Present understanding of RFP dynamics: theory and simulation

Present understanding of RFP dynamics: theory and simulation

D.F. EscandeUMR 6633 CNRS-Université de Provence, Marseille, France

Type of talkNot a pedagogical introductionStress on important facts which are often overlooked

SummaryAlpha confinement might not be crucial for the RFPA non stationary ohmic RFP might be reactor-relevantUniversal (F,) diagramSingle helicity paradigm

CTS Meeting – 14.01.2008 – Padova P.Martin

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An important fact for the reactor relevance of the RFPElectron Temperature increases with current: no signs of saturation

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 currentsseparated by a small time interval (few tens of ms)

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f-RFX-shellf shell MSTf-shell TPE20f shell T1F2F4F10F

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3D MHD

- Taylor’s prediction- Experimental points- 3D MHD results

Universal (F,) diagramComputed at the shell

Cappello 2004Try for MST/RFX comparison!

Single helicity paradigm: introductionSingle helicity paradigm: introduction

Change of paradigm for the RFP: Taylor relaxation theory (TRT) Single helicity paradigmThomas 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 modelSingle helicity paradigm: basic MHD model

Simplest (visco-resistive) MHD model describing the RFP:

Navier-Stokes

Faraday-Ohm

In 1974 no way to perform a convincing numerical simulationTaylor 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 experimentscontradict 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).

)()v( JBtB

vddv 2 BJt

Main features of the single helicity paradigmIn cylindrical geometry a RFP with only good magnetic surfaces may exist: single helicity (SH) RFP S. Cappello & Paccagnella1990Toroidicity 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 raisedProspect of an almost SH long pulse RFPNeither magnetic turbulence nor magnetic chaos are essential to the RFP state.

MAC-302

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 systemIn contrast to TRT where relaxation is of the “closed system” typeNo threat of an ohmic death of the magnetic configuration after relaxationResistivity helps the relaxation, as occurs for the tearing mode.



2. The RFP is essentially an open ohmic system3. 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 helicalFulfills the requirement of Cowling’s theorem

in contrast with TRT’s Bessel function model.

MAC-302





4. All these states are essentially helical5. 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. 2005Identical 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”.

zzzz

Indeed, in contrast to what suggested by TRT, no resistive death threatens the magnetic configuration!RFP relaxation is ohmic and comes with a dynamo

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 Quasi-Single 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.


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 tokamakMechanism = part of a global nonlinear processNot sure that an instability is involved in any of both transitionsNumerical check difficult, because of strong sensitivity of growth rates to small changes in current profilesLack of dynamic range to prove an exponential growth in simulations and experiments.

SEPARATRIX EXPULSION FOR SH STATESSEPARATRIX EXPULSION FOR SH STATES

Increasing amplitudeIncreasing amplitudeof the dominant modeof the dominant mode

SADDLE-NODESADDLE-NODEBIFURCATIONBIFURCATION

Bean ShapeBean ShapeKink-like structureKink-like structure

More about the SH paradigm1. 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 understood3. 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 energym=0 modes

H= 1/() ½

P = constant

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MH

QSH


1. Numerical simulations show the transition from SH to MH states is continuous2. The mechanisms bringing intermittently the plasma from QSH to MH, and vice-versa are not yet understood3. 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 plasmasOpen issue: does the Hartmann number rule the transition?

CaveatThe 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 eitherP. Piovesan, M. Zuin et al.,

QSH persistence

I (MA)


1. Numerical simulations show the transition from SH to MH states is continuous2. The mechanisms bringing intermittently the plasma from QSH to MH, and vice-versa are not yet understood3. 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) saturatesThe existence of this nonlinear saturation explains why the RFP configuration is disruption-free: the kink instability already occurred!

Conclusion 1/3Conclusion 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 experimentAs 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 2/3Conclusion 2/3Change 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 ohmicallyAnother 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 magnetsNo additional heating is necessaryNo risk of disruption

Dramatic simplifications with respect to ITER

Conclusion 3/3Conclusion 3/3

Further points in favor of the RFP are: high engineering betalow force at the coilsfree choice of aspect ratiohigh mass power densityno 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 pathOhmic dissipation is not a waste, but is useful to reach and maintain thermonuclear temperaturesEngineering is a lot simpler!

These results put Taylor paradigm in a difficult cornerfrom Varenna paper, Cappello et al. 2008

Two further remarks

1. There is a continuity from RFP to ULq Bonfiglio et al. 2008SH or almost axis-symmetryImportant 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 linesµ is constant in the chaotic radial domain.

Assumptions simpler than TRT’scloser to the present understanding, where magnetic

chaos is more important than magnetic turbulence. Straightforward derivationResult in full agreement with the fact that µ is almost constant in the

region of magnetic chaos, but not outsideThe 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

Documents

Present understanding of RFP dynamics: theory and simulation