Kinetic Effects on MHD Modes in NSTX

1
Kinetic Effects on MHD Modes in NSTX

• C. Z. (Frank) Cheng,
• N. N. Gorelenkov, G. Kramer, E. Fredrickson
• Princeton Plasma Physics Laboratory
• Princeton University
• Presented at NSTX Results Review/NSTX Research
  Forum

2
Outline

• Motivation
• MHD Model Advantage and Limitation
• Characteristic Scales in NSTX Plasmas
• Kinetic Stabilization of Ballooning Modes
• Destabilization of TAEs by Energetic Particles
• Kinetic-Fluid Model
• Summary

3
Motivation

• Multiscale coupling small-scale particle kinetic
  physics couples with large-scale MHD phenomena.
• Energetic particle physics global instabilities
  (TAE and fishbone modes) are driven by fast ions
  via wave-particle interaction and can cause
  serious fast ion loss in toroidal plasmas.
• Disparate scales in plasmas traditionally
  global-scale phenomena are studied using MHD or
  multi-fluid models, while small-scale phenomena
  are described by kinetic theories.
• Kinetic-MHD model and simulation codes through
  joint theory-experimental studies, understanding
  of energetic particle physics phenomena in major
  tokamak experiments has improved.
• Thermal particle kinetic physics thermal
  particle kinetic effects are as important as fast
  ions in determining global phenomena in burning
  plasmas e.g., kinetic ballooning modes, TAEs,
• ? Need to include kinetic effects of both fast
  and thermal particles in studying MHD phenomena
  in NSTX.

4
Global Phenomena MHD Model

• Momentum Equation
• r / t Vr V rP J B
• Continuity Equation
• / t Vr r rrV 0
• Maxwell's Equations
• B/ t rE, J rB , rB 0
• Ohm's Law E VB ?J
• Adiabatic Pressure Law / t Vr (P/r5/3)
  0
• ? important to understand advantages and
  limitations of MHD model.

5
Advantages of MHD Model

• Retains properly global geometrical effects such
  as pressure gradient, magnetic field gradient
  curvature.
• Covers most long wavelength, low-frequency EM
  waves and instabilities
• -- Fast Magnetosonic branch (w ' kVA)
  compressional Alfven waves (CAEs), mirror modes,
  etc.
• -- Shear Alfven branch (w kVA) shear
  Alfven waves (TAEs, GAEs), ballooning modes,
  kink modes, tearing modes, etc.
• -- slow magnetosonic modes (w kCs)
• Simpler to perform theoretical analysis and
  simulations than gyrokinetic/Vlasov models.

6
Limitations of MHD Model

• Ohm's law for small resistivity, plasma fluid is
  almost frozen in B and moves perpendicularly with
  EB velocity, and parallel electric field is
  negligible except in resistive boundary layer.
• Pressure law pressure changes adiabatically via
  EB convection and plasma compression.
• Gyroviscosity are ignored.
• ? No particle kinetic physics!

7
Characteristic Scales in NSTX Plasmas

• Characteristic scales of particle dynamics and
  low-frequency (?, k) perturbations
• For B 1 T, Te,i 1 keV, eh 100 keV, LB, Lp
  0.5 m,
• then we have ?i ' 0.3 cm, vi ' 108 cm/s,
• ?ci ' 108 sec-1, ?bi ' 106 sec-1, ?di ' 105
  sec-1
• -- temporal scale ordering
• ?ci ?be ?bh gt ?bi ?i,e
  ?di,e
• -- spatial scale ordering
• ?bh gt ?bi gt ?h gt ?i c/?pi gt
  ?e
• To describe low-frequency (?, k) phenomena, MHD
  model is a good approximation only if (a) ?ci gtgt
  ? gtgt ?t, ?b, ?d and (b) k??i ltlt 1 are satisfied
  for all particle species that have significant
  contributions in density or momentum or pressure.
  

