Reversed Field Pinch:
equilibrium, stability and transport
Piero Martin
Consorzio RFX- Associazione Euratom-ENEA sulla fusione, Padova, Italy
Department of Physics, University of Padova
Notes for the lecture at the European Ph.D. Course (Garching, 28 September 2009)
European Ph.D. course . - Garching 29.09.08)
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Note for users
These slides are intended only as tools to accompany the
lecture. They are not supposed to be complete, since the
material presented on the blackboard is a fundamental part of
the lecture.
European Ph.D. course . - Garching 29.09.08)
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Outline of the lecture
1) MHD equilibrium basics
2) 1d examples
1) Q-pinch
2) Z-pinch
3) Screw pinch
3) RFP equilibrium basics
4) RFP Stability
5) RFP dynamics and the dynamo.
6) Effects on transport
European Ph.D. course . - Garching 29.09.08)
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The RFP: what and why
RFP exist: e.g. RFX-mod
The largest RFP in the world, located in Padova, Italy
A fusion facility for MHD mode control
a=0.459 m, R=2 m, plasma current up to 2 MA
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A dynamic and well-integrated community
Stockholm
Madison
Padova
RFX-mod
Kyoto
EXTRAP T2R
RELAX
MST
The RFP: a tight link with University
(all experiments in University environment)
and a nursery for the fusion community
2008 IAEA Fusion
Energy
Conference,
GenevaColloquium
- P. Martin - June 4th, 2009
Princeton
Plasma
Physics
Laboratory
RFP: exploiting the weak field
The distinctive feature of the RFP that motivates its interest as a fusion
energy system is the weak applied toroidal magnetic field.
The RFP configuration is similar to a tokamak…
– like to the tokamak, the RFP is obtained by driving a toroidal electrical current in a
plasma embedded in a toroidal magnetic field  pinch effect.
…..but the applied toroidal field is 10 – 100 times weaker !
Most of the RFP magnetic field is generated by
current flowing in the plasma (driven also by a
dynamo mechanism)
DOE Fusion Science, Germantown, MD - 23/07/2008
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Why the RFP ?
A current-carrying low magnetic field configuration as the RFP:
– has several technological advantages as a potential reactor configuration and
will therefore contribute to the development of a viable reactor concept
– has unique capabilities to contribute to fusion energy science and
technology research
DOE Fusion Science, Germantown, MD - 23/07/2008
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Fusion potential of the low magnetic field
volume - averaged pressure
high engineering beta  
surface - averaged magnetic pressure (at the coils)
– For configurations like the tokamak the maximum field at the magnet is of order twice the
field in the plasma, whereas in the RFP the field at the magnet is less than in the plasma.
– The engineering beta in an RFP reactor might be as much as twice the physics beta (up to
26% in present experiments)..
Use of normal (rather than superconducting) coils,
High mass power density,
Efficient assembly and disassembly,
Possibly free choice of aspect ratio
DOE Fusion Science, Germantown, MD - 23/07/2008
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A comprehensive understanding of toroidal magnetic confinement, and
the possibility of predicting it, implies that plasma behavior would be
predictable over a wide range of magnetic field strengths.
The RFP provides new information since it extends our understanding to
low field strength, testing the results derived at high field with the
tokamak.
DOE Fusion Science, Germantown, MD - 23/07/2008
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MHD Equilibrium basics: just to
refresh our knowledge
The MHD equilibrium problem
Time-indpendent form of the full MHD equations with v=0
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Linear vs. toroidal configurations
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Magnetic flux surfaces
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Current, magnetic and pressure surfaces
The angle between J and B is in general arbitrary
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Rational, ergodic and stochastic
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Revisiting stochastic magnetic fields in present
day fusion devices
Coils like these are presently under consideration in ITER to produce, by
purpose, stochastic magnetic field for ELM suppression (Resonant
Magnetic Perturbation)
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Surface quantities
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One-dimensional configurations
Even if the magnetic configurations of fusion interest are toroidal, some physical
intuition can be obtained by investigating their one-dimensional, cylindrically simmetric
versions.
This separates:
– Radial pressure balance
– Toroidal force balance
For most configurations, once radial pressure balance is established, toroidicity can be
introduced by means of an aspect ratio expansion, from which one can then investigate
toroidal force balance.
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 pinch
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A simple example: -pinch
Configuration with pure toroidal field
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A simple example: -pinch
The sum of magnetic and kinetic pressure is constant throughout the plasma
The plasma is confined by the pressure of the applied magnetic field
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Experimental -pinch
Experimental -pinch devices among the first experiments to be realized
End-losses severe problem
A -pinch is neutrally stable, and can not be bent into a toroidal equilbrium
Additional field must be added to provide equilibrium
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European Ph.D. course . - Garching 29.09.08)
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Z-pinch
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Z-pinch
Purely poloidal field
All quantities are only functions of r
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Z-pinch
In contrast to the -pinch, for a Z-pinch it is the tension force and not the magnetic pressure
gradient that provides radial confinement of the plasma
The Bennet pinch satisfies the Z-pinch equilibrium
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Bennet Z-pinch
Tension force acts inwards, providing radial pressure balance.
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Experimental Z-pinch
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Z-machine
The Z machine fires a very powerful electrical discharge (several tens
million-ampere for less than 100 nanoseconds) into an array of thin,
parallel tungsten wires called a liner.
Originally designed to supply 50 terawatts of power in one fast pulse,
technological advances resulted in an increased output of 290 terawatts
Z releases 80 times the world's electrical power output for about seventy
nanoseconds; however, only a moderate amount of energy is consumed
in each test (roughly twelve megajoules) - the efficiency from wall
current to X-ray output is about 15%
At the end of 2005, the Z machine produced plasmas with announced
temperatures in excess of 2 billion kelvin (2 GK, 2×109 K), even reaching a
peak at 3.7 billion K.
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The general screw pinch
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General Screw Pinch
Though the momentum equation is non-linear, the -pinch and Z-pinch forces ad as a
linear superposition, a consequence of the high degree of symmetry
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