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Advances in Damping Physics of Rotating Wall Modes: Theory and Experiment

Explore the latest developments in damping physics of Rotating Wall Modes (RWM) including theoretical foundations, experimental findings, and toroidal modeling. Topics cover ideal and non-ideal MHD damping mechanisms, kinetic effects, Alfven continuum damping, and self-consistent drift-kinetic-MHD models.

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Advances in Damping Physics of Rotating Wall Modes: Theory and Experiment

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  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

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