judging the importance of the tokamak enterprise, the Reader should recall the
warning given in the Preface:
“Burning fossil fuels and using the atmosphere as an open sewer has turned out to
be a recipe for disaster. The Earth is warming and the pace is quickening.”
Some may consider that I have occasionally been too positive in advancing my theory of
tokamak transport, but I have been greatly encouraged by the many agreements between this
theory and observations over a wide range of distinct phenomena. However, ‘certainty’ is
neither possible nor expected and I hope that errors will be brought to my attention. A true
scientific attitude was well described by the greatest 19th century natural philosopher, James
Clerk Maxwell (see Niven (1890), p. 486):
“. . . I venture to say that anyone who understands the provisional and temporary
character of this hypothesis, will find himself rather helped than hindered by it in
his search after the true interpretation of the phenomena.”
Magnetoplasma dynamics is based on a synthesis of Maxwell’s kinetic and electromagnetic theories, but not the least of his bequests to the subject is the concept of second-order
transport theory, which is the basic mechanism developed in this text. Prompted by an idea
of Reynolds, he found the solution to the radiometer problem, which was of great interest in
the 1870s. A radiometer is an evacuated jar, carrying a spindle on which are mounted vanes,
silvered on one side and blackened on the other; the spindle rotates when a light is shone on
it. Light pressure was initially thought to be the cause, but Maxwell showed that this pressure was far too small. Later he found what is today accepted as the explanation. He showed
that temperature differences between the surfaces of the vanes would induce a surface velocity
flowing over the vanes that was proportional to ∇T , and when this surface velocity is sheared,
a viscous force proportional to the second-order term, ∇∇T , acts on the vanes and causes
them to spin. The fascinating story of the Reynolds–Maxwell quarrel over this Victorian ‘toy’
is described by Brush (see footnote, p. 112).
References
177
References
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Biskamp, D. & Welter, H. (1979). III, Paper Al.
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New York, Oxford.
Campbell, D.J. et al. (1985). V, Pt I, 130.
Campbell, D.J. et al. (1986). VII, Paper 47/A-VII-5.
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Connor, J.W., Hastie, R.J., & Taylor, J.B. (1978). Phys. Rev. Lett., 40, 396.
Deane, G.B. (1989). “The transport of mass and energy in toroidal fusion machines.”
D.Phil. thesis, University of Oxford.
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Murakami, M., Callen, J.D., & Berry, L.A. (1976). Nuclear Fusion, 16, 347.
Niven, W.D. (Ed.) (1890) The Scientific Works of James Clerk Maxwell. Vol. I, Cambridge
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Ohyabu, N. (1979). Nuclear Fusion, 19, 1491.
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TFR Group (1985). Nuclear Fusion, 25(8), 919-30.
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Vershkov, V. A. & Mirnov, S. V. (1974). Nuclear Fusion, 14, 383.
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References
A Plasma Physics Notes
The plasma physics notes in this appendix are self-contained with a little repetition; they cover
basic topics, an understanding of which is required at various points in the main text and can
be read in any order. Some ‘forward’ referencing is occasionally unavoidable. The symbols
are defined on pages XV and XVI.
A.1
Equations of fluid motion
The MHD equations are based on the assumption of charge neutrality, in which case the macroscopic
length and time scales, L and T say, satisfy
λ,D /L 1;
−1
ωpe
/T 1 ,
where λ,D = (0 kB Te /ne e2 )1/2 is the Debye length and ωpe = (ne e2 /0 me )1/2 is the plasma frequency (see Section A.2), and Maxwell’s equations relating the electric field E, magnetic field B and
current density j, reduce to Ampère’s law,
∇ × B = µ0 j,
and the induction equation
∂B
∇×E = −
,
∂t
∇·j = 0,
(A.1)
∇·B = 0.
(A.2)
Let denote the plasma density, v the plasma velocity, p the pressure tensor, and D = ∂/∂t+v · ∇
the convective derivative, then the equation of plasma motion reads:
9
´
`
∂` ´
v + ∇ · vv + p = j × B , =
∂t
(A.3)
;
Dv + ∇ · p = j × B ,
the second form of which follows from the plasma continuity equation
` ´
∂
+ ∇ · v = 0 .
∂t
(A.4)
Using equations (A.3) and the following vector relation
´
`
´
`
´
`
∇ × B × B = B · ∇B − ∇B · B = ∇ · BB − ∇ 12 B 2 11 ,
in which 11 denotes the unit tensor, we get
“
”
1
1
T ≡ BB −
B 2 11 ,
j×B = ∇·T
µ0
2µ0
Theory of Tokamak Transport: New Aspects for Nuclear Fusion Reactor Design. Leslie Colin Woods
Copyright © 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
ISBN: 3-527-40625-5
(A.5)
180
Appendix A Plasma Physics Notes
where T is called the electromagnetic stress tensor. The pressure tensor is conveniently divided into its
scalar (thermodynamic) pressure p and its viscous stress tensor :
p = p11 + .
(A.6)
Now (A.3) can be written
n
“
∂` ´
1 2”
B 11 +
v + ∇ · vv + p +
∂t
2µ0
−
o
1
BB = 0 ,
µ0
(A.7)
which makes clear how the concept of magnetic pressure, B 2 /2µ0 , arises. When the plasma fluid
velocity is negligible and the magnetic field is steady, the system if stable, is in static equilibrium and
so to obtain the magnetostatic equations, we set v and ∂/∂t equal to zero. As the viscous stress tensor
depends on the existence of fluid shear, we can also set equal to zero. Thus
∇p = j × B,
µ0 j = ∇ × B,
∇ · B = 0,
∇·j = 0,
(A.8)
and therefore
B · ∇p = 0,
and
j · ∇p = 0 ,
“
1 2”
1
∇ p+
B =
B · ∇B .
2µ0
µ0
(A.9)
(A.10)
It follows from (A.9) that B and j lie on constant pressure surfaces, which if closed, appear as
continuous windings of intersecting magnetic field and current lines; these properties are said to lie on
‘magnetic surfaces’ and p is termed a ‘surface quantity’. Figure 1.6 shows a set of nested surfaces, with
a limit line at their center, known as the ‘magnetic axis’. If p increases towards the axis, its negative
gradient is balanced by the j × B force directed inwards. The plasma is thus confined by the magnetic
force.
A.2 Collision intervals and Spitzer resistivity
The Debye length λD is the distance over which Coulomb forces can influence the trajectory of an
electron and therefore the number of particles in a Debye sphere able to do this is
nD = 43 πλ3D ne ,
λD ≈ 69(Te /ne )1/2 ,
and with typical tokamak values of T̂e = 3 keV (see notation on p. XVI) and ne = 5 ×1019 m−3 , the
number involved is nD ∼ 4 ×107 , which means that interactions in a plasma are almost all ‘grazing’
collisions.
The ‘collision interval’ τ is defined to be the average time that it takes the sequence of grazing
collisions to deflect a particle P through 90◦ ; in this time it will transfer all its momentum along its
initial path to those particles with which it has interacted. The average electron collision frequency for
momentum transfer, 1/τe1 say, is typically a hundred times greater than the frequency with which the
occasional head-on collision deflects it through 90◦ , so the accumulated effect of grazing collisions is
much more important than the occasional abrupt collision. P interacts with a vast number of particles
per momentum transfer time and the binary event called a ‘collision’ in a neutral gas is replaced by an
essentially continuous process in a plasma. Energy transfer takes a time τe2 , which is a little longer than
τe1 .
On average, neutral particles take only two or three collisional interactions to give up all their directed momentum or energy, and by the distribution of their velocities, they transport the imprint of
A.2 Collision intervals and Spitzer resistivity
181
the temperature and momentum of their source to the collided particles; thus, if they have a peculiar
velocity1 c, collectively they transport to the particles at (r, t) the earlier temperature T (r − τ c, t − τ ).
Particles are ‘labeled’ by their history, which lags one collision interval behind the present.
In a plasma this model of impulsive transfer is replaced by a continuous process of transfer via
grazing collisions, although still taking what is called a ‘collision interval’ to complete. A typical ‘test’
particle at a point Q(r, t) has arrived there along a random walk path that ‘commenced’, i.e. was moving
at 90◦ to its final direction — at a point Q1 a time τ say earlier. Like a neutral particle, its peculiar velocity will evince its recent history, the temperature of its random component being the ambient temperature
near Q1 , and the velocity of its fluid component likewise belonging (roughly) to Q1 .
Figure A.1: Particle trajectories in a plasma
Consider a large group of test particles passing through Q in the same direction and with the same
speed. On tracing their separate random walks back a collision interval in time, we find a scatter of
origins Q1 , with an average position Q̄1 on the common tangent to their paths at Q as shown in Fig. A.1.
As with neutral particles, the group’s peculiar velocity has a random component whose distribution can
be assigned a temperature (T1 ) and a fluid velocity (v1 ) taken from ambient conditions near Q̄1 . The
fluid velocity enters via the rate of stain tensor, e in the term −c · e occurring in (A.75) below. The
relaxation or memory time for the group is then the transit time τ for a particle moving at the bunch
speed between Q̄1 and Q.
Since pressure is a force transmitted via collisions, it follows that in a plasma a particle must experience a continuous non-random force −∇p/n, albeit time-delayed, due to the pressure gradient. The
particle will also experience a steady, non-random drag per unit mass, say R/, due to its interaction
with particles of other species. And there will be a small random force f due to the microfields.
When a particle experiences what is called a ‘collision’, it continuously yields up properties acquired
τ seconds earlier, and becomes randomized via the force f , its peculiar velocity ‘ending’ in an arbitrary
direction, roughly normal to its initial motion. Particles can also ‘collide’ in this sense with magnetic
fields, quite small uncertainties in B being sufficient to scatter an initially coherent bunch. And when the
relaxation time associated with magnetic microfields is less than the corresponding time for electrostatic
microfields, it is the former that determines the transport of macroscopic properties.
A precise approach to transport theory in a plasma deals with particle trajectories only in the immediate vicinity of Q — the domain of small scattering angles shown in Fig. A.1. In this domain the
1 The distinction between the particle velocity w relative to the laboratory frame L and its peculiar velocity
c ≡ w − v, where v is the fluid velocity — the average velocity of all the particles at a given infinitesimal volume element — is central to the whole topic of transport. Its neglect, common in works on tokamak physics, is a fatal
mistake so far as transport is concerned.
182
Appendix A Plasma Physics Notes
distribution function f satisfies the Fokker–Planck kinetic equation, which has a collision term C representing diffusion in velocity space (see, e.g. Woods, 2004).
Let Z denote the ionization number and e the charge on an electron, then by charge neutrality
ne = Zni and
`
´
j = eZni vi − ene ve = ene vi − ve .
(A.11)
The ion and electron fluids are subject to the Lorentz force, Q(E + v × B), where Q is the particle
charge; thus their equations of fluid motion are
)
e De ve + ∇ · pe = −ene (E + ve × B) + Re
,
(A.12)
i Di vi + ∇ · pi = ene (E + vi × B) − Re
where Re is the force density acting on the electron fluid due to collisions with the ions. By Newton’s
third law, −Re is the corresponding force acting on the ion fluid. The time derivatives on the left-hand
side of (A.12) are the appropriate convective derivatives, i.e. Dα = ∂/∂t + vα · ∇, where α = e, i for
the electron and ion fluids. By (A.11) the sum of these equations yields (A.3).
The time required for an electron to be scattered by a sequence of Coulomb collisions with ions
until its velocity vector is deflected through 90◦ relative to its original motion is written τei , and its
−1
is the electron-ion ‘collision frequency’. There is also a smaller contribution due
reciprocal νei = τei
to electron–electron collisions and the combined collision interval and collision frequency are denoted
by τe and νe . By Newton’s 2nd law, provided the relative velocity (vi − ve ) is parallel to the magnetic
field, the average ‘slowing-down’ force on an electron of mass me is me νe (vi − ve ), and therefore the
total impeding force per unit volume is
`
´
(A.13)
Re = α0 ne me νe ve − vi = ene η j ,
where
η =
α0 me
,
e2 ne τe
(A.14)
is termed the ‘parallel resistivity’ and α0 is defined below.
The theory is readily extended to accommodate several types of ion, specified by the subscript s. Let
X
X
ns Zs2
ne =
ns Zs ,
(A.15)
ne Zeff ≡
s
s
where Zs is the charge number for the s-type ion and Zeff is known as Z-effective, then with
α0 = 3π/32, the collision interval appearing in (A.14) is given by (Lorentz 1905)
´3/2
`
3/2
` ´3/2 20 me1/2 kB Te
2.75 ×105 Te
τe = 3 2π
=
,
(A.16)
e4 ln Λ ne Zeff
ln Λ ne Zeff
where kB is Boltzmann’s constant, 0 is the free-space permittivity, and
(
16.34 + 1.5 ln Te − 0.5 ln ne
(Te < 1.16 ×105 K)
ln Λ =
(Te > 1.16 ×105 K)
22.81 + ln Te − 0.5 ln ne
,
(A.17)
is known as the ‘Coulomb logarithm’. (In tokamak calculations ln Λ = 17 is usually a satisfactory
approximation.)
When the electron–electron collisions are taken into account, the electron distribution function is
distorted by the electric field, which has the effect of increasing the collision frequency between the ions
A.3 Energy in the electron and ion fluids
183
and electrons and the resistivity is increased from its Lorentzian value with the value of α0 changed as
shown in (A.18) below (Spitzer, 1962).
The perpendicular resistivity, η⊥ , depends on the value of the dimensionless number e ≡ ωe τe ,
where ω = eB/me is the electron cyclotron frequency (see Section A.4). For a plasma containing a
single species with charge number Z,
9
α0 me
0.39
>
, α0 = 0.295 +
η = 2
,
>
=
e ne τe
Z + 0.85
(A.18)
>
β0 me
5.88
>
η⊥ = 2
, β0 = 1 − 2
(all Z) , ;
e ne τe
e + 12
where the expressions for α0 and β0 are fair approximations to their calculated values (Woods 2004).
At Z = 1, in strong magnetic fields, Spitzer’s resistivity becomes
me
,
η = 0.51 2
e ne τe
η⊥ =
me
.
e2 ne τe
(A.19)
A generalized Ohm’s law follows from (A.12)1 and (A.13),
η · j = E + ve × B −
1
∇pe ,
ene
`
´
η = η bb + η⊥ (11 − bb) ,
(A.20)
where η is the resistivity tensor (see (A.46) below), b is unit vector parallel to B, and the relatively
small contributions from the electron inertia and viscosity have been omitted. Alternatively, from
ve = vi + j/ene ≈ v + j/ene and σ · η = 11:
“
j×B
∇pe ”
.
j = σ · E+v×B+
−
ene
ene
A.3
(A.21)
Energy in the electron and ion fluids
We shall start with the energy equation for a single fluid,
´¯
˘ `
´
¯
∂ ˘ `1 2
2 v + u + ∇ · v 12 v 2 + u + p · v + q = F · v − L ,
∂t
(A.22)
where u is the internal (thermal) energy, q is the heat flux vector, F is the body force per unit mass and
L is the rate at which energy density is lost through radiation.
