1. 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, 2003 Albuquerque, New Mexico

2. 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

3. 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.

4. Kinetic-MHD Model Momentum Equation (P c » P h ):  [  /  t + V ¢r ] V = – r P c – r¢ P h + J £ B Continuity Equation (n ' n c, n h ¿ n c ) : [  /  t + V ¢r ]  +  r¢ V = 0 Maxwell's Equations:  /  t = – r£ E, J = r£ B, r¢ B = 0 Ohm's Law: E + V £ B = 0, E ¢ B = 0 Adiabatic Pressure Law: [  /  t + V ¢r ] (P c /  5/3 ) = 0 Hot Particle Pressure Tensor: P h = {m h /2} s d 3 v vv f h (x,v) where f h is governed by gyrokinetic or Vlasov equations.

5. 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.

6. 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 E £ B drift velocity and E k =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)  ci À  À  t,  b,  *,  d for all particle species (b) kL > 1 and k  i ¿ 1

7. 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.

8. Kinetic-Fluid Model [Cheng & Johnson, J. Geophys. Res., 104, 413 (1999)] Consider high-  multi-ion species plasmas Consider  <  ci, k ?  i » O(1) Mass Density Continuity Equation: [  /  t + V ¢r ]  +  r¢ V = 0 Momentum Equation: (  /  t+ V ¢r ) V = J £ B – r ¢  j P j cm P j cm = m j s d 3 v (v – V)(v – V) f j Particle distribution functions f = F(x, , , t) +  f,  f = –[(q/m)  F/  (q/mB)  F/  (1– J 0 2 )(  – v k A k ) – (v ? J 1 /2k ? )  B k ]  g 0 e iL and g 0 is determined from gyrokinetic equation:  /  t + (v k +v d ) ¢r ] g 0 = –[(q/m)  F/   /  t – (B/B 2 ) £r (F+ g 0 ) ¢r £ ] [J 0 (  – v k A k ) + (v ? J 1 /2k ? )  B k ]; or  f can be solved by particle code. Maxwell's equations in magnetostatic limit are employed.

9. Pressure Tensor and Gyroviscosity: P = P ? (I - bb) + P k bb +  where I is the unit dyadic and b = B/B. P k = m s d 3 v v k 2 f, P ? = (m/2) s d 3 v v ? 2 f For k ? À k k, gyroviscosity tensor contribution r¢  ¼ b £ ( r  P c £ b) + b £r ?  P s  P c =  P c1 +  P c2,  P c1 = s d 3 v (m v ? 2 /2) g 0 (J 0 – 2 J 1 0 );  P c2 = s d 3 v (m v ? 2 /2) (q/mB)}  F/  [(  – v k A k )(2J 0 J 1 0 – J 0 2 ) – (v ?  B k /k ? )(J 0 J 1 – 2 J 1 J 1 0 )];  P s = s d 3 v (i mv ? 2 / 2 ) £ [(qF/T)(  0 -  * T ) /  c – (q/mB) (  k k v k   d )  F/  /  c ] £ {( J 0 J 1 + J 0 2 - 1)(  – v k A k ) – [ (1 – 2 J 1 2 ) – 2 J 0 J 1 ](v ?  B k /2k ? )};  0 =  (T  /m)  ln F/ , = k ? v ? /  c,  F(x, , , t) ´ averaged over fluctuation scales when necessary,  /  t  =  i  and r = ik operate on perturbed quatities.

10. Low-Frequency Ohm's Law E + V £ B = (1/n e e)[ J £ B – r¢ ( P e cm –  i (q i m e /e m i ) P i cm )] +  i (m i /  q i – 1/n e e)(B/B) £ ( r¢ P i 0 £ B/B) + (m e /n e e 2 ) [  J/  t + r¢ (JV + VJ)] +  J where P i 0 = m i s d 3 v vv f i Main Features: -- The kinetic-fluid model retains most essential particle kinetic effects in low frequency phenomena (  <  ci ) 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 n h /n c ratio -- Nonlinear

11. 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

12. Integration of Burning Plasmas Physics Auxiliary Heating Fueling Current Drive P(r), n(r), q(r) Confinement, Disruption Control MHD Stability Heating Power: P  > P aux Fast Ion Driven Instabilities Alpha Transport  interaction with thermal plasmas is a strongly nonlinear process. Must develop efficient methods to control profiles for burn control!  Need nonlinear kinetic-fluid simulation codes! Fusion Output  -Heating  -CD

13. Summary A nonlinear kinetic-fluid model has been developed for high-  plasmas with multi-ion species for  <  ci. 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  <  ci. 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.