Stability Properties of Field-Reversed Configurations (FRC)

E. V. Belova

PPPL

2003 International Sherwood Fusion Theory ConferenceCorpus Christi, TX, April 2003

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

FRC parameters:

R

Zφ

R

SZ

SR

radius.Larmor toradius separatrix of ratio the toequals parameter, kinetic - /

number; mode toroidal-

;elongation separatrix -

iS

SS

RS*

n

/ RZE

Ψ

inφe~B

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

),(0 pf

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 equilibriaanalytic 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

I. Linear stability: Hall effect

Growth rate is reduced by a factor of two for S*/E1.

To isolate Hall effects Hall-MHD simulations of the n=1 tilt mode

Hall-MHD simulations (elliptic separatrix, E=6)

0γ/γrω-

1/S*

- Compare with analytic results:

Stability at S*/E1 [Barnes, 2002]

Hall stabilization: not sufficient to explain stability; FLR and other kinetic effects must be included.

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

φR V,V

φR V,V In Hall-MHD simulations tilt modeis 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

I. Linear stability: FLR effect

Hybrid simulations with and without Hall term; E=4 elliptic separatrix.

Without Hall

With Hall

0γ/γ

- 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

r

Z

R

Z

R

ZVZV),( VVR ),( VVR

Hybrid simulation without Hall term Hybrid simulation with Hall term

FLR: Mode is MHD-like, FLR & Hall: Mode is Hall-MHD-like, 0r0r

I. Linear stability: FLR vs Hall

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.

mhdγ/γ

*/ SE 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

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

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*

)ω(Ω β,

5.1*

2.1

E

S

s

12*

4.9

E

S

s

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

Lines correspond to resonances:

3/)(

and ,1/)(

Stochasticity of ion orbits – expected to reduce growth rate.

MHD-like

Kinetic

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

exists. invariant adiabatic

another parameter small a is 2/1~/ ERZ

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

Regularity condition

2

2

2

)(),(

R

pZRVeff

Regularity condition:

(%) regularN

*/1 S

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*.

||

p

0

regular

stochastic

|| 0 p || 0 0 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 p

(Betatron orbit) (Betatron or drift, depending on )

2|| 2|| 0000 RpR Number of regular orbits ~ 1/S*

Elliptic, E=6, 12

Racetrack, E=7

I. Linear stability: Resonant effects

In f simulations evolve not f , but , where =>simulation particles has weights , which satisfy:

)2/( 02

00 vmff i0fff ffw /

)(ln

)()(ln 00 f

dt

fd

dt

dwδEv

xδEδj

δEv

i3

2

0

/1

ln)(

dw

fw

mm

mm

t

m

mwTE 2/20

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 resonantnature of instability.

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)

))/-( ; /( ffw

/)(

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 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.

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

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

A34tt

A42tt

A46tt

A54tt

A50tt

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.

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 V0.3v .

A

A

2n |V|

At/t

extz 0.5BB

A

R

Z

RZ B,B

0 10 20 30

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.