When this equation is applied to one component of a mixture, it is necessary to add terms to allow
for the frictional interaction between the components and for the transfer of energy between the species
due to temperature differences. Applying (A.22) to the species Pα of a plasma we have
´¯
˘
`
´
¯
∂ ˘ `1 2
α 2 vα + uα + ∇ · α vα 12 vα2 + uα + pα · vα + qα
∂t
¯
˘
= Qα nα (E + vα × B) + Rα · vα + Fαβ + Qαβ − Lα ,
(A.23)
where the collisional drag force (−Rα = Re = −Ri ) has been added to the Lorentz force (see (A.12)),
Fαβ is the energy dissipated in Pα by the friction between the species Pα and Pβ and Qαβ is the energy
gained by Pα due to a temperature difference (Tβ − Tα ).
As Pβ moves with a velocity (vβ − vα ) relative to Pα and exerts a force Rα on it, the friction
force does work at the rate Rα · (vβ − vα ), which is dissipated in both Pα and Pβ . If we assume that
184
Appendix A Plasma Physics Notes
the collisions are isotropic and elastic, the dissipated energy will be distributed inversely as the particle
, where ταβ
is the energy equipartition time. Hence
masses. Also Qαβ is proportional to (Tβ − Tα )/ταβ
`
´
`
´ mβ
Fαβ =
Rα · vβ − vα ,
.
(A.24)
Qαβ = 32 kB Tβ − Tα /ταβ
mα + mβ
We shall generalize (A.13) to Re = ene η · j, where η is defined in (A.20), then as me mi , when
applied to a plasma (A.24) gives
`
´
(A.25)
Fie ≈ 0 ,
Fei = j · η · j = j · E + v × B − ∇pe /ene ,
i.e. the dissipated energy goes almost entirely into the electron fluid.
By using the α-fluid conservation equations,
and
`
´
∂ α
+ ∇ · α vα = 0 ,
∂t
(A.26)
`
´
´
`
∂( α vα )
+ ∇ · α vα vα + pα = Qα nα E + vα × B + Rα ,
∂t
(A.27)
in (A.23), we can reduce it to
´
`
∂( α uα )
+ ∇ · α hα vα + qα − vα · ∇pα = Fαβ + Qαβ − Lα ,
∂t
(A.28)
where the viscosity tensor α has been omitted and hα = uα + pα / α is the enthalpy.
For the plasma taken as a single fluid, we find similarly from (A.22) that
`
´
`
´
∂(u)
+ ∇ · hv + q − v · ∇p = j · E + v × B − L .
∂t
(A.29)
A.4 Cyclotron frequencies
Consider the motion of a particle P of mass m, charge Q, and velocity w moving in (macroscopic) fields
E and B. The electromagnetic force acting on P is the Lorentz force, Q(E + w × B), and if in addition
there is a body force f per unit mass but no collisions, the acceleration of P is
ẇ =
Q
m
(E + w × B) + f .
(A.30)
The velocity w is measured relative to the laboratory frame L and as such its magnitude and direction
are ‘frame-dependent’. The average taken over all particles in a small element dr of space, w, defines
the fluid velocity v, then c ≡ w − v, defines a frame-indifferent velocity, termed the peculiar velocity.
The fluid acceleration Dv is the average of the particle accelerations, i.e. Dv = ẇ and
ẇ = Dv + ċ. The average of (A.30) is
´
Q`
E+ v×B + f ,
(A.31)
Dv =
m
where, since we are ignoring collisions, the pressure force, ∇ · p/, is absent. Subtracting (A.31) from
(A.30) we get
ċ = ωc c × b ,
c = 0 ,
(A.32)
where
ωc ≡ QB/m.
(A.33)
A.5 Dimensional analysis applied to energy confinement time
185
spin = - ωcb
c
P
G
a
X
B
r
B
Pc
O
Figure A.2: Gyration of a charged particle
It follows from (A.32) that the particle P moves with a velocity c about a point G of position vector
X (see Fig. A.2). The point G(X, t) is termed the guiding center of P, and the angular velocity ωc is
the cyclotron frequency. The radial distance between G and P, a = |a|, where a = r − X, is called the
Larmor radius of P and it is related to P’s perpendicular speed by a = |c|/|ωc |. The integral of (A.32)
is
´
`
(A.34)
a = r − X , c = bb · c ,
c = ȧ = ωc a × b + c
where c is the integration constant. These properties are illustrated in Fig. A.2. When a particle average
is required for the Larmor radius, we replace c by the thermal speed C = (2kB T /m)1/2 and denote the
Larmor radius by rL , i.e. rL = C/|ωc | .
A.5
Dimensional analysis applied to energy confinement time
The derivation of similarity laws by applying scale transformations to the basic equations is a widely
used technique in fluid dynamics (see Sedov 1959). It has been applied to plasma theory by Lacina
(1971), Kadomtsev (1975a), and Connor and Taylor (1977). In circumstances where the basic equations
are unknown, or uncertain, an elementary but more general procedure is to use dimensional analysis.
The dependency being sort is first postulated in terms of the expected dimensional variables, which are
then arranged into non-dimensional groups. The underlying principle is simply that each term in an
equation describing a physical relation must have the same dimensions.
Theorem. Suppose that in a steady-state, ohmically-heated tokamak plasma the electron-energy confinement time depends on the set of variables n̄e , a, R, B, Te , qa , Zeff , and no others. (This choice
excludes radiation losses.) Then τEe must have the functional form,
`
´
(A.35)
BτEe = F n̄e a2 , Te a1/2 , Ba5/4 , qa , R/a, Zeff .
Proof . Since τEe is a time interval, we start by identifying the time-scales implicit in the given independent variables. There are three, namely the electron gyration time, (me /eB), the transit time for
a sound wave, a/(2kB Te /me )1/2 , and the local energy replacement time, (ne kB Te /ηjϕ2 ), (cf. (1.22)).
Ignoring Zeff (which is separately listed) and very small variations in ln Λ, we adopt jϕ ∝ B/a and
5/2
η ∝ T −3/2 to write the last of these three times proportional to ne a2 Te /B 2 . Using (me /eB) to
186
Appendix A Plasma Physics Notes
non-dimensionalize the other time-scales, and ignoring molecular constants, we obtain the set of ‘di−1/2
5/2
mensionless’ parameters, BτEe , BaTe
, ne a2 Te /B. The original set of variables also yields the
2
dimensionless numbers 2µ0 ne kB Te /B and R/a; qa and Zeff are already dimensionless. The principle
of uniformity of physical dimensions now yields the functional relation
`
´
BτEe = Φ ne a2 Te5/2 B −1 , BaTe−1/2 , ne Te B −2 , qa , R/a, Zeff .
Since the arguments are dimensionless (in the sense that we are using this word here) they can be rearranged
by multiplying, dividing, or raising
to a power; this process yields the equivalent function
´
`
F ne a2 , Te a1/2 , Ba5/4 , qa , R/a, Zeff , and it remains to replace ne and Te by their averages to obtain (A.35).
The first three variables on the right of (A.35) can also be obtained by seeking transformations that
leave the equations of Boltzmann, Maxwell, and charge neutrality invariant (Connor and Taylor 1977).
A.6 Divergence and curl in cylindrical coordinates
Figure A.3: Cylindrical coordinates
Let (r̂, θ̂, ẑ) be unit vectors in the radial, azimuthal and axial directions, then for a vector F,
“ ∂
´
∂
∂ ”`
r̂Fr + θ̂Fθ + ẑFz .
+ θ̂
+ ẑ
∇F = r̂
∂r
r∂θ
∂z
As r̂, θ̂, ẑ are independent of r and z, and
∂r̂
∂ θ̂
∂ẑ
= θ̂,
= −r̂,
= 0,
∂θ
∂θ
∂θ
we find that
∂Fr
∂r
“ ∂F
Fθ ”
r
+ θ̂r̂
−
r∂θ
r
r̂r̂
∇F =
+ ẑr̂
∂Fr
∂z
+ r̂θ̂
+ θ̂ θ̂
+ ẑθ̂
∂Fθ
∂r
“ ∂F
θ
r∂θ
∂Fθ
∂z
+ r̂ẑ
+
Fr ”
r
+ θ̂ẑ
9
∂Fz >
>
>
∂r >
>
>
>
>
=
∂Fz >
r∂θ > .
>
>
>
∂Fz >
>
>
+ ẑẑ
>
∂z >
;
(A.36)
A.7 Tensorial form for Ohm’s law
187
In order to calculate ∇ · F and ∇ × F we introduce the operators ‘ · ’ and ‘ × ’ between the vector
pairs appearing in (A.36) and find
´ ∂Fθ
∂Fz
1 ∂ `
+
,
(A.37)
rFr +
∇·F =
r ∂r
r∂θ
∂z
and
“ ∂F
“ 1 ∂(rF )
“ ∂F
∂Fθ ”
∂Fz ”
∂Fr ”
z
r
θ
+ θ̂
+ ẑ
.
(A.38)
∇ × F = r̂
−
−
−
r∂θ
∂z
∂z
∂r
r ∂r
r∂θ
For a scalar φ,
F = ∇φ = r̂
∂φ
∂φ
∂φ
+ θ̂
+ ẑ
,
∂r
r∂θ
∂z
and
∇ · F = ∇2 φ =
A.7
(A.39)
1 ∂ “ ∂φ ” 1 ∂ 2 φ
∂2φ
r
+ 2 2 +
.
r ∂r ∂r
r ∂θ
∂z 2
(A.40)
Tensorial form for Ohm’s law
The transport tensors like κ and η have lateral isotropy about a unit vector b parallel to the magnetic
field B, by which is meant that in a plane orthogonal to b their properties are independent of direction.
So our first task in generalizing Ohm’s law is to determine the structure of typical second-order transport
tensors.
We shall start by describing transverse isotropy in terms of the unit Cartesian vectors i, j, k and
various polyadic combinations of them like ii, ikj, jiji, . . ., with k chosen to be the preferred direction.
Tensors of second order may be represented as sums over 9 dyads like ab, where a = i, j, or k and
similarly for b; likewise tensors of higher order are sums over polyads like abc (third order, 27 terms),
abcd (fourth order, 81 terms), etc., where a, b, c, d, . . . = i, j, k.
First consider the transformation of a polar vector a into a by a rotation about a unit vector k
through a small angle θ. The rotation will displace the point represented by a in a direction at right
angles to both a and k. Therefore a = a + θk × a. The dyad ab transforms according to
(ab) = a b = (a + θk × a)(b + θk × b) ≈ ab + θ(k × ab + ak × b),
and in general for small θ,
(abc . . .) = abc . . .+ θ (k × abc . . . + ak × bc . . . + abk × c . . . + abck × . . .).
If the polyad is invariant under the transformation, then (abc . . .) = abc . . . + . . ., so that
k × abc . . . + ak × bc . . . + abk × c . . . + abck × . . . = 0 .
(A.41)
While this condition for the invariance of abc . . . is based on small angles of rotation, since a large
rotation can be divided into a large number of small rotations in each of which (A.41) applies, this
equation holds generally.
Suppose that we reverse the magneto-fluid system P by a 180◦ rotation about any axis normal
to the preferred direction k. Transverse isotropy ensures that changes in the orientation of P in the
plane transverse to k cannot affect the constitutive relations, so apart from a change in the sign of B,
phenomenological tensors will be invariant under the proposed transformation. Thus choosing a rotation
about j, we have
i → −i , j → j , k → −k ; L → L :
L(B) = L (−B) ,
where L denotes a phenomenological tensor of any order.
(A.42)
188
Appendix A Plasma Physics Notes
Scalars are unaffected by coordinate transformations, so are isotropic. Among vectors, k itself is the only one satisfying (A.41). The set of dyads possessing lateral isotropy about k satisfy
k × ab − ab × k = 0, and it is readily verified that only the dyads
kk,
ji − ij = k × 11,
ii + jj = 11 − kk,
(A.43)
meet this constraint. It follows that the most general second-order tensor possessing k-symmetry is the
linear combination
„
«
α , α⊥ , even in B
(A.44)
α = α kk + α∧ (ji − ij) + α⊥ (ii + jj),
odd in B
α∧ ,
where the dependence of the coefficients α , α⊥ , and α∧ on the sign of B follows from (A.42).
If instead of k an arbitrary unit vector b is the preferred direction, then (A.43) is replaced by
α = α bb + α∧ (b × 11) + α⊥ (11 − bb) .
It follows that the resistivity tensor has the form
`
´
η = η bb + η∧ b × 11 + η⊥ 11 − bb ,
(A.45)
(A.46)
and similarly for the other second-order phenomenological tensors, κe , κi and the thermoelectric tensor
δ. The anisotropy is due to the effect of the magnetic field on the particle trajectories and its strength is
measured by the ratio of the cyclotron frequency ωc (see (A.33)) to the collision frequency, ν = 1/τ .
1, the magnetic field is said to be ‘strong’ and the cross-field mean
Thus for the electrons, if |ωce τe |
free path (viewed as a displacement rather than a distance travelled) is of the order of the Larmor radius.
The perpendicular and transverse resistivities, η⊥ and η∧ , are functions of ωce τe , whereas the parallel
resistivity, η is unaffected by the presence of the field with the value given in (A.18). If |ωce τe | 1,
1, kinetic theory shows that
the approximation η ≈ η11 is adopted; in the other limit, |ωce τe |
η⊥ ≈ 2η = 2η and η∧ ≈ 0.
Ohm’s law is given in (A.20) to which we shall add the thermoelectric contribution,
η ·j = E+v×B+
´
1 `
j × B − ∇pe + δ · ∇T .
ene
(A.47)
For a physical explanation of the thermoelectric phenomenon e.g. see Woods (2004).
A.8 Constants of the motion of gyrating particles
Because of their gyratory motion, each charged particle behaves like a dipole, with a magnetic moment
M given by
M = −M b,
M=
mc2⊥
.
2B
(A.48)
Provided the magnetic field changes slowly enough, the particles tend to move so as to enclose a constant
magnetic flux within their Larmor orbits; thus if rL = c⊥ /ωc is the Larmor radius (see Section A.4),
then πr2L B ∝ c2⊥ /B is approximately constant. We can show this as follows.
From (A.32), in a frame convected with the fluid,
mċ = mċ⊥ + mċ = Qc⊥ × B ,
(A.49)
A.8 Constants of the motion of gyrating particles
189
where in general B is a function of r and t. In the fluid frame only convective changes due to the peculiar
velocity remain, hence
`
´
ḃ = (c + c⊥ ) · ∇b
b = B/b .
