2003 International Sherwood Fusion Theory Conference Corpus Christi

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

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? “usual” (racetrack) FRCs vs long, elliptic-separatrix FRCs II. Nonlinear effects - nonlinear saturation of n=1 tilt mode for small S* - nonlinear evolution for large S*

Ψ R R φ Z FRC parameters:

Numerical Studies of 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

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

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.

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: Hall-MHD • modest reduction in (50% at most) • rotation (in the electron direction ) • significant change in mode structure Change in linear mode structure from MHD and Hall-MHD simulations with S*=5, E=6.

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 With Hall Growth rate reduction is mostly due to FLR; however, Hall effects determine linear mode structure and rotation.

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

I. Linear stability: Elongation and profile effects E=4 E=6 Elliptical equilibria (special p( ) profile) E=12 - 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. 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].

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

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

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 Drift orbit For drift orbit at the FRC ends stochasticity. Betatron orbit (regular) Drift orbit (stochastic)

Regularity condition can be obtained considering particle motion in the 2 D effective potential: Shape of the effective potential depends on value of toroidal angular momentum Regularity condition: (Betatron orbit) (Betatron or drift, depending on ) 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.

I. Linear stability: Resonant effects In f simulations evolve not f , but simulation particles has weights , where , 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.

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 = l ω β , l=1, 3, … 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.

I. Linear stability Wave-particle resonances are shown to Scatter plot of resonant particles in phase-space. • 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. 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 ext (S*=20).

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.

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 A. 0 10 20 30 R Z (a) Energy plots for n=0 -4 modes, (b) Vector plots of poloidal magnetic field, at t=32 t A. Effects of particle loss: • About one-half of the particles are lost by t=30 t. A. • 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. 3 v. A.

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?

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.