Stability Properties of Field-Reversed Configurations (FRC)

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

• E. V. Belova
• PPPL

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OUTLINE

• I. Linear stability (n1 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 n1 tilt mode for
  small S
• - nonlinear evolution for large S

usual (racetrack) FRCs vs long,
elliptic-separatrix FRCs

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?
R
R
f
Z
FRC parameters
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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

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I. Linear stability
- Concentrate on n1 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
equilibria analytic p(?)
special
p(?) Barnes,2001 racetrack-like

• always global mode
• ? scales as 1/E
• more stochastic

• end-localized mode
• ? saturates with E

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I. Linear stability Hall effect
To isolate Hall effects ? Hall-MHD simulations of
the
n1 tilt mode
Hall-MHD simulations (elliptic separatrix, E6)
- 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.
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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 S5, E6.
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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
E4 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.
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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,
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I. Linear stability Elongation and profile
effects
E4

• Elliptical equilibria (special p(??) profile)
• - For S/Egt2 growth rate is function of S/E.
• - For S/Elt2 growth rate depends on both E and
  S ,
• and resonant particles effects are
  important.

E6
E12
Racetrack equilibria (various p(??) profiles)
- S/E-scaling does not apply.
Hybrid simulations for equilibria with elliptical
separatrix and different elongations E4, 6,
12. For S/Elt2, resonant ion effects are
important.
S/E scaling agrees with the experimental
stability scaling M. Tuszewski,1998.
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I. Linear stability Resonant effects
Betatron resonance condition
Finn79.
O ? ??
ß
Growth rate depends on 1. number of resonant
particles
2. slope of distribution function
3.
stochasticity of particle orbits
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I. Linear stability Resonant effects
Particle distribution in phase-space
for different S
MHD-like
(E6 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.
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Stochasticity of ion orbits
For majority of ions µ is not conserved in
typical FRC For elongated FRCs with Egtgt1,
Two basic types of ion orbits (Egtgt1)
Betatron orbit (regular)
Betatron orbit
Drift orbit
Drift orbit (stochastic)
For drift orbit at
the FRC ends ? stochasticity.
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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, E7
regular
Elliptic, E6, 12
stochastic
Regular versus stochastic portions of particle
phase space for S20, E6. Width of regular
region 1/S.
Fraction of regular orbits in three different
equilibria.
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I. Linear stability Resonant effects
In ?f simulations evolve not f , but
, where
gt 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 (lt5) contributes more
than ½ into calculated growth rate which proves
the resonant nature of instability.
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I. Linear stability Resonant effects
Hybrid simulations with different values of
S10-75 (E6, elliptic)
Scatter plots in plane resonant particles have
large weights.
w
O ? l ? , l1, 3,
ß
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, E12, case is similar, but
resonant effects become important at larger S ?
smaller number of regular orbits, and smaller
growth rates.
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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 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 (S20).
?
?
ext
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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 E4, s2, elliptical
separatrix.
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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 t30 t .
• Effects of particle loss
• About one-half of the particles are lost by
• t30 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 n0-4 modes, (b) Vector
plots of poloidal magnetic field, at t32 t .
A
A
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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?

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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/Egt2.
• 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.

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