1. Stability Properties of Field-Reversed Configurations (FRC) E. V. Belova PPPL 2003 International Sherwood Fusion Theory Conference Corpus Christi, TX, April 2003

2. OUTLINE: I. Linear stability (n=1 tilt mode, prolate FRCs) - FLR stabilization - Hall term versus FLR effects - resonant particle effects - is linearly-stable FRC possible? II. Nonlinear effects - nonlinear saturation of n=1 tilt mode for small S* - nonlinear evolution for large S* “usual” (racetrack) FRCs vs long, elliptic-separatrix FRCs

3. FRC parameters: R Z φ R Ψ

4. FRC stability code – HYM (Hybrid & MHD): 3-D nonlinear Three different physical models: - Resistive MHD & Hall-MHD -large S* - Hybrid (fluid e, particle ions) -small S* - MHD/particle (fluid thermal plasma, energetic particle ions) For particles: delta-f /full-f scheme; analytic Grad-Shafranov equilibria Numerical Studies of FRC stability

5. I. Linear stability - Concentrate on n=1 tilt mode (most difficult to stabilize, at least theoretically) - Three kinetic effects to consider: 1. FLR 2. Hall 3. Resonant particle effects stabilizing destabilizing, and obscure the first two Long FRC equilibria : “Usual” equilibria Elliptical equilibria analytic p(ψ) special p(ψ) [Barnes,2001] & racetrack-like end-localized mode γ saturates with E always global mode γ scales as 1/E more stochastic

6. I. Linear stability: Hall effect Growth rate is reduced by a factor of two for S*/E  1. To isolate Hall effects  Hall-MHD simulations of the n=1 tilt mode Hall-MHD simulations (elliptic separatrix, E=6) 1/S* - Compare with analytic results: Stability at S*/E  1 [Barnes, 2002] Hall stabilization: not sufficient to explain stability; FLR and other kinetic effects must be included.

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

8. I. Linear stability: FLR effect Hybrid simulations with and without Hall term; E=4 elliptic separatrix. Without Hall With Hall - cannot isolate FLR effects without making FLR expansion  hybrid simulations with full ion dynamics, but turn off Hall term Growth rate reduction is mostly due to FLR; however, Hall effects determine linear mode structure and rotation. Without Hall With Hall

9. Z R Z R Hybrid simulation without Hall termHybrid simulation with Hall term FLR: Mode is MHD-like,FLR & Hall: Mode is Hall-MHD-like, I. Linear stability: FLR vs Hall

10. I. Linear stability: Elongation and profile effects Elliptical equilibria (special p(  ) profile) - For S*/E>2 growth rate is function of S*/E. - For S*/E<2 growth rate depends on both E and S*, and resonant particles effects are important. Hybrid simulations for equilibria with elliptical separatrix and different elongations: E=4, 6, 12. For S*/E<2, resonant ion effects are important. Racetrack equilibria (various p(  ) profiles) - S*/E-scaling does not apply. S*/E scaling agrees with the experimental stability scaling [M. Tuszewski,1998]. E=4 E=12 E=6

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

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

13. Stochasticity of ion orbits Betatron orbit Drift orbit 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) Drift orbit (stochastic) For drift orbit at the FRC ends  stochasticity.

14. Regularity condition Regularity condition: Fraction of regular orbits in three different equilibria. Regular versus stochastic portions of particle phase space for S*=20, E=6. Width of regular region ~ 1/S*. regular stochastic 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  ) Number of regular orbits ~ 1/S* Elliptic, E=6, 12 Racetrack, E=7

15. I. Linear stability: Resonant effects In  f simulations evolve not f, but, where => simulation particles has weights, which satisfy: It can be shown that growth rate can be calculated as: Here - plays role of perturbed particle energy. Simulations with small S* show that small fraction of resonant ions (<5%) contributes more than ½ into calculated growth rate – which proves the resonant nature of instability.

16. I. Linear stability: Resonant effects Hybrid simulations with different values of S*=10-75 (E=6, elliptic) w w Larger elongation, E=12, case is similar, but resonant effects become important at larger S*  smaller number of regular orbits, and smaller growth rates. -1 0 1 2 3 4 5 6 7 8 9 Scatter plots in plane; resonant particles have large weights. Ω – ω = l ω, 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.

17. I. Linear stability Investigated the effects of weak toroidal field on MHD stability - destabilizing (!) for B ~ 10-30% of external field growth rate increases by ~40% for B =0.2 B (S*=20).  Scatter plot of resonant particles in phase-space. Wave-particle resonances are shown to occur only in the regular region of the phase-space; highly localized. Possibilities for stabilization: Non-Maxwellian distribution function. Reduce number of regular-orbit ions.   ext

18. Hybrid simulations with E=4, s=2, elliptical separatrix. I. Non-linear effects: Small S* Nonlinear evolution of tilt mode in kinetic FRC is different from MHD: - instabilities saturate nonlinearly when S* is small [Belova et al.,2000]. Resonant nature of instability at low S* agrees with non-linear saturation, found earlier. Saturation mechanisms: - flattening of distribution function in resonant region; - configuration appear to evolve into one with elliptic separatrix and larger E.

19. II. Non-linear effects: Large S* Nonlinear hybrid simulations for large S* (MHD-like regime). (a) Energy plots for n=0-4 modes, (b) Vector plots of poloidal magnetic field, at t=32 t. Linear growth rate is comparable to MHD, but nonlinear evolution is considerably slower. 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 A A R Z 0 10 20 30 A

20. Future directions (FRC stability) Low-S* FRC stability is best understood. Can large-S* FRCs be stable, and how large is large? Which effects are missing from present model: - The effects of non-Maxwellian ion distribution. - The effects of energetic beam ions. - Electron physics (e.g., the traped electron curvature drifts). - Others?

21. Summary Hall term – defines mode rotation and structure. FLR effects – reduction in growth rate. S*/E scaling has been demonstrated for elliptical FRCs with S*/E>2. Resonant effects – shown to maintain 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 lnearly with 1/S*. Nonlinear saturation at low S* – natural mechanism to evolve into linearly stable configuration. Larger S* - nonlinear evolution is different from MHD: much slower; ion spin-up in diamagnetic direction.