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Multi-Scale Coupling Wave-Particle Interaction

• Multi-spatial scale coupling
• Finite ion Larmor radius effect (k? ?i O(1))
• (1) V? i ? V? e ' EB/B2
• (2) parallel electric field Ek
• (3) ion gyroviscosity
• Finite banana orbit width effect (?b k?-1)
• Small magnetic field scale length (LB ?i)
  particle magnetic moment is not conserved
• Multi-temporal scale coupling (?t gt ?b gt ?d)
• - ? - ?d - ?b,t ¼ 0, transit/bounce resonances
  will cause wave energy dissipation or growth
  (fast ion driven TAE instabilities)
• - ? lt ?b, trapped particles will respond to an
  bounce orbit-averaged field (trapped electron
  stabilization effects on kinetic ballooning
  instabilities)
• - ? ?d, wave-particle drift resonance effects
  are important for energy dissipation or release
  (fast ion driven fishbones)
• ? ltlt ?d, particle magnetic drift motion dominates
  over EB drift (sawtooth stabilization by fast
  ions)
• ? Particle kinetic effects must be included in
  studying NSTX burning plasma physics.

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Kinetic Effects on Ballooning Modes

• Consider finite dE due to kinetic effects of
  finite ion gyroradii and trapped electron
  dynamics.
• A finite dE enhances dJ which provides a
  strong stabilizing field line bending effect.
• Particle kinetic effects increase (decrease) the
  first (second) critical bC for ballooning
  instability over the MHD prediction.

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Kinetic Ballooning Mode Equations
Consider finite dE due to kinetic effects of
finite ion gyroradii and trapped electron
dynamics, and the approximate KBM equation is
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Stabilizing Kinetic Effects on Ballooning Mode

• For a given electric field perturbation electrons
  move across B differently from ions due to finite
  ion gyroradius effect and charge separation is
  created.
• Ions move much slower than the wave phase
  velocity along B and is essentially quasi-static.
• Electrons move much faster than the wave phase
  velocity along B and will play the role of
  keeping charge quasi-neutral.
• Trapped electrons do not contribute much to
  charge redistribution due to fast bounce motion.
• Untrapped electrons play the dominant role of
  maintaining charge quasi-neutrality.
• Untrapped electron density is much smaller than
  trapped electrons and thus an enhanced parallel
  electric field is created to move the untrapped
  electrons to maintain charge quasi-neutrality.
• Enhanced parallel electric field produces
  enhanced parallel current and thus enhanced field
  line tension, which stabilizes ballooning modes.

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KBM Stability
Local dispersion relation
At marginal stability ? ' ?pi and critical b is
given by
For low aspect ratio toroidal plasmas
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Trapped Electron Stabilization of KBMs in Large
Aspect Ratio Tokamaks
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Small Aspect RatioTorus NSTX
R/a 1.27, a 0.68 m, ? 1.63, ? 0.417, Te
Ti 1 keV, lt?gt 9, qmin 0.93 at r/a 0.3
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KBM Stability in NSTX (?i?e0)
n12
r/a0.35
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Stability of KBMs in NSTX
Temperature gradient effect R/a1.27, ?i?e1
Aspect ratio effect R/a1.67, ?i?e0
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Prediction of TAEs Based on MHD Model

• TAEs are discrete toroidal Alfven eigenmodes due
  to nonuniform q-profile and nonuniform magnetic
  field intensity along B.
• High-n TAE equation
• e r/R , s rq0/q , ?A VA/qR
• Coupling between neighboring poloidal harmonics
  produces Alfven continuous spectrum gap bounded
  by ?2 ' (1 ?) ?A2/4
• Magnetic shear allows discrete TAEs with
  frequencies in the gap
• ?2 ? ?-2 as s ? 0 ?2 ? ?2 as s ? 1
• TAEs exist because the periodicity in the wave
  potential is broken by magnetic shear similar
  to discrete energy states in a periodic lattice
  due to periodicity breaking by impurity in solid
  state physics.

