Present understanding of RFP dynamics: theory and simulation

1. 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

2. An important fact for the reactor relevance of the RFP Electron Temperature increases with current: no signs of saturation Te (KeV) Thomson Scattering All densities CTS Meeting – 14.01.2008 – Padova P.Martin

3. 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.

4. 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)

5. Universal (F,Q) diagram Computed at the shell Cappello 2004 - Taylor’s prediction - Experimental points - 3D MHD results Try for MST/RFX comparison! 3D MHD BFM

6. 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.

7. Single 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 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).

8. 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.

9. Main features of the single helicity paradigm • In cylindrical geometry a RFP with only good magnetic surfaces may exist: single helicity (SH) RFP • 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.

10. Main features of the single helicity paradigm • In cylindrical geometry a RFP with only good magnetic surfaces may exist: single helicity (SH) RFP • The RFP is essentially an open ohmic system • The RFP plasma may exist in a continuum of magnetic states ranging from laminar SH to the turbulent and partially chaotic multiple (MH) state.

11. Main features of the single helicity paradigm • In cylindrical geometry a RFP with only good magnetic surfaces may exist: single helicity (SH) RFP • The RFP is essentially an open ohmic system • The RFP plasma may exist in a continuum of magnetic states ranging from laminar SH to the turbulent and partially chaotic multiple (MH) state. • All these states are essentially helical • Fulfills the requirement of Cowling’s theorem • in contrast with TRT’s Bessel function model.

12. Main features of the single helicity paradigm • In cylindrical geometry a RFP with only good magnetic surfaces may exist: single helicity (SH) RFP • The RFP is essentially an open ohmic system • The RFP plasma may exist in a continuum of magnetic states ranging from laminar SH to the turbulent and partially chaotic multiple (MH) state. • All these states are essentially helical • 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.

13. In papers people never write: “A dynamo is necessary to sustain the saturated tearing mode against resistive diffusion”. A big relief comes with this! 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

14. More about the SH paradigm • 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.

15. 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.

16. SEPARATRIX EXPULSION FOR SH STATES Bean Shape Kink-like structure Increasing amplitude of the dominant mode SADDLE-NODE BIFURCATION

17. 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!

18. Dissipation rules the SH-MH transition Cappello & DFE 2000 • Magnetic energy • m=0 modes • H= 1/(nh) ½ • P = n/h= constant MH QSH SH

19. 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?

20. 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)

21. 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!

22. 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.

23. 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

24. 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!

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

28. Two further remarks • There is a continuity from RFP to ULq • Bonfiglio et al. 2008 • SH or almost axis-symmetry • Important to understand the Greenwald limit.

29. Two further remarks • There is a continuity from RFP to ULq • 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.

30. 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

31. 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.