YQ Liu, Peking University, Feb 16-20, 2009 Damping Physics of RWM

Yueqiang Liu

UKAEA Culham Science CentreAbingdon, Oxon OX14 3DB, UK

YQ Liu, Peking University, Feb 16-20, 2009OutlineDamping physics overview

Selected pieces of theory

Experimental results

Toroidal modelling

YQ Liu, Peking University, Feb 16-20, 2009Damping physics overview /1RWM traditionally treated in MHD framework

Rotational damping has been a major piece of physics investigated in MHD theoryIdeal 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]

YQ Liu, Peking University, Feb 16-20, 2009Damping physics overview /2Recent 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 ideasReactive fluid closure based modelTurbulence-induced RWM damping

YQ Liu, Peking University, Feb 16-20, 2009OutlineDamping physics overview

Selected pieces of theory

Experimental results

Toroidal modelling

YQ Liu, Peking University, Feb 16-20, 2009Selected pieces of theoryRWM dispersion relation extended energy principle

Alfven continuum damping in a cylinder

Kinetic dampingA toy model of drift-kinetic effects

YQ Liu, Peking University, Feb 16-20, 2009Extended energy principleDerived by several authors, most explicit form proposed by [Chu PoP 2 2236 (1995)]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 inertiaplasmavacuum+wallkineticNeglecting kinetic contribution, arrive at Haney&Freidbergs RWM dispersion relation [Haney PF B1 1637(1989)]

YQ Liu, Peking University, Feb 16-20, 2009Alfven continuum damping

YQ Liu, Peking University, Feb 16-20, 2009Alfven continuum dampingBondeson PPCF 45 A253(2003)

YQ Liu, Peking University, Feb 16-20, 2009Kinetic physicsMHD 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 dampingAt 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)]

YQ Liu, Peking University, Feb 16-20, 2009A self-consistent drift-kinetic-MHD modelThe model takes into account nonlinear mode eigenvalue formulation via kinetic integrals in self-consistent approachConsider precessional drift resonances onlywhereAssuming and , obtain a cubic dispersion relationwherewith all frequencies normalised by wall time.lumped over particle energy and pitch angle

YQ Liu, Peking University, Feb 16-20, 2009A self-consistent drift-kinetic-MHD modelFirst consider a perturbative approach w/o plasma rotationC11 full stabilisation0

YQ Liu, Peking University, Feb 16-20, 2009A self-consistent drift-kinetic-MHD modelSecondly consider self-consistent approach w/o plasma rotationSelf-consistent approachOne branch similar to perturbative oneOther two branches can be unstableComplex conjugate roots without rotation

YQ Liu, Peking University, Feb 16-20, 2009A self-consistent drift-kinetic-MHD modelFinally scan over rotation, for a case where perturbative approach predicts full stabilisationPerturbative approachStable root

Self-consistent approachOne stable root + two unstable rootsComplex conjugate roots at vanishing rotationThese two unstable branches resemble bursting mode (EWM) and RWM precursor observed in JT-60U [Matsunaga, IAEA FEC08]unstablestableunstablestable

YQ Liu, Peking University, Feb 16-20, 2009OutlineDamping physics overview

Selected pieces of theory

Experimental results

Toroidal modelling

YQ Liu, Peking University, Feb 16-20, 2009Experimental resultsCritical rotation with magnetic braking

Cross-machine comparison and scalingDIII-D + JET

RWM stability at slow plasma rotationDIII-DJT-60U

YQ Liu, Peking University, Feb 16-20, 2009Critical rotation due to braking

YQ Liu, Peking University, Feb 16-20, 2009Cross-machine comparison

YQ Liu, Peking University, Feb 16-20, 2009DIII-D experiments show very low critical rotation frequency for RWM stabilityRecent experimental data seem to suggest even lower critical rotation frequency [Strait, IAEA FEC08]critical rotation profileReimerdes, PPCF07, B349

YQ Liu, Peking University, Feb 16-20, 2009Field braking vs. Beam brakingOn DIII-D, critical rotation speed for RWM stability margin measured byusing magnetic field braking of the plasma rotation, and using counter-beam injection to control the toroidal rotation speedMagnetic braking produces much higher critical rotation speed than the balanced beam experiments

