Stability Properties of Field-Reversed Configurations (FRC)

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

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. Ψ R R φ Z FRC parameters:

4. Numerical Studies of FRC stability • 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

5. I. Linear stability - Concentrateonn=1 tiltmode(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”equilibriaElliptical equilibria analytic p(ψ) special p(ψ) [Barnes,2001] & racetrack-like • always global mode • γ scales as 1/E • more stochastic • end-localized mode • γ saturates with E

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

7. 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.

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

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

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. E=4 E=6 E=12 Racetrack equilibria(various p() profiles) - S*/E-scaling does not apply. Hybrid simulations for equilibria with elliptical separatrix and different elongations: E=4, 6, 12. For S*/E<2, resonant ion effects are important. S*/E scaling agrees with the experimental stability scaling [M. Tuszewski,1998].

11. 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

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

13. 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.

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

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) Scatter plots in plane; resonant particles have large weights. w 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. w -1 0 1 2 3 4 5 6 7 8 9 Larger elongation, E=12, case is similar, but resonant effects become important at larger S*  smaller number of regular orbits, and smaller growth rates.

17. I. Linear stability • 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. Scatter plot of resonant particles in phase-space.  Investigated the effects ofweak 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).   ext

18. 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. Hybrid simulations with E=4, s=2, elliptical separatrix.

19. II. Non-linear effects: Large S* Nonlinear hybrid simulations for large S* (MHD-like regime). • 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 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

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.