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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.