Kinetic Effects on the Linear and Nonlinear Stability Properties of Field-Reversed Configurations

1. 2003 APS DPP Meeting, October 2003 Kinetic Effects on the Linear and Nonlinear Stability Properties of Field-Reversed Configurations E. V. Belova PPPL In collaboration with : R. C. Davidson, H. Ji, M. Yamada (PPPL)

2. OUTLINE: I. Linear stability (n=1 tilt mode, prolate FRCs) - FLR stabilization - Hall term versus FLR effects - resonant particle effects - finite electron pressure and toroidal magnetic field effects II. Nonlinear effects - nonlinear saturation of n=1 tilt mode in kinetic FRCs - nonlinear evolution in the small Larmor radius regimes

3. Ψ R R φ Z FRC parameters:

4. FRC stability with respect to the tilt mode: Theory vs experiment Possible non-ideal MHD effects, which may be responsible for the experimentally observed FRC behavior: • Thermal ion FLR effects. • Hall term effects. • Sheared flows. • Profile effects (racetrack vs elliptical configurations). • Electron physics (finite P , kinetic effects). • Finite toroidal magnetic field. • Resonant ion effects, stochasticity of ion orbits. • Particle loss. • Nonlinear kinetic effects. Comprehensive nonlinear kinetic simulations are needed in order to study FRC stability properties. e

5. Numerical Studies of FRC stability • FRC stability code – HYM (Hybrid & MHD): • 3-D nonlinear • Three different physical models: • - Resistive MHD & Hall-MHD • - Hybrid (fluid e, particle ions) • - MHD/particle (fluid thermal plasma, • energetic particle ions) • For particles: delta-f /full-f scheme; analytic • Grad-Shafranov equilibria • Parallel (MPI) version for distributed memory parallel • computers. Scaled Fixed problem size

6. I. Linear stability: FLR effects FLR effects – determines linear stability of the n=1 tilt mode. Elliptical equilibria ( special p() profile[Barnes,2001] ) - For E/S*<0.5 growth rate is function of S*/E. - For E/S*>0.5 growth rate depends on both E and S*. Racetrack equilibria- S*/E-scaling does not apply. New empirical scaling: E=4 E=6 E=12 Hybrid simulations for equilibria with elliptical separatrix and different elongations: E=4, 6, 12. For E/S*>0.5, resonant ion effects are important. S*/E parameter determines the experimental stability boundary [M. Tuszewski,1998].

7. I. Linear stability: Hall effects Recent analytic results:stability of the n=1 tilt mode at S*/E1 [Barnes, 2002] To isolate Hall effects  Hall-MHD simulations FLR effects hybrid simulations with full ion dynamics, but turn off Hall term Without Hall With Hall 1/S* 1/S* Hall-MHD (elliptic separatrix, E=6): growth rate is reduced by a factor of two for S*/E1. Hall stabilization: not sufficient to explain stability. Growth rate reduction is mostly due to FLR; however, Hall effects determine linear mode structure and rotation.

8. I. Linear stability: Hall effect In Hall-MHD simulations tilt mode is more localized compared to MHD; also has a complicated axial structure. MHD • Hall effects: • modest reduction in  (50% at most) • rotation (in the electron direction ) • significant change in mode structure Hall-MHD Change in linear mode structure from MHD and Hall-MHD simulations with S*=5, E=6.

9. Finite electron pressure and toroidal field effects • Effects offinite P :increasing fraction of total pressure carried by electrons has a • destabilizing effect of the tilt mode due to effective reduction of the ion FLR effects. e 0.75 P =0.75P e 0.875 P =0.5P e 0.5 P =0.75P P =0.3P 0.3 e e P =0 P =0.5P e e P =0 P =0 e e • Effects ofweakequilibrium toroidal field(symmetric profile): - Destabilizing for B ~ 10-30% of external field; growth rate increases by ~40% for B =0.2 B (S*=20). - Reduction of average thermal ion Larmor radius. - Maximum beta is still very large β ~ 10-100.  ext 

10. I. Linear stability: Resonant effects Betatron resonance condition: [Finn’79]. Ω – ω = ω β Growth rate depends on: 1. number of resonant particles 2. slope of distribution function 3. stochasticity of particle orbits

11. I. Linear stability: Resonant effects Particle distribution in phase-space for different S* MHD-like 0.05 0.04 0.03 0.02 0.01 0.00 (E=6 elliptic separatrix) Lines correspond to resonances: -0.1 -0.05 0.00 0.05 0.1 As configuration size reduces, characteristic equilibrium frequencies grow, and particles spread out along  axis – number of particles at resonance increases. Kinetic 0.15 0.10 0.05 0.00 Stochasticity of ion orbits – expected to reduce growth rate. -0.4 -0.2 0.0 0.2 0.4

12. Stochasticity of ion orbits For majority of ions µ is not conserved in typical FRC: For elongated FRCs with E>>1, Two basic types of ion orbits (E>>1): Betatron orbit (regular) Betatron orbit Driftorbit Drift orbit (stochastic) For drift orbit at the FRC ends  stochasticity.