ċ = c ḃ + ċ b,
We shall denote averages over a gyration by an bar. Therefore, as c⊥ = 0, the scalar product of (A.49)
with c⊥ , followed by averaging over a gyration, yields
d `1 2 ´
mc = −mc c⊥ · ∇b · c⊥ .
dt 2 ⊥
(A.50)
Let r̂, θ̂ denote unit polar vectors orthogonal to b, then c⊥ = c⊥ θ̂. From the gyro-average of
11 · A = (r̂r̂ + θ̂ θ̂ + bb) · A,
where A is any second order tensor, it follows that
r̂r̂ · A = θ̂ θ̂ · A = 12 (A − bb · A) = 12 (11 − bb) · A,
i.e.
r̂r̂ = θ̂ θ̂ = 12 (11 − bb) .
(A.51)
Therefore, since ∇b · b is zero,
c⊥ · ∇b · c⊥ = 12 c2⊥ (11 − bb) ·· ∇b = 12 c2⊥ ∇ · b.
Now ∇ · b = ∇ · (B/B) = −b · ∇ ln B, whence (A.50) yields
d
d 1 2
( mc⊥ ) = 12 mc2⊥ c · ∇ ln B = 12 mc2⊥ (ln B),
dt 2
dt
or
dM
= 0,
dt
M=
mc2⊥
.
2B
(A.52)
(A.53)
It also follows directly from (A.50) that d( 21 mc2 )/dt = 0; thus there are two ‘constants’ of the
particle motion, namely
M=
mc2⊥
= const.,
2B
E = 12 mc2 = const.
(A.54)
In general neither of these quantities are exact constants; they require variations of B across the Larmor
radius a to be small, i.e. δ ≡ |a · ∇ ln B| 1. And of course collisions must be relatively rare, which
1.
in this context means that ωc τ
From (A.52) and (A.54),
d
d “ 1 2”
mc = − 12 mc2⊥ (ln B),
dt 2
dt
and therefore ċ = − 21 mc2⊥ ∇ ln B = −M ∇ B. Hence we have the so-called mirror force,
F = −M ∇ B ,
(A.55)
that repels particles from regions of increasing magnetic field strength. It is important to notice that as
F is calculated in the convected fluid frame, this force has no direct effect on the fluid velocity.
190
Appendix A Plasma Physics Notes
A.9 Equilibrium velocity distribution function
The velocity distribution function is f (r, w, t), with the meaning that in an element dr dw of
6-dimensional phase space (r, w), there are f dr dw particles. Here w is the particle velocity measured
in the laboratory frame. It is convenient to transform to the convected frame and therefore to replace w
by the peculiar velocity c. The entropy density is defined by (e.g. see Woods 1996, p. 35)
Z
f (ln f − 1) dc ,
(A.56)
s = −kB
all c
and the entropy within a volume V is
Z Z
Z
s dr = −kB
S=
V
V
all c
f (ln f − 1) dc dr .
Let f0 denote the equilibrium value of the distribution function, then at f = f0 , the entropy S of the
system is a maximum. Maximizing S is equivalent to maximizing s subject to any constraints applying
to the distribution function. We shall define f0 to be that distribution function which applies to the same
values of the number density n and energy density u as in the non-equilibrium case. Hence, as the
gradients relax towards zero and f → f0 , the functions
Z
Z
1
n=
f dc, u =
mc2 f dc ,
(A.57)
2
are held constant, restrictions that are accommodated by introducing Lagrangian multipliers α and β.
Z
Z
Thus
δ(s) = −kB δ{f (ln f − 1) + αf + βc2 f }dc = −kB (ln f + α + βc2 )δf dc.
Since the variation in f is now arbitrary, δ(s) is zero only if the integrand vanishes, i.e. if f has the
value f0 given by f0 = exp(−α − βc2 ). Using (A.57) to evaluate α and β we arrive at Maxwell’s
equilibrium distribution,
«3/2
«
„
„
mc2
m
.
(A.58)
exp −
f0 = n
2πkB T
2kB T
Let the particles be in an equilibrium velocity distribution, then the probability that a particle chosen
at random has a velocity in the range c, c + dc is
3
(f0 /n) dc = π − 2 C −3 exp(−ν 2 ) dc ,
where
C≡
„
2kB T
m
«1
2
,
ν=
c
,
C
ν = νĉ ,
(A.59)
and ĉ denotes unit vector along c. Transforming from the Cartesian coordinates (cx , cy , cz ) to the
spherical coordinates (c, θ, φ), we have dc = c2 dc sin θ dθ dφ = c2 dc 4π dΩ , where 4π dΩ is the
element of solid angle subtended at the origin. Thus the probability that the relative speed ν falls in
ν, ν + dν and the unit vector ĉ lies in dΩ is
4
f0
dc = √ ν 2 exp(−ν 2 ) dν dΩ
n
π
`
´
0 ≤ ν < ∞, 0 ≤ Ω ≤ 1 .
The average value of a function φ(ν) is therefore
Z Z ∞
4
φ(ν)ν 2 exp(−ν 2 ) dν dΩ .
φ = √
π Ω 0
(A.60)
A.10 Escape time for trapped particles
191
Figure A.4: Spherical coordinates for the velocity vector
From Fig. A.4, ĉ = i sin θ sin φ + j sin θ cos φ + k cos θ, from which we find that
Z
Z
ĉ dΩ = 0,
ĉĉ dΩ = 13 11,
Ω
Ω
where 11 is the unit tensor.
√
Two averages that will be required are c = 2C/ π and
1
mc2 = 34 mC 2 = 32 kB T .
2
(A.61)
When the system is disturbed from equilibrium, the concepts of parallel and perpendicular temperatures
defined by
`
´
1
k T ≡ 12 mc2 ,
kB T⊥ = 12 mc2⊥ T = 23 T⊥ + 13 T
(A.62)
2 B are useful. In equilibrium, T = T⊥ = T , i.e. c2 = 12 c2⊥ = 13 c2 .
A.10 Escape time for trapped particles
The particle deflection time is the time it takes grazing Coulomb collisions to deflect a typical test particle
through 90◦ . The deflection times for electrons being scattered by ions at the same temperature and by
other electrons are (e.g. see Woods (2004))
´3/2
`
1/2
kB Te
2 me
(D)
= 2π33/2 04
≈ 0.69τe ,
τei
e ln Λ
Zne
(D)
(D)
τee
≈ 1.40τei
,
τD ≈ 0.40τe ,
where τD is the net time obtained by adding the frequencies and τe is the standard collision interval for
1
electrons, given in (A.16). For ions the deflection time is ∼ (mi /me ) 2 τe .
For a deflection through a smaller angle α the time required is τα where
α2
τα
=
τD
(π/2)2
(A.63)
which follows from the constant value of the diffusivity — (displacement)2 /(time) — and the fact that
with a sequence of grazing collisions the accumulated angle is proportional to the displacement.
192
Appendix A Plasma Physics Notes
`
´1
−1 2
and in a
By (2.56) trapped particles escape when their pitch angle exceeds αc = sin−1 1 − Rm
tokamak it follows from (2.66) that Rm = (1 + ε)/(1 + ε cos θ0 ), ignoring the additional small terms
in ε∗ . Hence in this case (A.63) gives the following estimate for the escape time,
τes = τD
8 h −1 “ 2ε sin 12 θ0 ” 12 i2
sin
≈ 1.62ε sin 12 θ0 τD ,
π2
1+ε
where we have doubled the value, since for cross-field transport we require the capture and subsequent
escape to occur on opposite sides of the banana orbit, as illustrated in Fig. 3.5 with the points P and Q
for escape and capture. Thus the escape time for electrons is
τes = 1.62 sin 12 θ0 ετD = 0.65 sin 12 θ0 ετe ,
(A.64)
1
and the corresponding escape time for ions is ∼ (mi /me ) 2 times longer (see Section A.2).
A.11 Motion of a fluid element
Suppose that a convected point Pc (r, t) moves with a velocity v(r, t), then a neighboring convected
point Qc (r + R, t) has the fluid velocity
v = v + R · ∇v + O(R2 ) .
(A.65)
(see Fig. A.5). The relative velocity R · ∇v is the scalar product of the small distance R and the
velocity gradient tensor ∇v, and in order to analyze this product we require some acquaintance with
second-order tensors and their properties (see Section A.16); the relation that we need is given in (A.99):
◦
∇v = e − Ω × 11 = ∇v −Ω × 11 + 13 ∇ · v ,
(A.66)
where e is the rate of strain tensor (the symmetrical part of ∇v), Ω ≡ 12 ∇ × v is the fluid spin and the
term with the circle above is the deviator of ∇v (symmetric part with zero trace).
By using R · (Ω × 11) = R × Ω · 11 = R × Ω = −Ω × R, and R · 11 = R, we find from (A.65)
that
◦
v = v + Ω × R + 13 R∇ · v + R · ∇v +O(R2 ) .
V+ R
Ω
R
R
V
R
V
Qc
Pc
r+R
r
Figure A.5: Strain of a fluid element
V
(A.67)
A.12 Kinetic equations
193
A rigid body motion about an axis l, rotating through a small angle θ, changes a position vector R
fixed in the body to R + θl × R. The velocity of the point is therefore Ω × R, where Ω is the angular
velocity θ̇l. Hence the second right-hand term of (A.67) represents a rigid body motion of the fluid
element with an angular velocity equal to half the fluid vorticity, ∇ × v. We term Ω the spin of the fluid
element. Such motion does not strain (i.e. deform) the element, and it will not induce a stress, except
in fluids of unusual microstructure. The term Ω × R can be removed from (A.67) by transforming to a
convected reference frame Pc carried along ‘bodily’ with the fluid and spinning with the angular velocity
Ω. We shall call this a “fully convected” frame.
Let R̂ be the unit vector along R, then by (A.67) the ‘outwards’ speed of Qc relative to Pc is |R|
◦
times 13 ∇ · v + R̂R̂ ·· ∇v. If R̂ is distributed isotropically, the average of R̂R̂ taken over all directions
◦
radiating from Pc is 13 11 and as 11 ·· ∇v= 0, the average fluid speed outwards from Pc on the sphere
|R| = a is 13 a∇ · v. Thus the third right-hand term in (A.67) is due to the changing volume of the
fluid element; this type of strain is called dilatation. The remaining term in (A.67), representing pure
straining motion without volume change, is called the deviatoric rate of strain. It plays a central role in
transport theory.
A.12 Kinetic equations
The evolutionary equation for f is a balance equation for the number f dν of particles of a given type,
lying in a volume element dν of phase space; thus, if dν has a velocity w and an acceleration ẇ, the
rate of change of f dν due to ‘streaming’ is D(f dν), where
D≡
∂
∂
∂
+w·
+ ẇ ·
.
∂t
∂r
∂w
(A.68)
Let C dν denote the net rate at which the number of particles in dν are changed by collisions, then
conservation of particles requires the balance D(f dν) = C dν.
Euler’s relation between the rate of change of a volume element dr in physical space and the divergence of the (fluid) velocity v is
`
∂
∂ ´
D(dr) = dr ∇ · v
D≡
+v·
.
(A.69)
∂t
∂r
In phase space the 6-D ‘velocity’ is (w, ẇ) and the corresponding gradient operator is (∂/∂r, ∂/∂w),
so by generalizing (A.69), we get
«
„
∂
∂
∂
·w+
· ẇ = dν
· ẇ
D(dν) = dν
∂r
∂w
∂w
by the independence of w and r; thus, provided that
∂
· ẇ = 0 ,
∂w
(A.70)
the balance equation reduces to
Df = C ,
(A.71)
which is known as a ‘kinetic equation’.
A more useful form of the kinetic equation is obtained by transforming to the convected frame by
replacing (A.68) and (A.71) by
D≡
∂
∂
∂
+c·
+ ċ ·
,
∂t
∂r
∂c
Df (r, c, t) = C ,
(A.72)
194
Appendix A Plasma Physics Notes
where c is the peculiar velocity defined in Section A.4; this has the advantage of removing convective
effects from the theory. But we still need an expression for ċ.
Corresponding to the peculiar velocity c = w − v, we now introduce the ‘agitation’ acceleration,
F, defined as being the difference between the particle acceleration ẇ and its value averaged over all
the particles of the same species:
F ≡ ẇ − ẇ,
F = 0.
(A.73)
Long-range forces like those due to gravity, macroscopic electric potentials and so on, affect ẇ and ẇ
alike and cannot contribute to F; this acceleration is due to the short-range, impulsive forces resulting from molecular collisions. The fluid acceleration Dv is equal to the average value of the particle
acceleration, so that (A.73) can be written
∂
+ v · ∇) .
(A.74)
ẇ = Dv
(D ≡
F = ẇ − Dv,
∂t
A particle P leaves the convected frame Pc with an initial peculiar velocity c relative to it. Let dc
denote the change in P’s velocity after a time dt, as seen in Pc , a frame that rotates with angular velocity
Ω relative to the laboratory frame L. Thus measured in L, the total change in c is dc + Ω × (c dt), which
equals dw − dv, i.e.
„
«
∂v
+ w · ∇v dt
dc + Ω × c dt = dw − dv = ẇ dt −
∂t
= ẇ dt − (Dv + c · ∇v) dt,
or
dc = (ẇ − Dv) dt − c · (∇v + Ω × 11)dt .
By (A.73) and (A.99) below this can be written dc = −c · e dt + F dt , where e is the rate of strain
tensor; this result applies to a neutral gas. In a magnetoplasma there is the additional acceleration due to
the gyroscopic motion given in (A.32). Hence in a plasma we have
ċ = ωc c × b − c · e + F
(A.75)
giving the rate of change of c in the frame Pc . The term −c · e arises from the definition of c; as P
moves through the sheared fluid, there are continuous changes in the velocity of the ambient fluid, so
that the origin from which c is measured is likewise changing.
The agitation acceleration, or equivalently the scattering force per unit mass, can be split into a
friction term opposing P’s motion and a diffusion term orthogonal to it. The friction term is proportional
to P’s velocity relative to the mean velocity of all the other molecules. If at time t = 0, P starts its
trajectory with velocity c relative to Pc , then immediately prior to t = τ say, when collisions start
to moderate its motion, the speed relative to the average molecular motion is altered by dilatation to
c(1 − 13 τ ∇ · v). For example, if ∇ · v is positive, then on average P will experience a reduced velocity
relative to the average molecular motion owing to the expansion of the fluid element. It follows that the
friction term has the form −τ −1 c(1− 13 τ ∇ · v). In general τ will depend on |c| = c, but for the present
we shall ignore this and treat τ as being constant, but having different values, depending on whether it
is the relaxation of momentum or energy flux that is under consideration. The diffusion term, Fd say,
randomizes P’s lateral motion (measured from its original trajectory), and as our present interest is in
the transport of momentum in the direction of the original motion, we can drop Fd .