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NOVA Alfvén Continuum (n 3) and TAEs in NSTX
(q0 0.7, q1 16)
ltbgt 10
ltbgt 33

• Large continuum gaps due to low aspect ratio
• Many TAEs with different ns
• TAEs can be driven unstable by fast ions if
  nq(Vh/VA) rLh/R?h

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Kinetic-MHD Model Energetic Particle Physics

• Two-component plasma core and hot components
  with nh nc , n ' nc, Pc Ph
• Core plasmas are treated as MHD-fluid
• Hot particles are governed by kinetic models such
  as gyrokinetic equations or full Vlasov equations
• Coupling between core plasmas and hot particles
  is via pressure (or current) term in momentum
  equation
• No parallel electric field

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Kinetic-MHD Model

• Momentum Equation
• r / t Vr V rPc rPh J B
• Continuity Equation
• / t Vr r rrV 0
• Maxwell's Equations
• B/ t rE, J rB , rB 0
• Ohm's Law E VB 0, EB 0
• Adiabatic Pressure Law / t Vr (Pc/r5/3)
  0
• Hot Particle Pressure Tensor
• Ph mh/2 s d3v vv fh(x,v)
• where fh is governed by gyrokinetic or Vlasov
  equation.

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PPPL Kinetic-MHD Codes

• Linear Stability Codes
• -- NOVA-K global TAE stability code with
  perturbative treatment of thermal particle
  and fast ion kinetic physics
• -- NOVA-2 global kinetic-MHD code with
  non-perturbative treatment of fast ion kinetic
  effects
• -- HINST high-n kinetic-MHD code with
  non-perturbative treatment of fast ion kinetic
  effects
• Nonlinear Simulation Codes
• -- M3D-K global kinetic-MHD code with fast ion
  kinetic physics determined by gyrokinetic
  equation.
• -- HYM-1 global kinetic-MHD code with fast ion
  kinetic physics determined by full equation of
  motion.
• -- HYM-2 global hybrid code with ions treated
  by full equation of motion and electrons treated
  as massless fluid.
• ? Through joint theory-experiment efforts, we
  have gained understanding of energetic particle
  physics phenomena in major tokamak experiments.

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Fast ions excite large amplitude bursting TAEs,
which cause fast ion loss in NSTX

• Fast neutron drops correlated with H-alpha
  bursts fast ions hitting wall?
• Small impact on soft x-ray emission.

(Fredrickson et al.)
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Bursting TAEs in NSTX NBI Experiments
(Fredrickson et al.)

• NSTX shot with B 0.434T, R 87 cm, a 63cm,
  PNB 3.2MW.
• Single dominant mode being n2 or 3, mode
  amplitude modulation represents "beating" of
  multiple modes.
• Bursting TAEs lead to neutron drop and cause 5
  10 fast ion loss.

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NOVA-K Study of TAEs in NSTX
NOVA-K Results
NSTX Results
Including plasma rotation
25
Plasma Rotation in NSTX
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Limitations of Kinetic-MHD Model

• Negligible fast particle density
• Ek 0
• No thermal particle kinetic effects (except in
  the perturbative version of linear NOVA-K code)
• ? Need a hybrid kinetic-fluid model that treats
  kinetic physics of both thermal and fast
  particles and retains single-fluid frame work.

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Thermal Particle Kinetic Effects

• Finite ion Larmor radius (k? ?i O(1))
• Finite banana orbit width (?b k?-1)
• Trapped particles
• Wave-particle resonances
• ?
• - finite parallel electric field Ek ¹ 0
• - wave damping or drive
• - Kinetic Alfven waves, KTAEs
• - Radiation damping of TAEs and KTAEs
• - stochastic ion heating by large amplitude
  Alfven waves
• - boundary layer physics in kink and tearing
  modes
• - current layer physics in magnetic reconnection
• - stabilization of ballooning mode, kink modes

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Kinetic-Fluid Model

• High-b multi-ion species plasmas
• Ordering w lt wci, k?ri O(1)
• No ordering on nh , nc , Pc , Ph
• Single-fluid equations consisting of mass density
  and momentum equations, and generalized Ohms law
• Closure of single-fluid equations by determining
  pressure tensor (including gyroviscosity) from
  particle distributions
• Particle dynamics governed by kinetic models such
  as gyrokinetic equations or full Vlasov equations
• Finite parallel electric field

Cheng Johnson, 1999
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Single-Fluid Equations

