Damping Physics of RWM Yueqiang Liu UKAEA Culham Science Centre Abingdon, Oxon OX14 3DB, UK

1. Damping Physics of RWM Yueqiang Liu UKAEA Culham Science Centre Abingdon, Oxon OX14 3DB, UK

2. Outline • Damping physics overview • Selected pieces of theory • Experimental results • Toroidal modelling

3. Damping physics overview /1 • RWM traditionally treated in MHD framework • Rotational damping has been a major piece of physics investigated in MHD theory • Ideal MHD: • sound wave continuum damping [Bondeson94, Betti95] • shear Alfven resonance damping [Bondeson94, Zheng05] • Non-ideal MHD: • resistive layer damping [Finn95, Gimblett& Hastie00] • viscous boundary layer damping [Fitzpatrick96]

4. Damping physics overview /2 • Recent understanding of importance of kinetic effects • Parallel viscous force model for sound wave damping [Chu95] • Semi-kinetic model: mode resonance with bounce motion of thermal ions [Bondeson96, Liu04] • Drift kinetic damping at slow rotation [Hu& Betti04]: mode resonance with precession drifts of trapped ions and electrons (perturbative approach) • Self-consistent inclusion of toroidal drift-kinetic effects in MHD [Liu08] • New ideas • Reactive fluid closure based model • Turbulence-induced RWM damping

5. Outline • Damping physics overview • Selected pieces of theory • Experimental results • Toroidal modelling

6. Selected pieces of theory • RWM dispersion relation • extended energy principle • Alfven continuum damping in a cylinder • Kinetic damping • A toy model of drift-kinetic effects

7. Extended energy principle • Derived by several authors, most explicit form proposed by [Chu PoP 2 2236 (1995)] inertia plasma vacuum+wall kinetic • Recovers standard free-boundary MHD energy principle (w/ or w/o an ideal wall) • Neglecting inertia term, arrive at a general toroidal dispersion relation for RWM • Neglecting kinetic contribution, arrive at Haney&Freidberg’s RWM dispersion relation [Haney PF B1 1637(1989)]

8. Alfven continuum damping

9. Alfven continuum damping Bondeson PPCF 45 A253(2003)

10. Kinetic physics • MHD predicts unphysical resonant behaviour of sound waves, subject to strong ion Landau damping, modelled by a viscous force along parallel motion[Chu PoP 2 2236(1995)] • Depending on plasma rotation speed, RWM can be in resonance with drift motions of ions/electrons of bulk plasma, resulting in kinetic damping • At fast plasma rotation, mode resonant with bounce motion of passing/trapped thermal ions [Bondeson PoP 3 3013(1996), Liu NF 45 1131(2005)] • At slow rotation, mode resonant with magnetic precession drift of trapped ions/electrons [Hu PRL 93 105002(2004), Liu PoP15 092505(2008)]

11. A self-consistent drift-kinetic-MHD model • The model takes into account nonlinear mode eigenvalue formulation via kinetic integrals in self-consistent approach • Consider precessional drift resonances only where ’lumped’ over particle energy and pitch angle • Assuming and , obtain a cubic dispersion relation where with all frequencies normalised by wall time.

12. A self-consistent drift-kinetic-MHD model • First consider a perturbative approach w/o plasma rotation • Perturbative approach C1<0 destabilisation C1>1 full stabilisation 0<C1<1 partial stabilisation

13. A self-consistent drift-kinetic-MHD model • Secondly consider self-consistent approach w/o plasma rotation • Self-consistent approach • One branch similar to perturbative one • Other two branches can be unstable • Complex conjugate roots without rotation

14. A self-consistent drift-kinetic-MHD model • Finally scan over rotation, for a case where perturbative approach predicts full stabilisation • Perturbative approach • Stable root • Self-consistent approach • One stable root + two unstable roots • Complex conjugate roots at vanishing rotation unstable unstable stable stable • These two unstable branches resemble ‘bursting mode’ (EWM) and RWM precursor observed in JT-60U [Matsunaga, IAEA FEC08]

15. Outline • Damping physics overview • Selected pieces of theory • Experimental results • Toroidal modelling

16. Experimental results • Critical rotation with magnetic braking • Cross-machine comparison and scaling • DIII-D + JET • RWM stability at slow plasma rotation • DIII-D • JT-60U

17. Critical rotation due to braking

18. Cross-machine comparison

19. DIII-D experiments show very low critical rotation frequency for RWM stability • Recent experimental data seem to suggest even lower critical rotation frequency[Strait, IAEA FEC08] critical rotation profile Reimerdes, PPCF07, B349

20. Field braking vs. Beam braking • On DIII-D, critical rotation speed for RWM stability margin measured by • using magnetic field braking of the plasma rotation, and • using counter-beam injection to control the toroidal rotation speed • Magnetic braking produces much higher critical rotation speed than the balanced beam experiments • Present explanations … • Braking experiments explained by Fitzpatrick’s induction motor model (non-linear effect of resonant field caused momentum damping leads to rotation bifurcation) • Balanced beam experiments explained by kinetic effects on RWM stability

21. Rotation threshold in recent RWM experiments • Using NBI-torque to control plasma rotation, both JT-60U and DIII-D report a low rotation threshold for RWM stability, about 0.3% of Alfven frequency at q=2 surface