From (A.99) we can now write (A.75) as
◦
ċ = ωc c × b − e · c − c/τ .
(A.76)
A.12 Kinetic equations
195
The time-scales for the right-hand terms in (A.76) are ωc−1 , T , and τ , where T is the macroscopic timescale for changes in the fluid velocity. The Knudsen number is kN = τ /T = τ ||e||, hence the orders of
(kN , 1), we may
the three terms in τ ċ are ωc τ, kN , and 1, from which it follows that provided ωc τ
adopt the approximation
`
´
Q
(ωc τ
kN , 1) .
(A.77)
ċ⊥ = c × B
m
For calculations correct to first order in the Knudsen number, a simple but sufficiently accurate
expression for the collision operator C in (A.72) is given by the BGK relaxation model (Bhatnager,
Gross and Krook 1954),
C = (f0 − f )/τ ,
(A.78)
where the relaxation time is a collision interval that depends on what property is being transported. In
(A.78), −f /τ is the rate at which particles are lost from a small element of phase space by collisions,
and f0 /τ is the corresponding rate at which particles are acquired by scattering from the immediate
environment.
From (A.72), (A.76) and (A.78) we have
Df = Df − ωc b × c ·
◦
∂f
∂f
f0 − f
−c· e ·
=
.
∂c
∂c
τ
From (A.58) and p = kB nT ,
`
1´
C ≡ (2kB T /m) 2 .
ln f0 = ln p − 52 ln T − c2 /C 2
(A.79)
(A.80)
Let ϕ ≡ (f − f0 )/f0 define the ‘relative’ distribution function, then since ∂f0 /∂c = −2cf0 /C 2 ,
the kinetic equation becomes
◦
◦
∂ϕ
∂ϕ
− 2τ c · e · c(1 + ϕ) − τ c · e ·
,
ϕ = −τ (1 + ϕ)D ln f0 − τ Dϕ + b × c ·
∂c
∂c
where ≡ τ ωc . In a frame F moving with the fluid velocity and acceleration, the equation of motion for
the fluid (electrons or ions) is already incorporated and in particular the acceleration due to the pressure
gradient force is zero in F. Hence by (A.80), in steady conditions,
`
´
`
´
ν ≡ c/C .
D ln f0 = ν 2 − 52 c · ∇ ln T
Now expand ϕ in a Knudsen number power series:
`
´
n
ϕn = O(kN
) ,
ϕ = ϕ1 + ϕ2 + · · · ,
(A.81)
which reduces solving the kinetic equation to finding the solutions of a series of equations for
ϕ1 ϕ2 · · · . The operation τ D is O(kN ) and therefore the leading equation is
◦
`
´
`
´
∂ϕ1
= −τ2 ν 2 − 52 c · ∇ ln T − 2τ1 νν ·· ∇v ,
ν ≡ c/C , (A.82)
ϕ1 − b × c ·
∂c
where the collision interval τ2 is appropriate for energy transport and τ1 is appropriate for momentum
transport. In a neutral gas, τ2 = 3τ1 /2, a choice that yields the correct value for the Prandtl number.
The solution of (A.82) is (see Woods 1993)
`
´
◦
(A.83)
ϕ1 = −τ2 ν 2 − 52 c · k · ∇ ln T − 2τ1 C −2 cc ·· W ·· e ,
where from (3.43):
k ≡ bb −
1+
2
b × 11 +
1
1+
2
`
´
11 − bb ,
196
Appendix A Plasma Physics Notes
and the fourth-order tensor W is defined in Section A.23. The BGK collision operator is only accurate
to first order in the Knudsen number and determining how to modified it to allow an accurate form of
the second-order term ϕ2 to be calculated is not straightforward (see Woods 1993, Chapter 11). For this
reason we have preferred the physical arguments in Section 3.2.3 and Section A.22 instead of modified
kinetic theory.
A.13 Drift kinetic equation
From (A.30) and (A.68) the kinetic equation (A.71) can be written
´ ∂f
∂f
Q`
∂f
+w·
+
E+w×B ·
= C(f ) .
∂t
∂r
m
∂w
(A.84)
The gyro-average of (A.84) under the constraints kL 1, (ωc τ )−1 1 , and E⊥ /B ∼ O(kL C),
where C is the thermal velocity, is termed ‘the drift kinetic equation’ (see Hazeltine and Meiss, 1992,
pp. 108–111). The gyro-average of w × B · ∂f /∂w vanishes and the result is a kinetic equation for the
guiding center distribution function, f¯, of the form (e.g. see Wesson, 2004, p. 161):
Q
∂ f¯
∂ f¯
+ vg · ∇f¯ + E
= C(f¯)
∂t
m ∂w
`
´
vg = u + δu ,
(A.85)
where from (3.12) and (3.15) vg is the guiding center velocity. Equation (A.85) is adopted as the starting
point for the derivation of the neoclassical transport equations described in Section 3.4.
The most obvious mistake in the derivation of (A.85) is to ignore the distinction between w and the
peculiar velocity c ≡ w − w = w − v, where v is the fluid velocity, which amounts to ignoring the
difference between convection and diffusion (Section A.15). From (A.12), upon dropping the collisional
terms, since they are all represented in the collision operator, C(f¯) (recall that the pressure gradient also
depends on there being collisions, see Section A.2), we get Dv = (Q/m)(E + v × B). Thus the third
right-hand term in (A.84) is
o ∂f
o ∂f
n Dv
n
c
∂f
Q
Q
= ωc C
+ ×b ·
≈ c×B·
,
Dv + c × B ·
m
∂c
ωc C
C
∂c
m
∂c
where |Dv/ωc C| ∼ O(kL ) 1 (using time intervals instead of displacements). This reduction is
standard kinetic theory (e.g. see Chapman and Cowling (1970), pp. 367–368, or Ferziger and Kaper
(1972), pp. 433–435). In fact the last approximation is unnecessary, since in a frame convected with the
plasma, which includes its acceleration, Dv = 0, so that the fluid component of the Lorentz force, (E +
v × B), disappears and hence has no role in the kinetic theory of diffusive transport (cf. (A.79)). This
correction removes the E terms from (3.80) and (3.81) and with zero Ware pinch, according to (3.84)
the bootstrap current all but vanishes (see Section A.19). (Other arguments wipe it out completely.)
A second error in drift kinetic theory is perhaps a little more subtle; it is that the ratio of the time
for a particle to complete one gyration to the collision interval, viz. 2π/(ωc τ ), is ignored when taking
the gyro-average of the left-hand side of the kinetic equation, i.e.
≡ ωc τ is in effect set equal to
infinity in evaluating the operator on the left-hand side of the drift kinetic equation, but is held finite
when averaging the collision operator on the right-hand side. For this reason the O( )−1 terms in ϕ1
are missed (see (A.83) and (A.137)). In fact the terms on the right of equations (3.80) to (3.82) all
2
/τe ) = Ce2 τe /(ωce τe )2 , which is an order smaller than the neglected terms.
contain the factor (rLe
A final point is that there was never any need for gyro-averaging, since — although not subject to
the banana orbital constraint — the original kinetic equation had been solved exactly to O(kN ) and to
all orders in −1 decades before the appearance of the drift kinetic equation (e.g. see Chapter 18 of
Chapman and Cowling, (1958)).
A.14 Guiding center drifts
197
A.14 Guiding center drifts
Let x denote the position vector of the particle P relative to the convected point Pc (r, t), then ẋ = c
and (A.77) can be expressed
ẍ = ωc ẋ × b + ẍ .
(A.86)
Now assume that B is steady and uniform, in which case the solution of (A.85) is
c = ẋ = ωc (x − X) × b + c
(c ≡ bb · c),
(A.87)
where X is arbitrary. Choose X so that a = x − X is always perpendicular to b, then ȧ = 0, i.e.
Ẋ = c b, and the derivative of (A.87), i.e.
ẍ = ωc (ẋ − Ẋ) × b + ċ ,
is the same as (A.86) if Ẋ = Ẋ and ċ = c b = 0. Hence Ẋ⊥ = 0 and Ẍ = 0. It follows from
(A.86) and (A.87) that
ä = −ωc a,
a = ωc−1 b × c,
a = c⊥ /ωc ,
(A.88)
which describe the motion of a vector rotating about an end point G at X (relative to Pc ), with angular
velocity ωc . The point G is the guiding center, introduced in Section A.4.
The situation is depicted in Fig. A.2, p. 185. The point G follows p along the magnetic field lines, but
relative to the fluid particle centered on Pc , has no motion perpendicular to the field. Because rL τ C,
the particle motions are effectively collisionless and hence pressure gradients do not appear in particle
orbit theory. Instead of a ‘fluid’ velocity, we now have a ‘particle average’ velocity in its place. The
motion of Pc perpendicular to the field follows from the collisionless form of (A.12) for each of the
components, namely
`
´
Dv = Qne E + v × B + g ,
where we have included the gravitational acceleration g. Hence the velocity perpendicular to the field is
v⊥ =
´
m `
E
×b+
g − Dv × b .
B
QB
(A.89)
Let
u = Ẋ + ṙ = Ẋ + v
(A.90)
denote the guiding center velocity in the laboratory frame, then, since Ẋ⊥ = 0, we have established that
in a steady, uniform magnetic field,
u⊥ = v⊥ ,
u = v + c b,
u = v + c .
(A.91)
The guiding center motion perpendicular to the magnetic field is independent of the particle velocity
and is equal to the fluid velocity, v⊥ . The physical reason for this is that each guiding center is never
more than a Larmor radius, rL , from its particle and because rL is very small compared with any of the
macroscopic length scales, the speed at which particle mass is transported, i.e. the fluid velocity, must
be the same as the transport of guiding centers. Therefore, provided the constraint on rL applies, the
condition u⊥ = v⊥ is generally true whatever forces are responsible for the fluid and guiding center
motions.
198
Appendix A Plasma Physics Notes
Perpendicular motions of guiding centers are called ‘drifts’, and these are named according to the
forces that cause them. Thus the first two drifts in (A.89) are ‘electric’ and ‘gravitational’. In general a
force F causes a drift u⊥ , and conversely a drift u⊥ requires the presence of a force F , where
F =
QB
m
b × u⊥ ,
u⊥ =
m
QB
F × b.
(A.92)
In Section 3.1.2 it is shown that gradients in the magnetic field cause guiding centers to drift away
from the magnetic surface, drifting outwards while in one transit between the reflection points and in
the opposite direction on the return path. The banana shaped orbits that result have an basic role in the
transport of energy from tokamaks.
A.15 Convection and diffusion
Convection is the transport of a macroscopic property, such as density, momentum, energy, the concentration level of a contaminant, and so on, by the fluid motion. Let Φ(r, t) be such an attribute, specified
as an amount per unit mass of fluid—known as a ‘specific’ property—then a volume element of mass
dr will possess an amount Φ dr of it. As the volume of fluid crossing a stationary surface n dS in
one second is v · n dS, it follows that the transport of Φ due to convection occurs at the rate Φv · n dS
across this surface. Hence
convective flux of Φ = Φv · n.
This description can be generalized by introducing a specific property φ(r, w, t), whose value may
depend on the velocity w = v + c of the particles involved in its transport. Suppose that the average
value of φ taken over particles at a macroscopic point is Φ, i.e.
Φ(r, t) = φ(r, w, t),
then the local transport of Φ across n dS is the average of φw · n , dS and gives a total flux of
φw · n = φ(v + c) · n, or
total flux of Φ = Φv · n + Jφ · n,
(A.93)
where Jφ ≡ φc . We term Jφ the diffusion vector for φ. Diffusion is thus the transport of a property
by the purely random component of molecular motion. It is very important to distinguish between
diffusion and convection and a central task in kinetic theory is that of obtaining the diffusive flux Jφ · n
for various properties φ.
In some circumstances an expression for Jφ of the form
Jφ = −χφ ∇Φ
(A.94)
can be found; χφ is termed the coefficient of diffusion for Φ. It follows from (A.94) that χφ has the
dimensions: (length)2 /time, and since it is due to particle transport, we may write
χφ = α λ2 /τ ,
(A.95)
where α is a constant of order unity.
When the particle kinetic energy 12 mc2p is transported along a relatively straight path, the diffusivity of the energy takes the form
χφ = c2p τ .
(A.96)
A.16 The decomposition of second-order tensors
199
Sometimes there exist several ‘channels’ for the diffusion of Φ, i.e. a number of distinct processes each
contribute to Jφ . If these are independent, the total flux is obtained by summation and χφ in (A.95)
becomes
χφ =
X j
j
κφj =
X j
j
αj
λ2j
,
τj
(A.97)
the subscript j denoting a particular process.
The convection term in (A.93), viz. Φ v · n, is dependent on the choice of reference frame in which
the velocity is measured, whereas the diffusion term is not. In fact ‘frame-indifference’ is the essential
property that distinguishes diffusion from convection. In some circumstances it is not evident from the
physics where to draw the line between these two transport processes, and a mathematical definition is
useful.
It follows from Maxwell’s electromagnetic equations
∇×E = −
∂B
,
∂t
µ0 j = ∇ × B,
and Ohm’s law in its simplest form, η j = E , that
`
∂B
= ξ ∇2 B
∂t
´
ξ = η/µ0 ,
where ξ is the magnetic diffusivity. It is also the diffusivity for the electric current. Thus the electric
current diffuses across a strong magnetic at a rate determined by (see (A.18))
ξ⊥ =
3
me
= 1.025 ×108 ln Λ/Te2 .
2
µ0 e ne τe
(A.98)
−3/2
, where T̂e is the temperature in keV. For example,
In typical tokamak conditions ξ⊥ ≈ 4.4 ×10−2 T̂e
the time for the current to diffuse half a JET tokamak radius, when the temperature is 3 keV, is ∼ 29 s.
A.16 The decomposition of second-order tensors
×
In general a second-order tensor A has a symmetric part As , an antisymmetric part Aa , a trace A, a
◦
vector Av , and a deviator A defined by
◦
×
×
As ≡ 12 (A + Ã), Aa ≡ 12 (A − Ã), A ≡ 11 ·· A, Av ≡ 12 11 × 11 ·· A, A ≡ As − 13 11 A ,
where 11 is the unit tensor, i.e. 11 · A = A and A · 11 = A, and the tilde denotes the transposed
◦
tensor. Since 11 ·· 11 = 3 and 11 ·· A = 11 ·· Ã, it follows that A has zero trace. With double
products like ab ·· A we shall adopt the convention that ab ·· A = b · A · a = A ·· ab, e.g. if
A = K ij, ab ·· A = K(a · j)(b · i). Hence
11 ·· ab = b · 11 · a = b · a,
11 × 11 ·· ab = b · 11 × 11 · a = b × a,
f = 2r · (ab)a ,
r · 11 × 11 × 11 ·· ab = −r × (a × b) = r · (ab − ab)
or
(ab)a = −11 × (ab).
