Present understanding of RFP dynamics:
theory and simulation
D.F. Escande
UMR 6633 CNRS-Université de Provence, Marseille, France
Type of talk
Not a pedagogical introduction
Stress on important facts which are often overlooked
Summary
Alpha confinement might not be crucial for the RFP
A non stationary ohmic RFP might be reactor-relevant
Universal (F,Q) diagram
Single helicity paradigm
An important fact for the reactor relevance of the RFP
Te (KeV) Thomson Scattering
Electron Temperature increases with current: no signs of saturation
1,6
1,4
1,2
1
0,8
0,6
0,4
0,2
0
0
0,5
1
1,5
2
I (MA)
All densities
CTS Meeting – 14.01.2008 – Padova
P.Martin
Alpha confinement might not be crucial
for the RFP
Ohmic heating to thermonuclear temperatures is likely.
Important corollary: in contrast to the tokamak and the stellarator,
good alpha confinement might not be crucial for the RFP.
Possibly it is even not desirable: a strong transfer of alpha energy
might perturb too much the central magnetic state obtained and
sustained through ohmic heating.
No need of producing a quasi symmetry for the magnetic field.
No need to bother about fast particle instabilities.
However some good alpha confinement will exist at the edge, if the
plasma boundary is axis-symmetric enough. This is necessary to
avoid damaging the wall with high energy particles.
A non stationary ohmic RFP might be reactor-relevant
The absence of disruptions makes possible a fast start up and ramp down of
the discharges
Worth considering a RFP operated
with a series of ohmic discharges
with alternatively positive and negative currents
separated by a small time interval (few tens of ms)
Universal (F,Q) diagram
Computed at the shell
Cappello 2004
Try for MST/RFX
3D MHD
comparison!
- Taylor’s prediction
- Experimental points
- 3D MHD results
F
f-RFX-shell
f shell MST
f-shell TPE20
f shell T1
F2
F4
F10
F
0
-0.1
-0.2
-0.3
-0.4
-0.5
-0.6
1.2
1.4
1.6
BFM
1.8
2
2.2
Q
2.4
Single helicity paradigm: introduction
Change of paradigm for the RFP:
Taylor relaxation theory (TRT) Single helicity paradigm
Thomas Kuhn: change of paradigm ~political revolution
Very unpleasant!
Aim of this talk:
- Stress the discontinuities
- Emphasize necessary changes in usual statements
- Stimulate a debate.
Single helicity paradigm: basic MHD model
Simplest (visco-resistive) MHD model describing the RFP:
Navier-Stokes
Faraday-Ohm
dv  J  B   2v
dt
 B    (v  B)   (  J )
t
In 1974 no way to perform a convincing numerical simulation
Taylor relaxation theory (TRT):
involves the magnetic field only and ideal MHD.
Since the beginning of the 90’s numerical simulations:
results in good agreement with experiments
contradict both assumptions and predictions of TRT.
Convergence of experimental and theoretical results:
Change of paradigm for the RFP, the single helicity paradigm
Now described (current-driven RFP not addressed here).
Main features of the single helicity paradigm
In cylindrical geometry a RFP with only good magnetic surfaces may
exist: single helicity (SH) RFP
S. Cappello & Paccagnella1990
Toroidicity only adds a weak chaos to this cylindrical state (Sovinec
2003).
RFX-mod is coming closer to the SH state when the magnetic
boundary is improved and the current is raised
Prospect of an almost SH long pulse RFP
Neither magnetic turbulence nor magnetic chaos are essential to the
RFP state.
Main features of the single helicity paradigm
1. In cylindrical geometry a RFP with only good magnetic surfaces
may exist: single helicity (SH) RFP
2. The RFP is essentially an open ohmic system
In contrast to TRT where relaxation is of the “closed system” type
No threat of an ohmic death of the magnetic configuration after
relaxation
Resistivity helps the relaxation, as occurs for the tearing mode.
Main features of the single helicity paradigm
1. In cylindrical geometry a RFP with only good magnetic surfaces
may exist: single helicity (SH) RFP
2. The RFP is essentially an open ohmic system
3. The RFP plasma may exist in a continuum of magnetic states
ranging from laminar SH to the turbulent and partially chaotic
multiple (MH) state.
Main features of the single helicity paradigm
1. In cylindrical geometry a RFP with only good magnetic surfaces
may exist: single helicity (SH) RFP
2. The RFP is essentially an open ohmic system
3. The RFP plasma may exist in a continuum of magnetic states
ranging from laminar SH to the turbulent and partially chaotic
multiple (MH) state.