Present explanations

Braking experiments explained by Fitzpatricks 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

YQ Liu, Peking University, Feb 16-20, 2009Rotation threshold in recent RWM experimentsUsing 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

YQ Liu, Peking University, Feb 16-20, 2009OutlineDamping physics overview

Selected pieces of theory

Experimental results

Toroidal modelling

YQ Liu, Peking University, Feb 16-20, 2009Toroidal modellingMHD continuum + semi-kinetic damping

Drift-kinetic damping (MARS-K)self-consistent toroidal MHD-kinetic hybrid simulationModelling for Soloviev equilibriaModelling for DIII-DModelling for ITER

YQ Liu, Peking University, Feb 16-20, 2009Semi-kinetic damping

YQ Liu, Peking University, Feb 16-20, 2009Example of DIII-D modelling

YQ Liu, Peking University, Feb 16-20, 2009Global nature of semi-kinetic damping

YQ Liu, Peking University, Feb 16-20, 2009ITER steady state Scenario-4

YQ Liu, Peking University, Feb 16-20, 2009Cirtical rotation with various assumptions

YQ Liu, Peking University, Feb 16-20, 2009Critical rotation with semi-kinetic damping

YQ Liu, Peking University, Feb 16-20, 2009MARS-K: self-consistent kinetic-MHDMARS-F basically solves single fluid linear MHD, with a few features

Assumptions made in this formulationNeglected anisotropy of equilibrium pressureNeglected perturbed electrostatic potentialNo FLR effect includedNeglected radial excursion of particle trajectory Eulerian frame (for resistive plasma) Shear toroidal rotationParallel sound wave dampingKinetic inclusionLiu PoP 15 112503(2008)

YQ Liu, Peking University, Feb 16-20, 2009Self-consistent formulation couples drift kinetic effects with linear fluid MHD via perturbed pressure tensors Self-consistent inclusion of perturbed kinetic pressure tensorsPerturbed kinetic pressure tensors derived analytically from drift kinetic equationsconsidering particle bounce and magnetic precession driftsassuming Maxwellian thermal particle distribution

YQ Liu, Peking University, Feb 16-20, 2009Perturbed kinetic pressure couples to displacement Solving analytically drift kinetic equation for perturbed distribution function gives perturbed kinetic pressuresSplit particle Lagrangian into secular and periodic partsFourier 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 vectorExample: precession drift resonance

YQ Liu, Peking University, Feb 16-20, 2009Perturbative and self-consistent approaches differ largely in three aspectsDrift kinetic energy perturbation [Antonsen82, Porcelli94]precessionbouncemode frequency

perturbativeself-consistenteigen-function ideal-kink or fluid RWMmodified by kinetic effects self-consistentlyeigen-frequency or nonlinear eigenvalue formulationother dampingshear Alfven damping included a-posterioricontinuum damping included via MHD terms

YQ Liu, Peking University, Feb 16-20, 2009Development of MARS-K codeIncludes both precession and bounce resonance damping from thermal particlesmost of results shown here include precession resonances alone: valid at slow plasma rotation

Has both perturbative and self-consistent optionssharing the same piece of code for evaluating kinetic integrals

Perturbative option benchmarked vs. HAGISHAGIS is a drift-orbit particle-following code, computing with ideal-kink eigenfunction compted by MHD code MISHKAHAGIS takes into account effect of finite banana width

YQ Liu, Peking University, Feb 16-20, 2009Test case: analytical Soloviev equilibriumplasma boundary

YQ Liu, Peking University, Feb 16-20, 2009MARS-K reproduces well large aspect ratio drift frequenciesLarge aspect ratio (cylinder) drift frequencies for a circular plasmaParticle bounce frequencyPrecession drift frequency(trapped)

(passing)(trapped)

YQ Liu, Peking University, Feb 16-20, 2009Toroidal effect shows up at lower A

YQ Liu, Peking University, Feb 16-20, 2009Benchmark between MARS-K and HAGIS shows good agreement for a wide range of rotation frequencyChoose a Soloviev equilibrium with circular-like shapeBoth codes run with perturbative approachValidates approximation of neglecting banana width for kinetic RWMLiu, PoP08, 112503