200
Appendix A Plasma Physics Notes
Since a tensor A can always be expressed as the sum of three dyads, e.g. A = ab + cd + ef , it
follows that
×
2Av = a × b + c × d + e × f ,
A = a · b + c · d + e · f,
Aa = −11 × Av = −Av × 11 = −11 × 11 · Av ,
and
◦
×
A = A −Av × 11 + 13 A 11 .
In particular
×
∇v = ∇ · v, (∇v)v = 12 ∇ × v = Ω ,
and
◦
∇v = ∇v − Ω × 11 + 13 11 ∇ · v .
(A.99)
It is easily verified that for any vector a, a × 11 = 11 × a, hence the second right-hand term in (A.99)
can be expressed as −11 × Ω.
Let B denote another second-order tensor, then as A ·· B = Ã ·· B̃, it follows that
As ·· Ba = As ·· (−Ba ) = 0.
Also
Aa ·· Ba = Av × 11 ·· 11 × Bv = −2Av · Bv ,
and therefore expanding each tensor, we obtain
◦
◦
××
A ·· B = A ·· B − 2Av · Bv + 13 A B .
◦
◦
◦
◦
Also note that A ·· B = A ·· Bs = A ·· B .
A.17 Div and curl in local toroidal coordinates
Let R̂, Ẑ , ϕ̂ be the unit vectors in a cylindrical coordinate system (see Fig. 1.4, p. 6), and suppose
that the dependent variables have axial symmetry (∂/∂ϕ = 0), then if A = AR R̂ + AZ Ẑ + Aϕ ϕ̂,
(A.37) and (A.38) give
∇·A =
´ ∂AZ
1 ∂ `
RAR +
,
R ∂R
∂Z
(A.100)
“ ∂A
´
∂AZ ”
1 ∂ `
∂Aϕ
R
−
ϕ̂ −
RAϕ Ẑ .
R̂ −
∂Z
∂Z
∂R
R ∂R
(A.101)
and
∇×A =
Referring to Fig. 1.4, we see that the point P is at the point
R = R0 + r cos θ,
Z = r sin θ,
ϕ = ϕ,
and hence
∂r
= cos θ,
∂R
∂r
= sin θ,
∂Z
∂θ
1
= − sin θ,
∂R
r
∂θ
1
= cos θ ,
∂Z
r
A.18 Knudsen numbers and local thermodynamic equilibrium
R̂ = r̂ cos θ − θ̂ sin θ
and
201
Ẑ = r̂ sin θ + θ̂ cos θ ,
AR = Ar cos θ − Aθ sin θ,
AZ = Ar sin θ + Aθ cos θ .
With these relations, equations (A.100) and (A.101) transform into
∇·A =
and
´
´
1 ∂ `
1 ∂ `
rRAr +
RAθ ,
rR ∂r
rR ∂θ
(A.102)
n1 ∂ `
´
´
´ 1 ∂Ar o
1 ∂ `
1 ∂ `
ϕ̂ ,
RAϕ r̂ −
RAϕ θ̂ +
rAθ −
rR ∂θ
R ∂r
r ∂r
r ∂θ
(A.103)
∇×A =
where r̂, θ̂, ϕ̂ is the triad of unit orthogonal vectors associated with r, θ, ϕ.
If A is a tensor, we find similarly that
∇·A =
´
´
1 ∂ `
1 ∂ `
rR r̂ · A +
Rθ̂ · A
rR ∂r
rR ∂θ
+
¯
1˘
(Aϕr cos θ − Aϕθ sin θ)ϕ̂ − Aϕϕ cos θ r̂ + Aϕϕ sin θ θ̂ .
R
(A.104)
A.18 Knudsen numbers and local thermodynamic equilibrium
For a given continuum variable ϕ, the macroscopic scales are defined by
˛
˛
˛−1
˛−1
Lϕ ≡ min˛∇ ln ϕ˛ ,
Tϕ ≡ min˛d ln ϕ/dt˛ ,
(A.105)
the minimizing being over all relevant values of (r, t) and all orientations at a point in the gas. The
Knudsen numbers
kNL ≡ λ/L,
kNT ≡ τ /T ,
kN ≡ max{kNL , kNT },
(A.106)
are a measure of how nearly the medium may be regarded as being a continuum. In a ‘true’ continuum kN
is zero, but in this limiting case, diffusion is completely suppressed by collisions, making it impossible
to transmit fluid momentum and energy through the gas except by the collective process of convection.
A thermodynamic system Pc must have the possibility of reaching uniform values for its macroscopic variables in a relaxation time τth much smaller than the macroscopic time-scale T for, as will be
explained shortly, only then can precise values be assigned to Pc ’s pressure and temperature. In a gas the
mechanism that tends to produce equilibrium is the interaction of particles via collisions. Since τth ≈ τ ,
for these thermodynamic variables we must have kNT 1. And as the molecular speed, c = λ/τ , and
the speed L/T at which macroscopic perturbations propagate are usually comparable, this constraint
entails kNL 1.
In continuum mechanics pressure is force per unit area and in a gas the existence of such a force
requires the particles to collide either with each other or with confining walls. Away from boundaries,
the ambient particles around a point Pc play the role of the confining ‘walls’; this condition implies
that our local thermodynamic system, Pc , must have a typical dimension that is at least a mean free
path in length. In this case adjacent systems, say Pc1 and Pc2 , can interact, with each one exerting a
pressure on the other. If Pc1 has a slightly higher pressure than Pc2 , a net force will result, with Pc2
experiencing more numerous or more energetic collisions with particles coming from Pc1 , than Pc1 does
with particles from Pc2 . On a continuum description, the force is said to be due to the component of
the pressure gradient directed from Pc2 to Pc1 , but the underlying mechanism is an imbalance in the
particle collisions. In a ‘collisionless’ gas, however great the difference between the values of 13 c2 202
Appendix A Plasma Physics Notes
at two neighboring points, there would be no pressure force in the gas. We therefore define the pressure
as being the force transmitted across a unit surface. The concept of ‘equilibrium’ is not required in this
definition but collisions are essential.
Gas temperature appears to be a variable that does not depend on the presence of collisions. Its
definition in terms of the average kinetic energy of particles, i.e.
3
k T = m 21 c2 2 B
(A.107)
is one of the most famous results of early kinetic theory. But in classical thermodynamics the notion
of ‘thermal equilibrium’ plays a central role in the definition of empirical temperature, a necessary
preliminary to the introduction of the absolute temperature. Thermal equilibrium between Pc1 and Pc2
requires a collisional interchange just as already described for the pressure. Now suppose that Pc1 is
hotter than Pc2 . The transfer of energy between these systems involves two stages. First the more
energetic particles from Pc1 move through a free path and then they deposit their excess energy in Pc2
by collisions. It is important to distinguish between mere energy flux, which like radiation, need not be
deposited locally and heat flux, which does require collisions.
The precise definition of pressure and temperature is therefore determined by the force and energy
transmitted between neighboring fluid elements. When the laws of conservation of momentum and
energy are obtained using these variables, their physical properties are invoked in the formulation. Alternatively, these and related variables like the viscous stress tensor and the heat flux vector q are
defined implicitly by their roles in the conservation laws. Macroscopic equilibrium is an unnecessary restriction in the definition, but we must ensure that the gas has an internal structure such that the symbols
p, , T, q, etc. really do have the properties implied by in their appearance in the conservation laws.
The local transport of momentum and energy sets a lower bound on the size of the thermodynamic
system Pc . If d is a typical dimension of Pc , then it follows from the above discussion that we need
<
d. Similarly, the macroscopic time element dt cannot be much less than τ , for this would imply
λ∼
that the pressure and temperature could respond to changes that occur much faster than the collisional
mechanism effecting these changes.
We can now define our thermodynamic system at Pc to be an element in which, for each of the
thermodynamic variables of interest,
kNL 1,
kNT 1,
<
d L,
λ∼
<
τ ∼
dt T .
(A.108)
The Knudsen number is evaluated directly from the transport equations at the stage where the expansion of the heat flux vector or viscosity tensor in a (kB ) power series needs to be terminated. For example
closure of the viscous transport equations in the electron gas leads to the condition |τe ∇ve | 1, and
therefore |τe ∇ve | is identified as being the appropriate Knudsen number. It is incorrect to evaluate
kB by comparing the mean free path of the electrons with the radius of the tokamak, as is sometimes
suggested. The fact that electrons encircle the torus many times before accumulating the deflection that
counts for a collision is irrelevant.
A.19 Onsager’s reciprocal relations in neoclassical transport
The starting point is equation (A.23) and the assumption that terms second order in the velocity can be
omitted. We shall ignore viscosity, write ne = ni = n and adopt the approximation, vi = v, in which
case with the help of (A.26) and (A.27) we get
α Dα uα + α pα Dα (1/ α ) = −vα · ∇pα + Qα n(E · vα − F · v) − ∇ · qα ,
(A.109)
A.19 Onsager’s reciprocal relations in neoclassical transport
203
where Qα is the particle charge and F = Rα /(Qα n) is the friction force acting on the electron fluid.
From the thermodynamics relation, T Ds = Du + pD(1/), defining the specific entropy s, (A.109) can
be written
`
`
´
´
α Dα sα + ∇ · qα /Tα = − vα · ∇pα /Tα + Qα n E · vα − F · v /Tα + qα · ∇(1/Tα ) ,
(A.110)
where the left-hand side is the reversible rate of change of the entropy due to convection and conduction,
while the right-hand side is the entropy production rate, Σα , which by the second law of thermodynamics
is non-negative and irreversible. The total entropy production rate, Σ, is the sum of (A.110) over α.
From (A.27) applied to the ion fluid, ∇pi = en(E−F+v × B), so that vi · ∇pi = en(E−F) · vi ,
and using this relation to simplify the sum of the right-hand sides of (A.110), we arrive at
`
´
(A.111)
Σ = qe · ∇θe + qi · ∇θi + j · E/Te − ve · ∇pe + vi · ∇pi /Te ≥ 0 ,
where θα ≡ 1/Tα . The radial component of j is zero, r̂ · ve = r̂ · vi = vD and p = pe + pi , so that
(A.111) yields
Σ = qe θe + qi θi + j E /Te − vD p /Te ≥ 0 ,
(A.112)
where the dash denotes the radial derivative.
Equation (A.112) is in the standard quadratic form for linear constitutive laws, which may be inferred
to have the form (see Woods 1986),
3 2
qe
a11
6 qi 7 6 a21
7=6
6
4 j 5 4 a31
vD
a41
2
a12
a22
a32
a42
a13
a23
a33
a43
3 2
3
Te θe
a14
7
6
a24 7
7 6 T i θi 7 .
a34 5 4 E 5
a44
−p
(A.113)
Let α, β = 1, 2, 3, 4 then the necessary and sufficient condition for Σ ≥ 0 is (i) that the determinant |aαβ + aβα | and all its principal minors are non-negative and (ii) that Onsager’s reciprocal relations
constrain the off-diagonal parameters:
`
´2
aαβ (Bθ ) = ηα ηβ aβα (−Bθ ) ,
(A.114)
aαα ≥ 1 , aαα aββ ≥ 14 aαβ + aβα , . . . ,
where ηα , ηβ are the parities of the associated thermodynamic fluxes under particle motion reversal
(ηα = 1 for qi and qe , and ηα = −1 for j and vD ).
By writing n (Te + Ti )/n = p /(kB n) − (Te + Ti ) in (3.80), (3.81) and (3.83), we can express the
neoclassical equations in the same form as in (A.113) and obtain the values of aαβ , α, β = 1, 2, 3, 4.
The results are:
a11 = 3.34Ape ,
a21 = 0 ,
1
2
a31 = −1.75ε pe /Bθ ,
a41 = −1.55A ,
1
a12 = 1.80Ap
e,
“ e Ti /T
´
i Te
a22 = 0.48 m
Ap
i,
m T
a13 = 1.75ε 2 pe /Bθ ,
a14 = −1.53A
a23 = 0 ,
a24 = 0 ,
a32 = −2.86ε p /Bθ ,
a42 = −1.31ATi /Te ,
a33 = gσ ,
1
a43 = −2.44ε 2 /Bθ ,
a34 = 2.44ε 2 /Bθ ,
a44 = 1.12A/pe .
e i
1
2 i
1
3
where A = qs2 r2Le /(ε 2 τe ).
Onsager’s relations are obeyed by a13 , a14 , and a34 , but not by the other coefficients, which are
incorrect. In any case, as jb is zero, the only non-zero, off-diagonal coefficient satisfying the reciprocal
relations is a14 and the inequality a11 a44 ≥ a214 is satisfied.
204
Appendix A Plasma Physics Notes
A.20 Putative role of turbulence in transport
The large radial losses of thermal energy from tokamaks and the failure of neoclassical theory to explain
this, has encouraged fusion physicists to believe that thermal diffusion is driven by strong turbulence.
However, it is shown below that were this the case, the electrical resistivity around the torus would be so
large that a toroidal discharge would not be possible. In fact currents of several MA are obtained with
drops of less than one volt around the complete torus of the JET tokamak (see Table 4.6).
In a strong magnetic field the perpendicular diffusivity in the electron component is (see (3.32) and
(3.44)
´
`
5kB Te
1
1 ,
(A.115)
χ⊥ = χ / 2 =
e ≡ ωce τee
2 τ
2me ωce
ee
where τee is the electron–electron collision interval. A range of possible turbulence generating mechanisms to change χ⊥ into an ‘anomalous’ value χ∗⊥ are described by Wesson (2004), but there are no firm
conclusions and all seem to require a phenomenological input of uncertain size. Of the several processes
one of the more plausible was that introduced by Kadomtsev and Pogutse (1979) with turbulence due
to fluctuations in the magnetic field. For a brief account of this idea and extensions of it by others see
Woods (1987), p. 430. The outcome is a relation of the type,
`
´
A ∼ 104 ,
(A.116)
χ∗⊥ = Aχ⊥
where A is an unknown function of the number density ne . The value of A in (A.116) is chosen to bring
‘theory’ and observation into rough agreement.
The model leads to
`
´
χ∗⊥ = α2 |B̃/B|2 χ = Aχ⊥ ,
A = e2 α2 |B̃/B|2 ,
where B̃ is the magnitude of a fluctuation in the strength of the magnetic field B, · · · denotes an averaged value and α depends on the phase shift of the waves. The theory is incomplete, so a phenomenological element is required. Experimental values of |B̃/B| ∼ 10−3 have been observed (McGuire and
Robinson 1980) and with e = 107 , a factor of α ∼ 10−2 yields χ∗⊥ ∼ 104 χ⊥ , as in (A.116).