4. All these states are essentially helical
Fulfills the requirement of Cowling’s theorem
in contrast with TRT’s Bessel function model.
Main features of the single helicity paradigm
1. In cylindrical geometry a RFP with only good magnetic surfaces
may exist: single helicity (SH) RFP
2. The RFP is essentially an open ohmic system
3. The RFP plasma may exist in a continuum of magnetic states
ranging from laminar SH to the turbulent and partially chaotic
multiple (MH) state.
4. All these states are essentially helical
5. The dynamo necessary to sustain the configuration is a natural
consequence of the helical structure of the magnetic field and of
ohmic dissipation
Bonfiglio et al. 2005
Identical to what occurs for the dynamo involved in a stationary
saturated tearing or resistive kink mode (same MHD model).
Velocity field (electrostatic potential) essential to the relaxation,
as for the tearing and the resistive kink modes.
In contrast to TRT’s assumption, the velocity field may not be
neglected: relaxation comes with a dynamo.
A big relief comes with this!
In papers people never write:
“A dynamo is necessary to sustain the saturated tearing mode
against resistive diffusion”.
Therefore it is no longer justified to write:
“A dynamo is necessary to sustain the RFP state against
resistive diffusion”.
Indeed, in contrast to what suggested by TRT, no resistive death
threatens the magnetic configuration!
RFP relaxation is ohmic and comes with a dynamo
zzzz
More about the SH paradigm
1. Numerical simulations show the transition from SH to MH states is
continuous
Cappello & DFE 2000
Transition through intermittent occurrence of MH and QSH states
During the non MH fraction of the time, the plasma is in a QuasiSingle Helicity (QSH) state characterized by non vanishing
secondary modes.
Secondary mode amplitude and MH duration decrease when
coming closer to the SH state.
Experimental QSH states are quite similar (not different!): they are
non stationary and interrupted by MH episodes.
More about the SH paradigm
1. Numerical simulations show the transition from SH to MH states is
continuous (Cappello & DFE 2000).
Intermittent occurrence of MH and QSH: numerical and
experimental
2. The mechanisms bringing intermittently the plasma from QSH to MH,
and vice-versa are not yet understood.
Similarity of the time scales involved in both processes
unlikely analogy with sawtoothing in the tokamak
Mechanism = part of a global nonlinear process
Not sure that an instability is involved in any of both transitions
Numerical check difficult, because of strong sensitivity of growth
rates to small changes in current profiles
Lack of dynamic range to prove an exponential growth in simulations
and experiments.
SEPARATRIX EXPULSION FOR SH STATES
Bean Shape
Kink-like structure
Increasing amplitude
of the dominant mode
SADDLE-NODE
BIFURCATION
More about the SH paradigm
1. Numerical simulations show the transition from SH to MH states is
continuous
Intermittent occurrence of MH and QSH
2. The mechanisms bringing intermittently the plasma from QSH to MH,
and vice-versa are not yet understood
3. QSH states may occur in two different ways: with a magnetic island
or without such an island (SHAx state)
Predicted in 2000 (DFE et al. 2000)
Seen in 2007 (Lorenzini et al. 2008)
Brings a factor 4 in the improvement of the confinement time (more
with pellets)
No reason to oppose QSH and SHAx:
SHAx is a special instance of QSH: QSH/MH intermittency stays
“Only” a change of magnetic topology… very important though!
Dissipation rules the SH-MH transition
Cappello & DFE 2000
Magnetic energy
m=0 modes
MH
QSH
SH
H= 1/() ½
P = / constant
More about the SH paradigm
1. Numerical simulations show the transition from SH to MH states is
continuous
2. The mechanisms bringing intermittently the plasma from QSH to MH,
and vice-versa are not yet understood
3. QSH states may occur in two different ways: with a magnetic island
or without such an island (SHAx state)
4. In the simplest visco-resistive MHD model describing the RFP, the
transition from SH to MH is ruled by dissipation mainly through the
product of resistivity and viscosity (or through the Hartman number)
Viscosity is very hard to estimate in fusion plasmas
Open issue: does the Hartmann number rule the transition?
Caveat
The Lundquist number S is used as an empirical control parameter to
plot experimental results concerning the SH/MH transition.
