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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 > 5 burning plasmas -particles are dominant heating source because P > P aux 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 (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.

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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 E £ B 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) ci À À t, b, *, d (b) kL > 1 and k i ¿ 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 ? i » O(1), ion motion is different from electron E £ B drift motion and large E k can be produced. -- For L ? » i, particle magnetic moment is not an adiabatic invariant, ion motion is stochastic. -- Banana orbit B >> boundary layer width and i Temporal scale coupling: -- If b > , trapped particles will respond to an bounce orbit- averaged field -- If b, t » , transit or bounce resonances are important for energy dissipation -- If d » , wave-particle drift resonance effects are important for energy dissipation -- If d À , 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, n e ' cm -3, T i,e ' 10 keV, L B, L p ' a ( » 1m), R/a ' 3, c » h » Characteristic Scales of Core Particle Dynamics i ' 3 mm, ci ' 3 £ 10 8 sec -1, te, be » 10 7 sec -1, ti, bi » sec -1, i » n £ 10 5 sec -1, di, de » n £ ( ) sec -1 Fast Ions (n h < n c, h » c ) -Particles: 3.5 MeV; NBI-Particles: ¸ 100 keV N-NBI-Particles: ¸ 350 keV; ICRF Tail Ions: » 1 MeV h ' cm, th, bh » sec -1, dh » n £ ( ) sec -1

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Temporal and Spatial Scale Orderings: -- TAE Modes: ' V A /2qR » 10 6 sec -1 ** For low-n modes: te, be, *h ¸ » th, bh > ti, bi, dh k ? Bh ¸ 1, k ? h » k ? Bi » 1, k ? i < 1 ** For high-n modes: te, be, *h ¸ » th, bh, dh, i > ti, bi k ? Bh À 1, k ? h » k ? Bi ¸ 1, k ? i » 1 -- Internal Kink and Fishbone Modes: n = 1 and $ ' i, dh » 10 5 sec -1 te, be ¸ th, bh > ti, bi, dh » > di,e In the inertial layer: k ? h > 1, k ? i » 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- multi-ion species plasmas in general magnetic field geometry 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_j are determined from gyrokinetic (for < ci ) or Vlasov (for » ci ) equation. Maxwell's equations in magnetostatic limit are employed.

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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) F/ ( k k v k d )/ c ] £ {( – v k A k ) ( J 0 J 1 + J ) – (v ? B k /2k ? )[ (1 – 2 J 1 2 ) – 2 J 0 J 1 ]} 0 = -(T /m) ln F/ , = k ? v ? / c

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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

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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 Auxiliary Heating Fueling Current Drive P(r), n(r), q(r) Confinement, Disruption Control MHD Stability 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! Fusion Output -Heating -CD

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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 the kinetic-fluid model agree with those derived from gyrokinetic equations for < ci. 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.

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