The transport of energy through the electron gas must ultimately be due to electron–electron collisions with the turbulence playing a facilitating role by greatly reducing the collision interval τee . This
∗
. Thus
allows us to interpret (A.116) in terms of an anomalous value for τee , say τee
χ∗⊥ =
5kB Te
5kB Te
1
=
2 τ∗
2
2me ωce
2m
e e
ee
„
2
τee
∗
τee
«
`
´
∗
.
= χ⊥ τee /τee
∗
= τee /A ∼ 10−4 τee , i.e. the turbulence must shorten the collision interval by four orders of
Hence τee
magnitude.
There is an obvious way of testing the result just obtained, namely by checking whether or not the
∗
affects the
large reduction in the electron–electron collision time from the classical value of τee to τee
current-voltage relationship around the torus. The voltage drop V around the complete torus is given by
V = 2πRE, where R is the major radius and the electric field E is related to the total current I, the
minor radius a and the Spitzer resistivity ηs by E = ηs RI/πa2 ; thus, ignoring the contribution due to
particle trapping (see Section 2.4.4), we get V = 2ηs RI/a2 .
Spitzer and Härm (1953) (see Spitzer 1962) showed that the resistivity is related to the Lorentz value
ηL by (see Section A.2)
„
«
3πme
1.32
−1
(A.117)
ηs = ηL /γE ,
ηL =
,
γ
≈
1
+
32e2 ne τe
Z + 0.85
A.21 Solution of a vector equation
205
The collision interval τe in ηL , is proportional to τei because the Lorentz resistivity is based on the
assumption that only electron-ion collisions are involved. The Spitzer value is a correction to ηL that
allows for electron–electron collisions, and it follows that the correct collision interval for the Spitzer
correction is τee .
If we define τei as the 90◦ defection time for the scattering of electrons moving at the r.m.s. speed
(kB Te /me )1/2 by ions, then (e.g. see Woods 1987, p. 253) τei = 0.69τe . Similarly, for 90◦ degree
electron–electron scattering, τee = 0.72τe . Therefore, in the absence of turbulence (A.117) reads
„
«
3π me 0.69
3π me 0.72
1.32
ηs =
+
,
(A.118)
32 e2 ne τei
Z + 0.85 32 e2 ne τee
whereas if τee has its turbulent value,
„
«
3π me 0.69
1.32
3π me 0.72
ηs∗ =
+
,
∗
32 e2 ne τei
Z + 0.85 32 e2 ne τee
(A.119)
ignoring the possibility that τei will also be reduced by turbulence.
Typical voltage drops measured in JET are given in Tables 4.4 and 4.5; the conclusion is that the
actual drops are at least within an order of magnitude the same as predicted by Spitzer resistivity. Were
there sufficient turbulence to increase χ⊥ to its observed values, the effect on ηs∗ would be to increase
it by a factor of ∼ 5000 with a corresponding effect on the voltage drop and no useful current would
flow. It follows that turbulence cannot be present in sufficient strength to account for the high values
of thermal transport in tokamaks. Some other phenomenon not involving turbulence and therefore not
‘anomalous’ must be responsible.
A.21 Solution of a vector equation
Let vectors A and B satisfy
ˆ
˜
α bb + α∧ b×11 + α⊥ (11 − bb) · A = B,
(A.120)
where b is unit vector and α , α∧ , α⊥ are scalar constants, then the solution of this equation is
ˆ
˜
A = β bb + β∧ b×11 + β⊥ (11 − bb) · B ,
(A.121)
where
β =
1
,
α
β∧ = −
α∧
,
α2⊥ + α2∧
β⊥ =
α⊥
.
α2⊥ + α2∧
This is readily verified by direct substitution.
Proof . The scalar constants must satisfy
ˆ
˜ ˆ
˜
α bb + α∧ b×11 + α⊥ (11 − bb) · β bb + β∧ b×11 + β⊥ (11 − bb) = 11 ,
which reduces to
α β bb + (α∧ β⊥ + α⊥ β∧ )b × 11 + (α⊥ β⊥ − α∧ β∧ )(11 − bb) = 11 ,
whence
α β = 1,
and (A.121) follows.
α⊥ β⊥ − α∧ β∧ = 1,
α∧ β⊥ + α⊥ β∧ = 0 ,
206
Appendix A Plasma Physics Notes
A.22 Viscous stress tensor
The transport of momentum through a plasma is determined by the pressure tensor p, which is comprised
of two parts, namely p11 due to the thermodynamic pressure p and due to viscosity: p = p11 + .
As with the (usual) transport of energy, which is described in Section 3.2.2, the transport of momentum takes a collision interval τ to complete and therefore that actually obtained at time t is due to
a cause m say, at (r − τ v, t − τ ). Thus, to a first approximation, p(r, t) ≈ m(r − τ v, t − τ )
= m(r, t) − τ Dm(r, t) , where (see (3.39)) the time derivative D is calculated in convected frame
spinning with the angular velocity Ω = 12 ∇ × v. In Section 3.2.2 we showed that for a vector a
embedded in a fluid element,
Da = Da − Ω × a = a · e ,
(A.122)
where D is the usual material derivation and e is the rate of strain tensor, namely the symmetrical part
of ∇v.
P
Like p, the tensor m may be expressed as a sum of dyads, (ab), where the vectors a and b are
chosen from the unit vectors i, j, and k. Now
D(ab) = (Da)b + aD(b) = a · eb + ab · e = (ab + ba) · e ,
where in the last expression we have used the symmetry of e. The tensor p and its cause m are symmetric
and therefore
Dm = 2e · m = 2m · e .
Hence
p(r, t) = m(r, t) − 2τ m · e(r, t) ,
(A.123)
and it remains to identify m. The dominant component of p is p11 and in the limit as the delay time
tends to zero, the viscosity vanishes and (A.123) reduces to p11 = m. It follows that correct to O(τ ), or
equivalently O(kN ) where kN is the Knudsen number (see (A.108)), this equation becomes
p = p11 − 2µp e
(µp = pτ ) ,
(A.124)
and the first-order viscous stress tensor is therefore given by
1 = −2µp
e.
(A.125)
From (A.99)
◦
g = e + 1 11 ∇ · v
e ≡ 12 (∇v + ∇v)
3
◦
(A.126)
◦
where e is the deviator of ∇v. In fact we could replace e in (A.124) by e, since dilatation is important
only when the medium can exhibit bulk viscosity, which plasmas do not. The above theory holds for
both the ion gas and the electron gas.
Now consider the case when the plasma lies in a magnetic field. The general principle adopted in
Section 3.2.2 and Section 3.2.3 for the heat flux vector, q, can also be applied to (A.124), that is the
value of actually obtained applies at a time τ later than the strain tensor e to which it is the response.
However, the strain tensor itself is the result of earlier collisions. Hence, applying these two time-delays,
we replace (A.124) by
(r + τ v, t + τ ) = −2µp
3
e(r − τ v, t − τ ) + O(kN
).
(A.127)
A.22 Viscous stress tensor
207
The time delay applied to e leads to transport theory accurate to second-order in the Knudsen number,
which case we shall develop shortly. For a theory correct to first-order in the Knudsen number we need
to solve
(r + τ v, t + τ ) =
(r, t) + τ D (r, t) = −2µp e(r, t) ,
(A.128)
where for a vector a , D∗ a = Da + ωc b × a is a time derivative in a frame F in which all ordered
motions have been eliminated, i.e. F spins with the gyrating particles (cf. (3.40)).
Now
D∗ (ac) = (D∗ a)c + a(D∗ c) = (Da)c + ωc b × (ac) + a(Dc) + ωc ab × c
= D(ac) + ωc b × ac − ωc ac × b ,
and more generally for a tensor A,
D∗ A = D A + ω c b × A − ω c A × b .
(A.129)
Hence (A.128) yields
τD +
+
b×
−
× b = −2pτ e .
In an O(kN ) treatment the time derivative can be omitted and with the expansion
(A.130) gives
`
´
b× 1 −
= ωc τ .
1 +
1 × b = −2pτ e ,
(A.130)
=
1 +
2 ··· ,
(A.131)
To solve this equation for 1 we need the analysis given in Section A.23.
It follows from (A.135) and (A.136) that the solution of (A.131) is
1 = −2µp
◦
W ·· e
(µp = pτ ) ,
(A.132)
where W is the fourth order tensor defined in the following section and we have replaced e by its
deviator.
2
To obtain an expression for correct to O(kN
) in an isothermal plasma we start from (A.127), the
strain tensor on the right-hand side of which, by an application of (A.122), reads
e(r − τ v, t − τ ) = e(r, t) − τ De(r, t) = e(r, t) − 2τ e · e .
Hence in place of (A.130) we have
τD +
+
b×
−
`
´
× b = −2pτ e · 11 − 2τ e .
Applying the solution used in (A.132) we get
`
´
`
´
= −2pτ W ·· e · 11 − 2τ e − τ D W ··
,
(A.133)
´
`
the last term of which can be approximated by −τ D W ·· 1 . If we ignore parallel gradients and
O( )2 terms in the time derivative, this term can be omitted from (A.133), in which case the secondorder solution becomes
2 = 4pτ
2
W ·· e · e .
(A.134)
208
Appendix A Plasma Physics Notes
A.23 Solution of a tensor equation
Let tensors A and D satisfy the equation
A+
b×A −
A×b = D,
(A.135)
then its solution is
A = W ·· D ,
(A.136)
where W is a fourth-order tensor with five distinct components:
W = W1 +
1
1+4
2
W2 +
1
1+
2
W3 +
2
1+4
2
W4 +
1+
2
W5 .
(A.137)
The tensors Wi , i = 1, 2, . . . , 5, can be represented as open products of the following second-order
tensors,
11 ≡ bb,
11∧ ≡ b×11,
11⊥ ≡ 11 − bb,
(A.138)
with the convention that a diamond implies that inner products are required, i.e. that for any second-order
tensor A, 11i 11j ·· A = 11i · A · 11j . The basic fourth-order tensors are:
9
>
W1 = 11 11 + 12 (11⊥ 11⊥ − 11∧ 11∧ ),
W1 = 11 11 − 12 11⊥ 11 , >
>
=
1
W2 = 2 (11 11 + 11∧ 11∧ ),
W3 = 11 11⊥ + 11⊥ 11 , > (A.139)
>
>
;
W4 = 12 (11⊥ 11∧ − 11∧ 11⊥ ),
W5 = 11 11∧ − 11∧ 11 ,
where the second form for W1 holds only for its contraction with a deviator, it being readily deduced
that
◦
◦
◦
◦
11⊥ · e · 11⊥ − 11∧ · e · 11∧ = 11⊥ 11⊥ ·· e = −11⊥ 11 ·· e .
The solution in (A.137) can be derived directly from (A.135); alternatively we can confirm it by substitution.
From
11∧ · 11 = 0, 11∧ · 11⊥ = 11∧ , 11∧ · 11∧ = −11⊥ ,
which are readily verified, and (A.139) we can show that
b×W1 − W1 ×b = 11∧ · W1 − W1 · 11∧ = 0,
b×W2 − W2 ×b = 2W4 ,
b×W3 − W3 ×b = W5 ,
b×W4 − W4 ×b = 2W2 ,
b×W5 − W5 ×b = W3 .
Hence by (A.137) and (A.139),
W+
b×W −
W×b = W1 + W2 + W3
= 11 11 + 11⊥ 11⊥ + 11 11⊥ + 11⊥ 11 = 11 11,
which is the unit fourth-order tensor and it follows that when (136) is substituted in (A.135), we get
W ·· D +
b×W ·· D −
W×b ·· D = 11 11 ·· D = D,
and the solution in (A.136) is verified.
A.24 MHD instabilities
209
B
C
D
(a)
k=k
2
Σ
(i)
1
1
2
1
2
(b)
Σ
(ii)
Σ
(iii)
Figure A.6: Thermodynamic instabilities; (a) flute instability, (b) interchange instability
A.24 MHD instabilities
In the tokamak literature certain concepts from the theory of plasma instabilities are frequently cited in
explanations of the various tokamak instabilities. In this note we shall list some of these instabilities,
giving references rather than mathematical details as they will not be required in this text. (For the details
see Woods (2004) or Wesson (2004), where many references are given.)
Exchange instability
In disturbances with crests parallel to the magnetic field the plasma is displaced without twisting
or bending the magnetic field lines, and therefore they offer no resistance to the displacement. In
Fig. A.6(a) the dashed curve CD represents an initial magnetic surface. If after a short time it is convected into the fluted surface indicated by the solid line, and the corrugations continue to develop, we
have what is termed a flute instability. In Fig. A.6(b) the fluting is distorted into a convective overturning
of the initial state (i); in the final state (iii), the flux tubes 1 and 2 have been interchanged and the instability is named accordingly. If there is no dissipation, we may assume that there is no flow across the
field lines and, provided the magnetic energy is unaltered by the field exchange, the question of stability
can be decided by thermodynamic considerations alone.
=c
on
st.
ps
increasing
Σ
p
o = const.
A
B
L
l
Σ
Figure A.7: A perturbed flux surface
210
Appendix A Plasma Physics Notes
Resonance surfaces
With a cylindrical magnetoplasma an interchange instability is possible if the displacements have crests
running parallel to the steady state magnetic
field, B ¯= Bθ θ̂ + Bϕ ϕ̂. We shall assume a standing wave
˘
with amplitude ξ(r, m, k) = ξ̂(r) exp i(mθ + kz) , m = 0, ±1, ±2, · · · , so that the wave-number
vector is k = (m/r)θ̂ + kẑ. Then the condition for an exchange instability is f = 0, where
“
“
Bθ
mν ”
1 ”
m
kBz
.
(A.140)
ν≡
=
f ≡ k · B = Bθ + kBz = 1 +
r
k
rBz
Rq
From (3.101) the pitch of the helical magnetic field is ℘ = 2π/ν and if this fits into the circumference of the torus, so that ℘ = 2πnR, n = 1, 2, · · · then the field lines will close after n
circuits of the torus, making a flute instability possible. In this case ν = 1/Rn = 1/Rq, so that
q = n, n = 1, 2, · · · , which values define ‘resonance surfaces’. From (A.140) it follows that the wave
number satisfies kR = −m/n.
At a given radius, r = r0 , k and m may be chosen so as to satisfy f (r0 ) = 0, making an instability
possible. But provided the pitch changes sufficiently rapidly with r, in distorting the lines of force an
interchange would require an investment of magnetic energy sufficient to offset the loss in potential
energy due to the change in the magnetic pressure. Thus a configuration in which the magnetic field
lines are sheared is more stable than one in which they are not.