YQ Liu, Peking University, Feb 16-20, 2009SolovievChoose a toroidal Soloviev equilibrium with E = 1.6

Run MARS-K with both perturbative and self-consistent options

Only partial stabilisation (destabilisation) achieved

YQ Liu, Peking University, Feb 16-20, 2009Kinetic effects do modify mode eigenfunction for a test toroidal equilibriumKinetic effects give only partial stabilisation, following both approaches. Sometimes even destabilisation [Liu, PoP08, 112503]For a test toroidal Soloviev equilibrium: R/a=3, = 1.6kinkfluid RWMkinetic RWM

YQ Liu, Peking University, Feb 16-20, 2009Kinetic modification of eigenfunctionsPerturbation more edge-localised in SC calculationsHigher number poloidal harmonics of perturbed kinetic pressure excited by kinetic resonances

YQ Liu, Peking University, Feb 16-20, 2009DIII-D 125701 used in MARS-K modelling

YQ Liu, Peking University, Feb 16-20, 2009Perturbative simulation for DIII-D 125701 predicts complete stabilisation over a wide parameter spaceConsider 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.exp.exp.no-wall limitideal-wall limit

YQ Liu, Peking University, Feb 16-20, 2009DIII-D: perturbative

YQ Liu, Peking University, Feb 16-20, 2009Strong stabilisation also observed with other assumptions in perturbative calculationsreference case

YQ Liu, Peking University, Feb 16-20, 2009Self-consistent approach seems to predict much less stabilisation for DIII-D plasmasPossible reasons for different results between two approaches:kinetic modification of eigenfunction ?nonlinear eigenvalue formulation through kinetic integrals ?perturbativeself-consistentexp.exp.exp.exp.unstable again at very slow rotation

YQ Liu, Peking University, Feb 16-20, 2009DIII-D: non-perturbativeInfluence of equilibrium ion/electron ratio

YQ Liu, Peking University, Feb 16-20, 2009DIII-D: non-perturbativeInfluence of plasma pressure

YQ Liu, Peking University, Feb 16-20, 2009No significant kinetic modification of RWM eigenfunction observed for DIII-D plasmasCompare radial distribution of poloidal Fourier harmonics for normal displacement, betweenno-wall kinkfluid RWMSC kinetic RWMSince 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

YQ Liu, Peking University, Feb 16-20, 2009Unstable kinetic RWM in DIII-D simulations qualitatively agree with analytic modelAll modes have (similar) RWM eigenstructureJT-60U also reports n=1 kink-ballooning structure for all modesunstableunstable

YQ Liu, Peking University, Feb 16-20, 2009ITER: perturbative One example: precession drift resonances at very slow plasma rotation [Liu, IAEA FEC08]Bounce resonance damping does not give a significant contribution

YQ Liu, Peking University, Feb 16-20, 2009ITER: perturbativeRadial distribution of drift kinetic energy perturbation with precessional resonances

YQ Liu, Peking University, Feb 16-20, 2009Self-consistent approach predicts less kinetic stabilisation of RWM in ITER plasmasperturbativeself-consistentTwo unstable branches also exist, similar to DIII-D results and analytic modelITER(estimated)ITERLiu NF 49 035004(2009)central rotation

YQ Liu, Peking University, Feb 16-20, 2009SummaryIn addition to MHD type of continuum damping, kinetic effects play a significant role in the RWM stability

Self-consistent modelling using MARS-K generally predicts less kinetic stabilisation on RWM compared to perturbative approaches Primarily due to nonlinear eigenvalue formulation through kinetic integrals in self-consistent approachSometimes also observe kinetic modification of mode eigenfunction

Self-consistent kinetic simulation does not fully explain DIII-D results. May require additional damping physics. This approach does reveal some features of unstable modes observed in JT-60U.

Kinetic effects may not fully stabilise RWM in ITERIt is necessary to consider the feedback option for ITER RWM suppression