Kinetic-Fluid Model for Modeling Fast Ion Driven Instabilities

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Kinetic-Fluid Model for Modeling Fast Ion Driven
Instabilities

• C. Z. Cheng, N. Gorelenkov and E. Belova
• Princeton Plasma Physics Laboratory
• Princeton University
• 45th Annual Meeting of the Division of Plasma
  Physics October 27-31, 2003Albuquerque, New
  Mexico

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Outline

• Energetic Particle Physics Issues
• Kinetic-MHD Model
• Advantages
• Limitations
• Linear and Nonlinear Kinetic-MHD codes
• Particle Characteristics and Kinetic Effects
• Nonlinear Kinetic-Fluid Model
• Summary

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Modeling Energetic Particle Physics

• The Modeling difficulty stems from disparate
  scales which are traditionally analyzed
  separately global-scale phenomena are studied
  using MHD model, while microscale phenomena are
  described by kinetic theories.
• The kinetic-MHD model was developed by treating
  thermal particles as MHD fluid and fast particles
  by kinetic theories, on which all present
  energetic particle codes are based.
• Kinetic physics of both thermal and fast
  particles involve small spatial scale and fast
  temporal scale, and can strongly affect the
  global structure and long time behavior of
  thermal plasmas and fast particles.
• ? A kinetic-fluid model has been developed to
  treat kinetic physics of both thermal and fast
  particles, but also retains the framework of
  kinetic-MHD model.

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Kinetic-MHD Model

• Momentum Equation (Pc Ph)
• r / t Vr V rPc rPh J B
• Continuity Equation (n ' nc, nh nc)
• / 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
  equations.

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Advantages of Kinetic-MHD Model

• Retains properly global geometrical effects such
  as gradients in P, B, etc.
• Covers most low-frequency waves and
  instabilities 3 Branches of waves and
  instabilities
• -- Fast Magnetosonic Branch compressional
  wvaes, mirror modes, etc.
• -- Shear Alfven Branch shear Alfven waves,
  ballooning, tearing, K-H instabilities, etc.
• -- Slow Magnetosonic Branch sound waves, drift
  wave instabilities, etc.
• Retains energetic particle kinetic physics.

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Limitations of Kinetic-MHD Model

• Assumes that fast particle density is negligible.
• Thermal particle dynamics is governed by MHD
  model.
• -- Ohm's law plasma is frozen in B and moves
  with EB drift velocity and Ek 0.
• -- Adiabatic pressure law thermal plasma
  pressure changes adiabatically through plasma
  convection and compression.
• -- Gyroviscosity (contains ion gyroradius
  effects) and pressure anisotropy are ignored.
• -- Thermal particle kinetic effects of
  gyroradii, trapped particle dynamics (transit,
  bounce and magnetic drift motions), and
  wave-particle resonances are ignored.
• Kinetic-MHD model for thermal plasmas is valid
  only when
• (a) wci À w À wt, wb, w, wd for all particle
  species
• (b) kL gt 1 and kri 1

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PPPL Kinetic-MHD Codes

• Linear Stability Codes
• -- NOVA-K code global TAE stability code with
  perturbative treatment of non-MHD physics of
  thermal and fast particles
• -- NOVA-2 code global stability code with
  non-perturbative treatment of fast particle
  kinetic effects
• -- HINST code high-n stability code with
  non-perturbative treatment of fast particle
  kinetic effects
• Nonlinear Simulation Codes
• -- M3D-K code global simulation code with fast
  particle kinetic physics determined by
  gyrokinetic equation.
• -- HYM-1 code global simulation code with fast
  particle kinetic physics determined by full
  equation of motion.
• -- HYM-2 code global hybrid simulation code
  with ions treated by full equation of motion
  and electrons treated as massless fluid.

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Kinetic-Fluid Model Cheng Johnson, J.
Geophys. Res., 104, 413 (1999)

• Consider high-b multi-ion species plasmas
• Consider w lt wci, k?ri O(1)
• 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
• Particle distribution functions f F(x, e, m, t)
  ?f,
• ?f (q/m)F/eF (q/mB)F/m (1 J02)(F
  vkAk) (v? J1/2k?) dBk g0 eiL and g0 is
  determined from gyrokinetic equation
• ?/?t (vk vd)r g0 (q/m)F/e ?/?t
  (B/B2)r(F g0)r J0(F vkAk) (v? J1 /2k?)
  dBk or ?f can be solved by particle code.
• Maxwell's equations in magnetostatic limit are
  employed.

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• Pressure Tensor and Gyroviscosity
• P P? (I - bb) Pk bb P
• where I is the unit dyadic and b B/B.
• Pk m s d3v vk2 f, P? (m/2) s d3v v?2
  f
• For k? À kk, gyroviscosity tensor contribution
• rP ¼ b (rdPc b) b r?dPs
• dPc dPc1 dPc2 , dPc1 sd3v (m v?2/2) g0
  (J0 2 J10)
• dPc2 s d3v (m v?2 /2) (q/mB) F/m
• (F vk Ak)(2J0J10 J02) (v?d Bk
  /k?)(J0 J1 2 J1 J10)
• dPs s d3v (i mv?2 /l2)
• (qF/T)(w0 - wT) /wc (q/mB) (w- kk vk -
  wd) F/m /wc
• (l J0 J1 J02 - 1)(F vkAk)
• l(1 2 J12) 2 J0 J1(v?dBk/2k?)
• w0 - (Tw/m) ln F/e, l k?v?/wc,
• F(x, e, m, t) ltfgt averaged over fluctuation
  scales when necessary,
• ?/?t -i? and r ik operate on perturbed
  quatities.

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• Low-Frequency Ohm's Law
• E VB
• (1/nee) JB r( Pecm åi (qi me/e mi)
  Picm)
• åi (mi/rqi 1/nee)(B/B) (r Pi0 B/B)
• (me/nee2) J/ t r(JV VJ) hJ
• where Pi0 mi s d3v vv fi
• Main Features
• -- The kinetic-fluid model retains most
  essential particle kinetic effects in low
  frequency phenomena (w lt wci) for all particle
  species
• -- Gyroviscosity is included so that ion Larmor
  radius effects are properly retained
• -- A new Ohm's law for multi-ion species
• -- No assumption on nh/nc ratio
• -- Nonlinear

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Kinetic-Fluid Codes

• Based on Kinetic-Fluid Model we will extend
  existing PPPL codes to include both thermal and
  fast particle kinetic effects
• Linear Stability Codes
• -- non-perturbative global NOVA-2 code
• -- high-n HINST code
• Nonlinear Global Simulation Codes
• -- M3D-K and HYM codes

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

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

• A nonlinear kinetic-fluid model has been
  developed for high-b plasmas with multi-ion
  species for w lt wci.
• Physics of wave-particle interaction and
  geometrical effects are properly included, and
  the kinetic-fluid model includes kinetic effects
  of both thermal and fast particles.
• Eigenmode equations for dispersive shear Alfven
  waves and kinetic ballooning modes derived from
  kinetic-fluid model are verified with those
  derived from full kinetic equations for w lt wci.
  
• Based on kinetic-MHD model global and high-n
  linear stability codes (e.g., NOVA-K, NOVA-2,
  HINST, etc.) and nonlinear simulation codes
  (e.g., M3D-K, HYM codes) have been developed to
  study effects of energetic particles on MHD modes
  such as TAEs, internal kinks, etc.
• Linear stability and nonlinear simulation codes
  based on kinetic-fluid model can be constructed
  by extending these existing kinetic-MHD codes.