Suydam (1958) found a necessary condition for the stability of a sheared magnetoplasma, which
Laval et al. (1971) later extended to a toroidal system; this condition is:
r “ d ln ν ”2
2µ0 dp
+
≥ 0.
(A.141)
(1 − q 2 ) 2
Bz dr
4
dr
The first term in this criterion is likely to be negative, so there is a minimum value for the radial variation
in the pitch below which the plasma is unstable.
Minimum B configuration
Let Σ be a magnetic surface enclosing a magnetoplasma of cross-section A(l), where l is the distance
measured along a field line L on Σ (see Fig. A.7). The volume within Σ is
Z
Z
dl
,
V = A dl = ψ0
B
since the flux ψ0 = AB is constant (cf. Section 2.1.1). The volume change due to a fluting of the surface
near L is
Z
dl
.
(A.142)
θ≡
δV = ψ0 δθ,
B
Let ps be the pressure just outside Σ and p the pressure within Σ, then in an isentropic perturbation,
mechanical stability requires that (ps − p)δV > 0, i.e. the configuration is stable if and only if
(ps − p) δθ > 0 .
(A.143)
Let ψ denote the unperturbed flux variable then (see Fig. A.8)
Z
Z
“ ∂θ ”
δψ0 =
dψ ,
dψ = 0,
δθ =
Σ0 −Σ
Σ0 −Σ ∂ψ
from which it follows that δθ and ∂θ/∂ψ have the same sign. And since ps is at a larger value of ψ than
p, we can write (A.143) as
∂p ∂θ
> 0.
∂ψ ∂ψ
(A.144)
A.24 MHD instabilities
t.
Σ0
ns
o
=
co
211
d
Σ
= const.
Figure A.8: Evaluation of δθ
n, B
B
B
stab
e
stabl
n, B
unstable
n, B
le
plasma
Figure A.9: Effect of curvature on stability
Pressures usually fall away with increasing distance from the axis of a magnetically confined plasma;
in this case the stability condition becomes
Z
Z
∂
dl
dl
< 0, or
< 0.
(A.145)
δ
B
∂ψ
B
When this is satisfied, the field strength increases outwards from the plasma and we have what is termed
a ‘minimum B configuration’.
From (A.10) and
“ 1
”
1 B2n
1
,
B · ∇B = ∇
B2 +
µ0
2µ0
µ0 R
where n is unit vector orthogonal to B, directed towards the field line center of curvature, of radius R,
we find
“
1 2”
B2n
∇⊥ p +
B =
,
2µ0
µ0 R
Hence, when the field is convex towards the plasma, B increases outwards and the contribution to θ is
negative, which is stabilizing; fields that are concave towards the plasma increase θ and are therefore
destabilizing. In Fig. A.9 we have illustrated this feature of field line curvature for mirror geometry.
Because the two stabilizing contributions to θ occur at large B, they are unable to offset the destabilizing
effect of the central region at low B. Thus, as a whole, the mirror geometry is unstable unless other
stabilizing effects can be included to modify the situation.
212
Appendix A Plasma Physics Notes
y
0
B
By
B
R
B
x
B=0
B
By
-
(a)
y
B
vy
B
vx
0
B
R
B
vx
x
vy
-
(b)
Figure A.10: The tearing mode; (a) initial perturbation, (b) developed perturbations, showing
‘magnetic islands’
Resistive tearing mode and magnetic islands
The tearing mode instability occurs in the current layer separating opposing magnetic fields, and is
triggered by a small amplitude standing wave lying in the interface. This is illustrated in Fig. A.10(a), in
which the dotted lines show the initial magnetic field configuration; in Fig. A.10(b) this has developed
to the stage when the field lines have ‘reconnected’, creating stationary loops that slowly fade away. The
instability strengthens when the magnetic field lines come close together at points like R. This produces
a large positive B̂y in the neighborhood of R, which in turn enhances the speed of convection towards
R, further increasing B̂y , and so on, until ‘reconnection’ occurs, forming what are termed ‘magnetic
islands’. The diffusion process tends to reduce the curvature of the B̂y profile, so it is essential that there
be an inwards flow, convecting the magnetic field, as shown in Fig. A.10(b).
The reconnection of sheared field lines in the manner described above, i.e. with the aid of the tearingmode instability, requires a plasma flow strongly converging towards the neutral points in the magnetic
field, a process known as ‘flux pile-up’. Magnetic energy is transformed into thermal energy via this
forced ohmic dissipation at rates much higher than with ‘normal’ ohmic dissipation. However, when
reconnection is invoked as a mechanism to explain sudden changes in tokamak variables, the issue is
usually the speed at which this happens — quite often it is much too slow.
A.25 The Catherine wheel fallacy
Perhaps the most incredible argument to be found in the tokamak literature is that illustrated in
Fig. A.11 — there is said to be a fluid flow at right angles to B with a magnitude proportional to
the number density gradient, even though the guiding centers G1 , G2 , . . . remain stationary, presumably because they are ‘nailed’ down by the magnetic fields lines. More particles move to the right in
the figure than to the left because of the number density gradient, but whether or not the kinematics of
this ‘Catherine wheel’ array corresponds to realistic stable plasma dynamics is not questioned (e.g. see
Spitzer (1962), p. 26, Chen (1974), p. 61, Helander and Sigmar (2002), p. 66). In contemplating the
figure, it should be remembered that the average Larmor radius in tokamak conditions for the electrons
A.25 The Catherine wheel fallacy
213
is only ∼ 0.1 mm and ∼ 5 mm for the ions, and by definition there is no way that guiding centers can be
detached from their particles. Any force that is said to be acting on the guiding centers — which have
no substance — can do so only by the agency of the associated particles.
It should be clear from the theory in Section A.4 that in a uniform magnetic field, only if the reference
frame for the orbits is convected with the fluid, will they be circular; in any other reference frame the
orbits will be cycloidal. The simultaneous presence of fluid motion and stationary circular orbits is an
oxymoron. There are two distinct resolutions of this triumph of geometry over physics. Firstly, assume
that the diagram represents a stable flow and determine the missing equilibrium condition, or treat it as
being unstable and allow for the increment δv in the fluid velocity that occurs in a gyro-time.
In the case of stable flow, if the magnetic field is uniform, equilibrium requires a constant pressure,
i.e. p = kB nT = const. Hence the particles arriving in the boxes shown in the figure from the less dense
region are hotter, that is on average they are moving faster and fewer particles moving at greater speeds
are equivalent in mass flux to more particles moving slower. A similar argument applies when there is
a gradient in the magnetic pressure, in which case the mass balance is achieved through the gradient in
the Larmor radii, which become smaller with increasing B.
When equilibrium conditions do not apply, the velocity increment δv is subtracted from the velocity
of the ‘G1 ’ particles as they arrive at the interface and added to the velocity of the ‘G2 ’ particles as they
arrive at the same interface. With δv chosen to reduce the relative mass flux at the interface to zero, we
can determine the acceleration required to make physical sense of the kinematics.
The description “diamagnetic drift” is often employed to describe the metaphoric phenomenon illustrated in Fig. A.11; in fact the misused adjective diamagnetic appears to be applied to any flux orthogonal
to the field, e.g. the term b × ∇pi /(ene B) in equation (2.22) for v⊥ might be called the “diamagnetic
drift velocity”, the transverse heat flux q∧ defined in (3.112) becomes the “diamagnetic heat flux”, and
even the equilibrium relation, j = b × ∇p/B, defines a “diamagnetic current”. Hazeltine and Meiss
(1992) define plasma diamagnetism as ‘the depression of magnetic fields by plasma pressure’, that is
the concept is based on the conservation of plasma momentum, equation (A.7). However, diamagnetism
and its opposite, paramagnetism, are constitutive properties of the medium, independent of the generally
valid conservation laws.
Figure A.11: Diamagnetic drift ‘explained’
214
Appendix A Plasma Physics Notes
The correct definition of “diamagnetic” is as follows; in the relation B = µH between the magnetic induction and the magnetic field, the permeability µ is written as (χm + 1)µ0 , where χm is the
magnetic susceptibility, then a medium is diamagnetic if χm < 0 and paramagnetic if χm > 0. The
Bohr–van Leewen theorem states that when Boltzmann statistics is applied completely to any dynamical
system, it is found to have zero magnetic susceptibility (see Van Vleck, 1932). Thus a magnetoplasma
is classically non-magnetic and when the spin magnetic moment of the electron is taken into account
(e.g. see Ginzburg 1970, p. 27), it is infinitesimally paramagnetic. Were a plasma really magnetic, a
magnetization current would be an essential part of Ampère’s law, but in plasma physics this is almost
always given its correct form, with free space permeability, namely µ0 j = ∇ × B. The problem with
the incorrectly applied word ‘diamagnetic’ is that the drifts of mass, charge and energy that are decorated
with this adjective are usually equilibrium relations that hold only in steady states. This appears to be
forgotten; any kinematic model in which these drifts are involved must always be checked for stability.
The analogy between the “Catherine wheels” of Fig. A.11 and the banana orbits of Fig. 3.11 should
be clear. The center points of the banana orbits (n1 and n2 ) are in effect super ‘guiding’ centers for
the particles (strictly, particle guiding centers) traversing their bananas, and — as described in the first
paragraph of Section 2.4.4 — viewed in the laboratory or tokamak frame, a force is required on the
particles traversing the banana orbits to hold these orbits stationary relative to the magnetic mirrors, and
then the kinematics represent a stable flow. In steady state conditions the argument given in Section 3.4.3
is valid and the bootstrap current vanishes.
An alternative treatment is as follows. We return to (3.88) and remove an obvious blemish in the
argument by allowing for the radial dependence of the number of trapped particles:
F = c r
Thus
F = c r
1 ´
d`
(2ε) 2 n
dr
“ 2 ”1
2
R
H,
(ε = r/R) .
(A.146)
“ 1 dn
n ”
.
H ≡ r2
+
1
dr
2r 2
Adopting the empirical distribution for ne given in (4.5)1 we get for the electrons
H=
n0
2r
1
2
¯
˘
(1 − y)αn −1 1 − (4αn + 1)y .
The average value for αn for the H-mode in Table 4.5 is 0.76, so with this value, H = 0 at y ≈ 0.25, i.e.
r ≈ 0.5 a, which happens to be the location of the inversion radius (see Fig. 4.4). It follows that in the
inversion radius region no transfer of momentum can occur from trapped to passing electrons. Similarly,
this is also true for the ions. This shows that, even without appealing to the force acting on the particles
imposed by the stationary mirrors, there is no bootstrap current in the mid region of the plasma.
What happens in the regions on either side of r ≈ 0.5 a is that the mirror force acts to maintain equilibrium by altering the particle speeds in much the same way as the pressure force did in the Catherine
wheel array. We conclude that the mechanism required to generate the bootstrap current does not exist.
Finally, while the mathematical derivation of the formula in (3.84) for jb from the ‘security’ of
kinetic theory appears to bypass the physical objections raised above, the velocity distribution functions
adopted do not reflect the existence of the mirror force necessary to maintain a steady state (e.g. see
Hellander and Sigmar 2002, p. 206). The model adopted in this text is that the trapped particles oscillate
back and forth in the banana orbits many times per electron collision time and any momentum transferred
to passing particles cancels out in direction. Therefore the passing particles obey the usual Spitzer law,
with the trapped particle modification described in Section 2.4.4, while the trapped particles carry no
current at all.
A.26 Limitations of Boltzmann’s kinetic equation
215
A.26 Limitations of Boltzmann’s kinetic equation
In this final note we shall present Boltzmann’s fundamental equation for the evolution of the velocity distribution function f (see Section A.12) and briefly explain why it is valid only to first order in
the Knudsen number kN . Boltzmann derived it in order to explain how f approached the equilibrium
distribution f0 discovered by Maxwell a few years earlier and also to deduce the second law of thermodynamics by a purely mechanical argument. Only about 10% of Boltzmann’s very long original paper
2
) and
was directly concerned with transport and the idea of developing transport theory correct to O(kN
even beyond was not entertained. Much later scientists applied Boltzmann’s equation over a wide range
of Knudsen numbers without being aware of its limitations. The approach to second-order transport
theory presented in Section 3.2 was chosen to avoid these limitations, and it implies that the theory of
this text cannot be reproduced by the kinetic theory usually adopted by tokamak physicists, namely the
Fokker–Planck equation (which in fact is a special case of Boltzmann’s equation).
Boltzmann’s integro-differential equation for f1 reads (cf. (A.68) and (A.78))
ZZ
∂f1
∂f1
∂f1
(A.147)
+w·
+ ẇ ·
=
(f1 f2 − f1 f2 )gα(k | k ; g) dk dw2 ,
∂t
∂r
∂w
where class ‘1’particles of the distribution f1 (r, w1 , t) are moving with a speed g = gk relative to class
‘2’particles of f2 (r, w2 , t) and colliding with them. The collisions scatter the class ‘1’particles into the
element dk of solid angle, the differential cross section for which is α(k|k ; g).
The basic assumption is that the velocity of a particle is uncorrelated, both with its position and with
the initial velocity of any particle with which it is about to collide. Thus the probability that a class ‘1’
molecule collides with a class ‘2’ molecule is assumed to be proportional to the product of their phasespace number densities, f1 f2 ; the neglect of correlations between the particles is termed the ‘hypothesis
of molecular chaos’, and this is the weak point in the theory.
The problem arises in its simplest form with the pressure gradient force P = −∇p/ per unit mass.
Because of the continuous nature of the collisional forces acting on particles (see Section A.2), during
the transit of a class ‘1’particle over one collision interval τ , the acceleration P alters its speed by τ P, so
that it now belongs to the distribution f1 + τ P · ∂f1 /∂c1 . The collision rate is therefore proportional to
(f1 + τ P · ∂f1 /∂c1 ) f2 , instead of f1 f2 . To see that redefining f1 to include the pressure gradient term
does not resolve the problem, we can approximate f1 in the O(kN ) term by its equilibrium distribution
to obtain (see (A.79))
o
n
P · c1
f10 f2 ,
f1 − τ
2
C
which shows that the collision rate depends on the direction from which the incident particle ‘1’approaches particle ‘2’. This makes it impossible to replace the distribution of class ‘1’particles by a
single function valid for all directions immediately before the collision. In other words the hypothesis of
molecular chaos is not valid, although of course it is a satisfactory approximation for first-order transport
theory.
Although it appears that this conclusion about molecular chaos depends on the relatively continuous
nature of the pressure gradient force in a plasma, it also holds with the impulsive collisions of neutral gas
particles. In the latter case, since the pressure gradient force is transmitted intermittently at the collisions,
its effect on Boltzmann’s collision operator is exactly the same as obtained above for a plasma. Thus
the integrand in (147) must be modified so that the average direction of colliding particles lies parallel
to the pressure gradient, with a magnitude sufficient to transmit the pressure force. In the second-order
transport theory associated with the names of Burnett and Cowling (Chapman & Cowling, 1970), its
neglect produces some physically impossible results, e.g. frame-dependent terms in the second-order
heat flux vector, q2 , and terms that yield heat transport in isothermal conditions.