However: - the other dimensionless numbers are not kept constant
- I, n and T cannot be varied independently enough to
check that their combination in S is the physical one
a.I ~ a2/3 T ~ a-2 n for any a ?
 S cannot be proved to be the physical control parameter either
P. Piovesan, M. Zuin et al.,
QSH persistence
I (MA)
More about the SH paradigm
1. Numerical simulations show the transition from SH to MH states is
continuous
2. The mechanisms bringing intermittently the plasma from QSH to MH,
and vice-versa are not yet understood
3. QSH states may occur in two different ways: with a magnetic island
or without such an island (SHAx state)
4. In the simplest visco-resistive MHD model describing the RFP, the
transition from SH to MH is ruled by dissipation through the product of
resistivity and viscosity: is it right?
5. In simulations field reversal comes because of toroidal flux
conservation while one (or more) tearing-kink mode(s) saturates
The existence of this nonlinear saturation explains why the RFP
configuration is disruption-free: the kink instability already occurred!
Conclusion 1/3
The main features of the single helicity paradigm were summarized.
For more information see the invited paper by S. Cappello at the 2008
Varenna meeting.
In particular section 7 “Criticism of Taylor relaxation theory” is to be criticized
It is non longer possible to say that TRT and the SH paradigm are
compatible!
Thesis: “TRT is in contradiction with the present knowledge about the RFP”
Corollary: « Stop teaching TRT as the reference model! »
In the future: evolution of the present view... or revolution?
Need of a strong dialog between theory and experiment
As was stressed by Popper, science is always in the making
As yet theory was quite successful in predicting the main self-organization
features of the present RFP’s.
Conclusion 2/3
Change of paradigm was felt in the magnetic confinement community:
during the TRT period: “The RFP is a disrupted tokamak”
now: “ The RFP is a bad stellarator”
No if it reaches thermonuclear temperature ohmically
Another criticism is “10 ms of confinement time at 1,5 MA are not much for a
tokamak”
However for a tokamak the magnetic field is one order of magnitude larger
than in the RFP.
Appealing reactor-relevant features of the RFP:
It uses normal magnets
No additional heating is necessary
No risk of disruption
Dramatic simplifications with respect to ITER
Conclusion 3/3
Further points in favor of the RFP are:
high engineering beta
low force at the coils
free choice of aspect ratio
high mass power density
no current limit because of stabilization by shear.
Two ways of producing strong magnetic fields with little heat dissipation:
- using currents in superconducting magnets
- or in hot plasmas (resistivity at 20 keV ~ 1/10 of Cu resistivity)
The RFP has the unique feature among confinement devices to choose the
second path
Ohmic dissipation is not a waste, but is useful to reach and maintain
thermonuclear temperatures
Engineering is a lot simpler!
These results put Taylor paradigm in a difficult corner
from Varenna paper, Cappello et al. 2008
Two further remarks
1. There is a continuity from RFP to ULq
Bonfiglio et al. 2008
SH or almost axis-symmetry
Important to understand the Greenwald limit.
Two further remarks
1. There is a continuity from RFP to ULq
2. Kadomtsev-Moffatt-Rusbridge picture of MH
Rusbridge 1991
In the radial domain where the magnetic field is chaotic, transport
is fast, and the equilibrum is almost force-free
j = µB
With div(j)=0 µ must be constant along field lines
µ is constant in the chaotic radial domain.
Assumptions simpler than TRT’s
closer to the present understanding, where magnetic chaos is
more important than magnetic turbulence.
Straightforward derivation
Result in full agreement with the fact that µ is almost constant in the
region of magnetic chaos, but not outside
The Kadomtsev-Moffatt-Rusbridge picture does not assume or predict
any axis-symmetry, in contrast to TRT.
Simple model for magnetic self-reversal
Escande and Bénisti, EFTC 1997
Inspired from Kadomtsev
Tokamak plasma, 1992
and from Verhage, Furzer, and
Robinson, “Observations of large
amplitude helical kink instabilities and
field reversal in a fast pinch experiment
(HBTX-1)”, NF 1978
Two important facts for the reactor relevance
of the RFP
1. Ohmic heating to thermonuclear temperatures is possible.
Good alpha confinement is not crucial for the RFP.
No need of producing a quasi symmetry for the magnetic field.
No need to bother about fast particle instabilities.
2. No conductive shell is necessary around the plasma.
Control of RWM’s and of the shape of the plasma boundary can be
made with saddle coils
ITER relevant
Use only 1/10 of the ohmic power.