22. Outline • Damping physics overview • Selected pieces of theory • Experimental results • Toroidal modelling

23. Toroidal modelling • MHD continuum + semi-kinetic damping • Drift-kinetic damping (MARS-K) • self-consistent toroidal MHD-kinetic hybrid simulation • Modelling for Soloviev equilibria • Modelling for DIII-D • Modelling for ITER

24. Semi-kinetic damping

25. Example of DIII-D modelling

26. Global nature of semi-kinetic damping

27. ITER steady state Scenario-4

28. Cirtical rotation with various assumptions

29. Critical rotation with semi-kinetic damping

30. MARS-K: self-consistent kinetic-MHD • MARS-F basically solves single fluid linear MHD, with a few features • Eulerian frame (for resistive plasma) • Shear toroidal rotation • Parallel sound wave damping • Kinetic inclusion • Assumptions made in this formulation • Neglected anisotropy of equilibrium pressure • Neglected perturbed electrostatic potential • No FLR effect included • Neglected radial excursion of particle trajectory Liu PoP 15 112503(2008)

31. Self-consistent formulation couples drift kinetic effects with linear fluid MHD via perturbed pressure tensors • Self-consistent inclusion of perturbed kinetic pressure tensors • Perturbed kinetic pressure tensors derived analytically from drift kinetic equations • considering particle bounce and magnetic precession drifts • assuming Maxwellian thermal particle distribution

32. Perturbed kinetic pressure couples to displacement • Solving analytically drift kinetic equation for perturbed distribution function gives perturbed kinetic pressures • Split particle Lagrangian into secular and periodic parts • Fourier decompose periodic part in particle bounce orbit = ’geometrical factor’ associated with Fourier projection in particle bounce orbit = ’geometrical factor’ associated with Fourier projection along poloidal angle = integral over particle energy = poloidal Fourier harmonics of solution vector Example: precession drift resonance

33. Perturbative and self-consistent approaches differ largely in three aspects • Drift kinetic energy perturbation [Antonsen82, Porcelli94] precession bounce mode frequency

34. Development of MARS-K code • Includes both precession and bounce resonance damping from thermal particles • most of results shown here include precession resonances alone: valid at slow plasma rotation • Has both perturbative and self-consistent options • sharing the same piece of code for evaluating kinetic integrals • Perturbative option benchmarked vs. HAGIS • HAGIS is a drift-orbit particle-following code, computing with ideal-kink eigenfunction compted by MHD code MISHKA • HAGIS takes into account effect of finite banana width

35. Test case: analytical Soloviev equilibrium plasma boundary

36. MARS-K reproduces well large aspect ratio drift frequencies Particle bounce frequency Precession drift frequency Large aspect ratio (cylinder) drift frequencies for a circular plasma (trapped) (trapped) (passing)

37. Toroidal effect shows up at lower A

38. Benchmark between MARS-K and HAGIS shows good agreement for a wide range of rotation frequency • Choose a Soloviev equilibrium with circular-like shape • Both codes run with perturbative approach Liu, PoP08, 112503 • Validates approximation of neglecting banana width for kinetic RWM

39. Soloviev • Choose a toroidal Soloviev equilibrium with E = 1.6 • Run MARS-K with both perturbative and self-consistent options • Only partial stabilisation (destabilisation) achieved

40. Kinetic effects do modify mode eigenfunction for a test toroidal equilibrium • For a test toroidal Soloviev equilibrium: R/a=3, = 1.6 • Kinetic effects give only partial stabilisation, following both approaches. Sometimes even destabilisation[Liu, PoP08, 112503] kink fluid RWM kinetic RWM

41. Kinetic modification of eigenfunctions • Perturbation more edge-localised in SC calculations • Higher number poloidal harmonics of perturbed kinetic pressure excited by kinetic resonances

42. DIII-D 125701 used in MARS-K modelling

43. Perturbative simulation for DIII-D 125701 predicts complete stabilisation over a wide parameter space • Consider precessional drift resonances of thermal particles • Use eigenfunction and eigenvalue of fluid RWM to evaluate perturbed drift kinetic energy • Ti/Te=1 • Scale amplitude of critical rotation profile from expt. ideal-wall limit exp. no-wall limit exp.

44. DIII-D: perturbative

45. Strong stabilisation also observed with other assumptions in perturbative calculations reference case

46. Self-consistent approach seems to predict much less stabilisation for DIII-D plasmas • Possible reasons for different results between two approaches: • kinetic modification of eigenfunction ? • nonlinear eigenvalue formulation through kinetic integrals ? perturbative self-consistent exp. exp. unstable again at very slow rotation exp. exp.

47. DIII-D: non-perturbative • Influence of equilibrium ion/electron ratio

48. DIII-D: non-perturbative • Influence of plasma pressure

49. No significant kinetic modification of RWM eigenfunction observed for DIII-D plasmas • Compare radial distribution of poloidal Fourier harmonics for normal displacement, between no-wall kink fluid RWM SC kinetic RWM • Since no significant difference in eigenfunction between fluid and SC kinetic RWM, difference caused by nonlinear eigenvalue formulation via kinetic integrals? • … confirmed by the three-roots toy model

50. Unstable kinetic RWM in DIII-D simulations qualitatively agree with analytic model • All modes have (similar) RWM eigenstructure • JT-60U also reports n=1 kink-ballooning structure for all modes unstable unstable