13. Regularity condition Number of regular orbits ~ 1/S* regular Racetrack, E=7 stochastic Elliptic, E=6, 12 Regular versus stochastic portions of particle phase space for S*=20, E=6. Width of regular region ~ 1/S*. Regularity condition: Regularity condition can be obtained considering particle motion in the 2D effective potential: Shape of the effective potential depends on value of toroidal angular momentum . (Betatron orbit) (Betatron or drift, depending on )

14. I. Linear stability: Resonant effects Hybrid simulations with different values of S*=10-75 (E=6, elliptic) Scatter plots in plane; resonant particles have large weights. Ω – ω = lω , l=1, 3, … β For elliptical FRCs, FLR stabilization is function of S*/E ratio, whereas number of regular orbits, and the resonant drive scale as ~1/S*  long configurations have advantage for stability. -1 0 1 2 3 4 5 6 7 8 9 Simulations with small S* show that small fraction of resonant ions (<5%) contributes more than ½ into energy balance – which proves the resonant nature of instability.

15. I. Non-linear effects: Large Larmor radius FRC Nonlinear evolution of tilt mode in kinetic FRC is different from MHD: - instabilities saturate nonlinearly when s is small. _ Possible saturation mechanisms: - flattening of distribution function in resonant region, - configuration appear to evolve into one with elliptic separatrix and larger E, - velocity shear stabilization due to ion spin-up. _ Hybrid simulations with E=4, s=2, elliptical separatrix.

16. I. Non-linear effects: Large Larmor radius FRC Energy plots from nonlinear hybrid simulations E=4, s=2 n=2 n=1 LSX [Slough, Hoffman, 93] n=3 Ion velocity at FRC midplane. n=4 0.2 0.1 0.0 • Nonlinear simulations show growth and saturation of • the n=1 tilt mode. • In the nonlinear phase, the growth of and saturation of • the n=2 rotational mode is observed. • Ion spin-up with V ~ 0.1-0.3 V at t ~ 40. • Similar behavior found for other FRC configurations • with different shapes and profiles. i A R Radial profile of ion flow velocity at t=53.

17. I. Non-linear effects: Large Larmor radius FRC Equilibrium with E=6 and s=2.3, elliptical shape. Contour plots of plasma density. t=44 n=0 n=1 n=2 t=60 t=76 R Vector plot of poloidal magnetic field. Z t=76

18. II. Non-linear effects: Small Larmor radius FRC _ Nonlinear hybrid simulations for large s (MHD-like regime). • Linear growth rate is comparable to MHD. • • No saturation, but • • Nonlinear evolution is considerably • slower than MHD. • Field reversal ( ) • is still present after t=30 t . • Effects of particle loss: • About one-half of the particles are lost by • t=30 t . • Particle loss from open field lines • results in a faster linear growth due • to the reduction in separatrix beta. • Ions spin up in toroidal (diamagnetic) • direction with V0.3v . A 0 10 20 30 A R Z (a) Energy plots for n=0-4 modes, (b) Vector plots of poloidal magnetic field, at t=32 t . A A

19. Summary • FLR effects – main stabilizing mechanism. • s/E scaling has been demonstrated for elliptical FRCs. • Resonant effects – shown to drive instability at low s. • Stochasticity of ion orbits is not strong enough to prevent instability; • regularity condition has been derived; number of regular orbits has been • shown to scale linearly with 1/s. • Hall term – defines mode rotation and structure. • Finite toroidal field and electron pressure are destabilizing. • Nonlinear evolution: saturation at low s, n=2 rotational mode; • Larger s - nonlinear evolution is slow compared to the MHD; • Ion spin-up in diamagnetic direction. _ _ _ _ _

20. Conclusions Pressure evolution form SSX-FRC simulations. • FRC behavior at low-s is best understood, more realistic theoretical studies provide explanation for experimentally observed FRC properties. • Large-s FRCs: new formation schemes (other than theta-pinch) and better theoretical understanding of large-s FRC stability properties are needed. • New formation methods: - Counter-helicity spheromak merging (U. Tokyo, SSX-FRC, SPIRIT). - RMF (U. Washington, PPPL). • Numerical studies using HYM code will guide development of SPIRIT program.