216
Appendix A Plasma Physics Notes
There are other problems with Boltzmann’s collision operator, for example fluid shear also imposes a
directional anisotropy similar to that described here for the pressure gradient. Then there is the difficulty
with the magnetic field, which leads to considerable algebraic complexity (see Chapter 11 of Woods
1993), which we have avoided by the mean free path approach adopted in this text. In judging the
accuracy of this method, which depends on empirical collision intervals τ , it should be remembered that
an essential part of second-order transport theory was that of finding appropriate values for τ , which for
thermal transport had to be replaced by an average bounce time (see (2.78)). In effect, this makes a mean
free path approach to tokamak transport the only possible route.
References
217
References
(For abbreviations for conference proceedings see page 25.)
Bhatnager, P.L., Gross, E.P., & Krook, M. (1954). Phys. Rev., 94, 511.
Chapman, S. & Cowling, T.G. 2nd. Ed. (1958), 3rd. Ed. (1970), The mathematical theory of non-uniform
gases. Cambridge University press, Cambridge.
Chen, F.F. (1977). Introduction to plasma physics. Plenum Press, New York.
Connor, J.W. & Taylor, J.B. (1977), Nuclear Fusion, 13, 1047.
Ferziger, J.H. & Kaper, H.G. (1972). Mathematical theory of transport processes in gases. NorthHolland Pub. Co., Amsterdam.
Ginzburg, V.L. (1970), The propagation of electromagnetic waves in plasmas. Pergamon Press, Oxford.
Hazeltine, R.D. & Meiss, J.D. (1992). Plasma confinement. Addison-Wesley Pub. Co.
Helander, P. & Sigmar, D.J. (2002). Collisional transport in magnetized plasmas. Cambridge University
press, Cambridge.
Kadomtsev, B.B. (1975a). Sov. J. Plasma Phys., 1, 295.
Kadomtsev, B.B. & Pogutse O.P. (1979). Plasma Physics and Controlled Fusion Research, O, Vol 1,
649.
Lacina, J. (1971). Plasma Physics, 13, 303.
Laval, G., Luc, H., Maschke, E.K., Mercier, C., & Pellat, R. (1971). Plasma Physics and Controlled
Fusion Research, Vol 11, 509. IAEA, Vienna.
McGuire, K.M. & Robinson, D.C. (1979). Nuclear Fusion, 19(4), 505.
Sedov, L.I. (1959). Similarity and dimensional methods in mechanics. Academic Press, New York.
Spitzer, L. (1962). Physics of fully ionized gases, 2nd edn., Interscience, New York.
Spitzer, L., & Härm, R. (1953). Phys. Rev., 89, 977.
Suydam, B.R. (1958), Proc. 2nd U.N. Conf. on the Peaceful Uses of Atomic Energy, Vol. 31, p. 157.
United Nations, Geneva.
Van Vleck, J.W., (1932). The theory of electric and magnetic susceptibilities, Oxford University Press,
Oxford.
Wesson, J., (2004). Tokamaks, (3rd edn.) Clarendon Press, Oxford.
Woods, L.C., (1986), The thermodynamics of fluid systems. Oxford University Press, Oxford.
Woods, L.C., (1987), Principles of magnetoplasma dynamics. Oxford University Press, Oxford.
Woods, L.C., (1993). An Introduction to the Kinetic Theory of Gases and Magnetoplasmas. Oxford
University Press, Oxford.
Woods, L.C., (1996). Thermodynamic inequalities in gases and magnetoplasmas. John Wiley & Sons,
Chichester.
Woods, L.C., (2004). Physics of plasmas. Wiley-VCH, Weinheim, Germany.
Index
ambipolar
constraint 12, 103–104, 156
flow 121, 123–124
Ampère’s law 179
aspect ratio 6, 34
ballooning instability 8, 143, 159–161
banana
bounce time 42, 47, 48
orbits 42, 43, 53–60
regime 71
beam 14, 24
beta β 3, 8
limiting values 162
poloidal 8, 87, 99, 143
toroidal 8
Boltzmann’s collision operator 215
Boltzmann’s kinetic equation 67, 215–216
bootstrap current 5, 74, 133, 136
absence of 73, 196, 214
boundary condition
L- and H-modes 99–101, 117
bremsstrahlung 10, 137
captured electrons
escape of 43, 191
resistivity due to 43–44
collision
frequency 182
interval 180, 195
of electrons 10, 180
in momentum transport 62
ions 180
collisionality 23, 71
collisions 180
Coulomb 182
electron–electron 182, 204
energy dissipated in 184
conductivity, see also diffusivity, thermal
classical 61–65
electrical, see resistivity
neoclassical 18, 60, 71–76
second-order 66–67
thermal 61
confinement time
electron energy 12, 18, 20–22, 79, 88,
93–94, 98, 104–106
energy, saturation of 21
Kaye–Goldston 105
L- and H-modes 101, 105
momentum 14, 141
particle 13, 119–121
theory & observation 104–105
total energy 11, 104
with auxiliary heating 22, 104–106
connection length 40
convection and diffusion 73, 196, 198
corona, solar 68
current
drive 5
non-inductive 5, 119, 132
Pfirsch–Schlüter 70
profile
hollow 136
instability of 113
cyclotron frequencies 184–185
cylindrical coordinates 27, 78, 186
Debye length 180
density limit 24, 152, 153
diamagnetic drift 213
diamagnetism, absence of 213
diffusion
dimensions of 198
frame indifference of 199
Theory of Tokamak Transport: New Aspects for Nuclear Fusion Reactor Design. Leslie Colin Woods
Copyright © 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
ISBN: 3-527-40625-5
220
of momentum 138
of particles 122
Pfirsch–Schlüter 72
diffusivity
angular momentum 139–141
frame indifference of 199
ion 61
magnetic 199
mass 70
neoclassical 61
Pfirsch–Schlüter 71
physics of 46–48
plasma 75, 120, 126
thermal 61, 91, 97, 116
trapped particles, due to 45–49
disruptions 24, 25
and radiation 153, 158
auxiliary heating 152, 157
causes of 156, 158
collapse phase of 156–158
density limit 158
Kadomtsev’s model 145
low qa limit 158
Murakami limit 152, 158
precursor sequence 162–163
precursor waves of 154
soft 157
stability diagram for 156
divertor 6, 22
drift kinetic equation 196
drifts, guiding center 53–58, 197
electric current density 37
electric field, constraint on 56
electrical resistivity, see resistivity
electron
cyclotron heating 17
energy confinement time 18, 91
energy losses 21
source term 120, 128
thermal diffusivity 76–79
viscosity tensor 32, 122
ELM-free H-mode 136
ELMs 166–168
ELMy H-mode 101, 104, 143
elongation 34, 104
empirical
profiles 88–90
scaling laws 19, 94, 104
Index
indices in 20
energy
confinement time 11, 39
at high beta 21
dimensional analysis for 185
losses 3, 20
energy equations 183–184
enthalpy 11
equations of motion 68, 179, 182
fishbone oscillation 51, 143
fluid element
rate of strain of 192, 193
spin of 193
fluid motion
equation of 11, 68, 179
shear of 66, 194
Fourier’s law 18, 62
in a magnetoplasma 62–65
frame indifference 58, 184, 199
fusion reactions 2
Grad–Shafranov equation 28–29, 34
integral constraints for 30, 31
solution of 32, 35–36
Greenwald limit 5, 153
guiding center 55, 185, 197
drift velocity of 57, 197–198
H-mode 22, 23
boundary condition for 23, 100
ELMy 101
heat flux 11, 61
physical mechanism for 79–82
second-order 67, 68
transverse 65
up temperature gradients 46, 77
heating
neutral beam 16–17, 22
ohmic and auxiliary 22
RF 17
Hugill diagram 152, 158
ignition curve 2
impurities 6, 9, 10
radiation due to 10, 153, 158, 173
transport of 120
instabilities
ballooning 159–161
Index
edge localized modes (ELMs) 101, 166
kink 159
MARFE 173–174
MHD 158, 209
Mirnov 143, 155
resistive 158
sawtooth, see sawtooth
Suydam criterion 210
tearing mode 143, 155, 159
thermal 159
internal energy 11
internal transport barriers (ITEs) 111
inversion radius 96, 99, 112
ion conductivity, measurements of 104
ion-cyclotron resonance heating 17
ionization 31, 128
kinetic equations 193–196
BGK 195
drift 71, 196
in convected frame 194
Knudsen number expansion of 195
Knudsen number 62, 67, 117, 201
constraint 63, 84
in tokamaks 84
L- and H-modes 99–101
L-mode 21, 22
boundary condition for 100
transition to H-mode 117, 164–166, 171
Larmor radius 54, 56, 184
limiter 6, 8, 92
ln Λ 182
loop voltage 107–108
induced 130
instability of 133–135
Lorentz current
bootstrap interpretation 135
instability due to 135
Lorentz voltage 131, 156
magnetic
axis 7, 8, 32, 34, 40
bottle 38
field 54, 64
island 145, 212
mirrors 38
energy sinks at 45
in tokamak field 41
221
moment 38
pressure 3, 180
reconnection 145, 212
shear 77, 106, 113
surface 9, 29, 30
throat 39
magnetization current 58
magnetoplasma transport theory 64
magnetostatics 27, 32, 180
major disruptions 143, 151–153
and radiation 153
ballooning limit 159–161
collapse phase of 151, 156–158
density limit 152, 153
Greenwald limit 153
Hugill diagram for 152
Murakami number in 152
MARFEs 173–174
Maxwell’s distribution function 39, 190
Maxwell’s EM equations 199
mean free path 47, 61
MHD
equations 27, 179–180
equilibrium 5, 27–32
instabilities 209–212
ballooning modes 150–161
exchange 209
flute 209
magnetic islands 212
minimum B 210
pitch 210
resistive tearing mode 212
stability 34
β limit 162
minimum B configuration 211
minor disruption 143
collapse phase of 156
Mirnov oscillations 143, 155
mirror force 189
mirror ratio 39, 41
momentum diffusion 119, 138
Murakami parameter 152
neoclassical
diffusivity 60–61
resistivity 43
transport theory 18, 21, 44, 71–120
failure of 74, 203
reciprocal relations for 203
222
tests of 73
neon puffing 22, 128, 129
neutral beam heating 16–17
neutral beam injection (NBI) 104
Ohm’s law
general form for 30, 183
tensorial form of 187–188
ohmic heating 15, 16
enhanced by particle trapping 44
limitations of 15, 16
steady state with 109–110
Onsager’s reciprocal relations 76, 202–203
particle
confinement time 127, 129
diffusivity 70–71, 127
spin 62
transport 125–127
pinch velocity 126
trapping 38–41
resistivity due to 44
particles
detrapped 49
magnetic moment of 188–189
passing 47
reflected from mirror 38
trapped, escape time of 191–192
peculiar velocity 181, 196
pedestals 166
pellets 23, 143, 168, 171
PEPs 23, 171–173
Pfirsch–Schlüter transport 68–70
plasma
beta 3, 7
contaminated 13
current 7
diffusion velocity 32
frequency 179
ignition 2
inductance 36, 161
radial flow of 120, 123, 127
sheared 210
plateau regime 72
poloidal beta 8, 22
poloidal flux 29
pressure 181, 202
tensor 180, 206
profile
Index
empirical 88–90
energy losses 23
instabilities 112–115
parameters 90
q, safety factor 8, 40, 77, 87
distribution of 89
gradient in 77
instability 112–114
profile 112
radiation 76
and major disruptions 153, 158
losses 12, 23, 153
radio-frequency heating 17
reconnection model 143, 212
reflection points 42
replacement time
energy 11, 109
momentum 14
plasma 13
resistive instabilities 158–159
resistivity
due to particle trapping 43
electrical 33
neoclassical 43, 137
parallel 10
Spitzer 180–183
tokamak 108
trapping factor 131
resonance surfaces 209
safety factor, see “q”
sawtooth
collapse phase 24, 147
double 146
energy losses 95
instability 68, 143
Kadomtsev’s model for 145
oscillations 87, 100, 144–151
period 144, 149–151
Lorentz voltage in 150
ramp phase 24, 147–150
reconnection model for 145
scaling laws, confinement 19, 185
second-order
heat flux 66
tensors 199
transport 48, 66–68, 80
Index
viscosity 138
Shafranov shift 34, 36, 42
sheared flow 81
snakes 168–171
conditions for 112, 143, 168
from pellet injection 143, 168
stream function 29
supershot regime 106
surface quantities 7, 28
Suydam criterion 210
tearing mode instability 143, 153, 155, 159
temperature 202
profiles 24, 87, 95–96
oscillation of 95
tensors, analysis of 200
thermal
barriers 101, 112
collapse 162
conductivity 63
electron 78
ion 65
diffusivity 96–98
electron 61, 78, 87, 91
equation for 79
ions 103
radial dependence of 12
radial distribution of 90
energy transport, electron 113
equilibrium 202
instability 77, 114–115
pumping 49–51, 155
quench 163
transport, ion fluid 102
thermodynamic
equilibrium
local 201
instabilities 210
thermoelectric term 33
tokamak
elongation 5
heat flux in 99
heating 14–18
mirrors 40
transport 12
typical parameters 4
tokamaks, see also Table 1.1
Alcator C 173
ALCATOR-C 105
223
ASDEC 168
D-III-D 162
DIII-D 91, 115, 158
DOUBLET III 4
Doublet III 104
ISX-B 24
ITER 174
JET 4, 112, 120, 123, 129, 135, 144,
146, 150, 157, 162
typical variables 94
JT-60U 111
Pulsator 153
pulsed operation of 5
T-3 4
TEXTOR 145
TFR 144, 146
TFTR 134, 141
toroidal coordinates 200
toroidal velocity 138–141
transport
diffusive 18
neoclassical 18
Pfirsch–Schlüter 68–70
unsteady conditions 47
transport barrier 166
internal 110–112
pedestal 164–166
transverse heat flux
deflection of 45, 65
trapped electrons
loss of conductivity 43
trapped particles
bounce time of 42
diffusivity of 45
energy losses due to 39
fraction of 39
parallel speed of 42
resistivity due to 10, 43
Troyon limit 159–162
turbulence 94, 128
anomalous transport 83
loop voltage problem with 204–205
magnetic field fluctuations 204
role of 82–83
transport due to 204–205
vacuum region, effect of 38
velocity distribution function 190–191
viscous stress tensor 33, 119, 121–122, 206
224
Index
voltage
loop 108
Lorentz 132–133
Ware pinch 73, 76
and bootstrap current 73
Z-effective 9, 12, 137