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

Reversed Configurations

E. V. Belova

PPPL

2003 APS DPP Meeting, October 2003

In collaboration with : R. C. Davidson, H. Ji, M. Yamada (PPPL)

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

FRC parameters:

R

Zφ

R

SZ

SR

).(ion configurat in radiiLarmor ion ofnumber - / s

);(depth skin ion toradius separatrix of ratio the toequals parameter, kinetic - /

number; mode toroidal-

;elongation separatrix -

iFLR

Hall

ρ

λ

a

R*Sn

/ RZE

iS

SS

=

=

=

Ψ

inöe~Bδ

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

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.

Numerical Studies of FRC stability

),(0 ϕε pfprocsN

1)[sec] step time( −

Fixed problem size

Scaled

I. Linear stability: FLR effects

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.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].

FLR effects – determines linear stability of the n=1 tilt mode.

New empirical scaling:

{ 44 344 21(FLR) kineticMHD

3 exp ⎟⎟⎠

⎞⎜⎜⎝

⎛−=

s

i

s

A

RE

ERV ρ

γmhdã/ã

si RESE /*/ ρ=

E=4

E=12

E=6

I. Linear stability: Hall effects

Hall-MHD (elliptic separatrix, E=6): growth rate is reduced by a factor of two for S*/E1.

To isolate Hall effects Hall-MHD simulations

0ã/ãrù-

1/S*

Recent analytic results: stability of the n=1 tilt mode at S*/E1 [Barnes, 2002]

Hall stabilization: not sufficient to explain stability. Growth rate reduction is mostly due to FLR; however, Hall effects determine linear mode structure and rotation.

FLR effects hybrid simulations with full ion dynamics, but turn off Hall term

Without Hall

With Hall

0ã/ã

1/S*

I. Linear stability: Hall effect

Change in linear mode structure from MHD and Hall-MHD simulations with S*=5, E=6.

MHD

Hall-MHD

1E

*S<

ZV

ZV

( )öR V,V

( )öR V,V In Hall-MHD simulations tilt modeis more localized compared to MHD;also has a complicated axial structure.

Hall effects:

• modest reduction in γ (50% at most)• rotation (in the electron direction )• significant change in mode structure

Finite electron pressure and toroidal field effects

• Effects of weak equilibrium 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.

ϕ

ϕ ext

• Effects of finite 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

as i //1 ρ=

mhdã/ã

0P

e =

P =0 e

P =0.5P e

P =0.75P e

0.875

P =0 e

0.5

0.3

0.75

P =0.5P e

P =0 e

P =0.3P e

P =0.75P e

21 || V

Att /

Betatron resonance condition: [Finn’79].

Ω – ω = ω β

I. Linear stability: Resonant effects

frequencybetatron axial - frequency,rotation toroidalparticle is -

number, odd is where, if ,resonances particle- waveobserve We

β

β

ω

ωω

Ω

=−Ω ll

Growth rate depends on: 1. number of resonant particles 2. slope of distribution function 3. stochasticity of particle orbits

I. Linear stability: Resonant effects

(E=6 elliptic separatrix)

Particle distribution in phase-space for different S*

)ù(Ù â,

βω

5.1*

2.1

=

=

ESs

12*

4.9

=

=

ESs

βω

Ω

As configuration size reduces,characteristic equilibrium frequencies grow, and particles spread out along Ω axis – numberof particles at resonance increases.

Lines correspond to resonances:

3/)(

and ,1/)(

=−Ω

=−Ω

β

β

ωω

ωω

Stochasticity of ion orbits – expected to reduce growth rate.

MHD-like

Kinetic

-0.4 -0.2 0.0 0.2 0.4

-0.1 -0.05 0.00 0.05 0.1

0.15

0.10

0.05

0.00

0.05

0.04

0.03 0.02

0.01

0.00

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,

)1(/ OLi =ρ

exists. invariant adiabatic

another parameter small a is 2/1~/ →ERZ ωω

Two basic types of ion orbits (E>>1):Betatron orbit (regular)

Drift orbit (stochastic)

For drift orbit at the FRC ends stochasticity. O(1)/ =RZ ωω

Regularity condition

ϕφ +

+=

2

2

2

)(),(

R

pZRVeff

Regularity condition:

(%) regularN

LS i /~*/1 ρ

Regular versus stochastic portions of particle phase space for S*=20, E=6. Width of regular region ~ 1/S*.

||

p

0φ

ε

regular

stochastic

|| 0φ ≥p || 0 0φ << p

Regularity condition can be obtained consideringparticle motion in the 2D effective potential:

Shape of the effective potential depends on value of toroidal angular momentum .φp (Betatron orbit) (Betatron or drift, depending on ε)

εε φ 2|| 2|| 0000 RpR −>>+

Number of regular orbits ~ 1/S*

Elliptic, E=6, 12

Racetrack, E=7

I. Linear stability: Resonant effects

12/*4.9=

=ES

s

5.1/*2.1=

=ES

s

Hybrid simulations with different values of S*=10-75 (E=6, elliptic)

))/-( ; /( βωωδ Ω= ffw

βωω /)( −Ω -1 0 1 2 3 4 5 6 7 8 9

Scatter plots inplane; 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.

Simulations with small S* show that small fraction of resonant ions (<5%)contributes more than ½ into energy balance – which proves the resonantnature of instability.

f

fδ

f

fδ

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

A34tt =

A42tt =

A46tt =

A54tt =

A50tt =

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.

_

_

At / t

I. Non-linear effects: Large Larmor radius FRC

R

A53tt =

2i |V|

Energy plots from nonlinear hybrid simulations E=4, s=2

Ion velocity at FRC midplane.

Radial profile of ion flow velocity at t=53.

• 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

n=1n=2

n=3

n=4

LSX [Slough, Hoffman, 93]

RV,Vφ

)(V Rφ

0.2

0.1

0.0

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

t=76

t=60

At / t

2i |V|

n=1n=0

n=2

R

Z

( )RZ B,B

t=76

Vector plot of poloidal magnetic field.

II. Non-linear effects: Small Larmor radius FRC

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.• 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 V0.3v .

A

A

2n |V|

At/t

extz 0.5BB −≈

A

R

Z

( )RZ B,B

0 10 20 30

A

_

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.

_

_

_

_

_

Conclusions

• 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.

Pressure evolution form SSX-FRC simulations.