• Mass Density Continuity Equation
• / t Vr r rrV 0
• Momentum Equation
• (/t Vr) V J B r åj Pjcm
• Pjcm mj s d3v (v V)(v V) fj
• Generalized Ohm's law for multi-ion species
• E VB (1/nee) JB r( Pecm åi (qi me/e
  mi) Picm)
• (me/nee2) J/ t r(JV VJ) hJ
• åi (mi/rqi 1/nee)(B/B) (r Pi0 B/B)
• Pi0 mi s d3v vv fi
• Pressure Tensor diagonal pressures and
  gyroviscosity
• P P? (I - bb) Pk bb P
• I is the unit dyadic and b B/B.
• Pk m s d3v vk2 f, P? (m/2) s d3v v?2
  f

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Pressure Tensor (one possible model)

• P? and Pk are determined by f, which is governed
  by low frequency gyrokinetic equations (w lt wci)
• Gyroviscosity P contains ion FLR and w/wci
  effects and is derived from general frequency
  gyrokinetic equation
• rP ¼ b (r?dPc b) b r?dPs dPc dPc1
  dPc2
• dPc1 sd3v (m v?2/2) (J0 2 J10) g0
• dPc2 s d3v (m v?2 /2) (q/mB) F/m
• (F vk Ak)(2J0J10 J02) (v?dBk
  /k?)(J0 J1 2 J1 J10)
• dPs s d3v (i mv?2 /l2)
• (qF/T)(w0 - wT) /wc (q/mB) (w- kkvk - wd)
  F/m /wc
• (l J0 J1 J02 - 1)(F vkAk) l(1 2J12)
  2J0J1(v?dBk/2k?)
• w0 - (Tw/m)ln F/e, l k?v?/wc,
• F(x, e, m, t) ltfgt averaged over fluctuation
  scales,
• ? i?/?t and k -ir operate on perturbed
  quantities.
• Further approximations on FLR can be made!

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Advantages of Kinetic-Fluid Model

• Retains framework of kinetic-MHD model
• Retains naturally global geometrical effects such
  as pressure gradient, magnetic field gradient,
  curvature, etc.
• Reproduces correct linear eigenmode equations
• Allows flexibility in modeling particle
  distributions of different species
• Easier to perform analysis than
  gyrokinetic/Vlasov model

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Integrated Modeling of Burning Plasmas
a interaction with thermal plasmas is a strongly
nonlinear process.
P?(r,q), Pk(r,q), n(r), q(r)
Global Stability, Confinement, Disruption Control
Fusion Output
a-Heating a-CD
Auxiliary Heating Fueling Current Drive

Heating Power Pa gt Paux
Fast Ion Driven Instabilities Alpha/Fast Ion
Transport
Must develop efficient methods to control
profiles for burn control! ? Need nonlinear
kinetic-fluid simulation codes!
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Summary

• Kinetic effects are significant for MHD modes,
  e.g.,
• -- kinetic stabilization of ballooning modes by
  trapped electron dynamics and ion FLR
• -- destabilization of TAEs by fast ions
• Kinetic-MHD codes (linear codes NOVA-K, NOVA-2,
  HINST nonlinear codes M3D-K, HYM) have been
  developed to study fast ion physics.
• A low frequency (w lt wci) nonlinear kinetic-fluid
  model has been developed to include coupling
  between global modes and kinetic physics of both
  thermal and fast particles.
• Physics of wave-particle interaction and global
  geometrical effects are properly included in the
  kinetic-fluid model.
• Extension of kinetic-MHD codes to include thermal
  particle kinetic effects will be developed based
  on kinetic-fluid model.

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Collaborators

• PPPL N. Gorelenkov, J. Johnson, E. Belova, G.
  Kramer, E. Fredrickson, G. Fu, R. Nazikian, S.
  Zweben
• JT-60U Y. Kusama, K. Shinohara, M. Takechi, M.
  Ishikawa, H. Kimura, T. Ozeki, M. Saigusa
• DIII-D/UCI W. Heidbrink
• NIFS N. Nakajima
• IFS J. Van Dam, H. Berk