A Kinetic-Fluid Model for Studying Thermal and Fast Particle Kinetic Effects on MHD Instabilities

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A Kinetic-Fluid Model for Studying Thermal and
Fast Particle Kinetic Effects on MHD
Instabilities

• C. Z. Cheng, N. Gorelenkov and E. Belova
• Princeton Plasma Physics Laboratory
• Princeton University

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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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Why is Energetic Particle Physics Important?

• Fast ions exist in all magnetic fusion devices
  and play essential roles in heating and current
  drive
• -- Fast ions in NBI, N-NBI, ICRH
• -- Alpha particles produced in D-T fusion
  reaction
• Significant loss of fast ions can lead to
  degradation of heating and current drive
  efficiency
• Lost fast ions tend to localize near outer
  midplane and may cause localized damage in first
  wall of toroidal reactors
• In Q gt 5 burning plasmas a-particles are dominant
  heating source because Pa gt Paux
• Control of fast ion pressure profile is important
  in controlling thermal plasma profiles, which
  affects global plasma stability and confinement
• ? Need to integrate energetic particle physics
  with global stability, confinement, and heating
  physics

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Modeling Energetic Particle Physics

• The difficulty of theoretical modeling stems from
  the disparate
• scales which traditionally are analyzed
  separately global-scale phenomena are generally
  studied using MHD model, while microscale
  phenomena are described by kinetic theories.
• The kinetic-MHD model was developed by treating
  thermal particles by MHD model and fast particles
  by kinetic theories.
• Kinetic physics of both thermal and fast
  particles involve small spatial and fast temporal
  scales 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, on which all present energetic
  particle codes are based.

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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 hot 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 parallel electric
  field vanishes.
• -- Adiabatic pressure law thermal plasma
  pressure changes adiabatically through plasma
  convection and compression.
• -- Gyroviscosity, that 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
• (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 Coupling Processes

• Spatial scale coupling
• -- For k? ri O(1), ion motion is different
  from electron E B drift motion and large dEk
  can be produced.
• -- For L? ri, particle magnetic moment is not
  an adiabatic invariant, ion motion is
  stochastic.
• -- Banana orbit rB gtgt boundary layer width db
  and ri
• Temporal scale coupling
• -- If wb gt w, trapped particles will respond to
  an bounce orbit- averaged field
• -- If wb, wt w, transit or bounce resonances
  are important for energy dissipation
• -- If wd w, wave-particle drift resonance
  effects are important for energy dissipation
• -- If wd À w, particle magnetic drift motion
  dominates over
• E B drift

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Typical Fusion Plasma Parameters

• Typical Parameters of Magnetic Fusion Devices
• B ' 5 T, ne ' 1014 cm-3, Ti,e ' 10 keV,
• LB, Lp ' a ( 1m), R/a ' 3, bc bh 10-2
• Characteristic Scales of Core Particle Dynamics
• ri ' 3 mm, wci ' 3 108 sec-1,
• wte, wbe 107 sec-1, wti, wbi 105-106 sec-1,
• wi n 105 sec-1, wdi, wde n (104-105)
  sec-1
• Fast Ions (nh lt nc, bh bc)
• a-Particles 3.5 MeV NBI-Particles 100
  keV
• N-NBI-Particles 350 keV ICRF Tail Ions
  1 MeV
• rh ' 1 - 5 cm, wth, wbh 106 - 107 sec-1,
• wdh n (105 - 106) sec-1

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• Temporal and Spatial Scale Orderings
• -- TAE Modes w ' VA/2qR 106 sec-1
• For low-n modes
• wte, wbe, wh w wth, wbh gt wti, wbi, wdh
• k? rBh 1, k? rh k? rBi 1, k? ri lt 1
• For high-n modes
• wte, wbe, wh w wth, wbh, wdh, wi gt wti,
  wbi
• k? rBh À 1, k? rh k? rBi 1, k? ri 1
• -- Internal Kink and Fishbone Modes
• n 1 and w ' wi, wdh 105 sec-1
• wte, wbe wth, wbh gt wti, wbi, wdh w gt wdi,e
  
• In the inertial layer k? rh gt 1, k? ri 1
• ?
• Both thermal and fast particle kinetic effects
  are important in determining energetic particle
  physics.

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

• Consider high-b multi-ion species plasmas in
  general magnetic field geometry
• 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_j are
  determined from gyrokinetic (for w lt wci) or
  Vlasov (for w wci) equation.
• 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) F/m (w- kk vk
  - wd)/wc
• (F vkAk) (l J0 J1 J02 - 1)
• (v?dBk/2k?)l(1 2 J12) 2 J0 J1
• w0 -(Tw/m) ln F/e, l k? v? / wc

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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

• Linear Stability Codes
• -- Extend non-perturbative global NOVA-2 code to
  include both thermal and fast particle kinetic
  effects
• -- Extend high-n HINST code with
  non-perturbative treatment of both thermal and
  fast particle kinetic effects
• Nonlinear Global Simulation Codes
• -- Extend M3D-K and HYM codes to include both
  thermal and fast particle kinetic effects.

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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

Pa gt Paux
Fast Ion Driven Instabilities Alpha Transport
Must develop efficient methods to control
profiles for burn control!
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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
  the kinetic-fluid model agree with those derived
  from gyrokinetic equations for w lt wci.
• Based on the 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 the kinetic-fluid model can be
  constructed by extending these existing
  kinetic-MHD codes.