# Presentation: 900255

## Stability Properties of Field-Reversed Configurations (FRC)

- E. V. Belova
*PPPL*

- 2003 International Sherwood Fusion Theory Conference
- Corpus 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

- Z

- φ

- R

**Ψ**

## Numerical Studies of FRC stability

- 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**

## 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 equilibria**
- analytic 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*/E*1.

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

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

- 1/S*

- - Compare with analytic results:
-  Stability at S*/E1  [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**

- In Hall-MHD simulations tilt mode
- is 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

-  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

## I.  Linear stability: FLR vs Hall

- Z

- R

- Z

- R

- Hybrid simulation without Hall term

- Hybrid simulation with Hall term

**FLR:** Mode is MHD-like,

**FLR & Hall**: Mode is Hall-MHD-like,

**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.

-  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

## I. Linear stability: Resonant effects

**Ω – ω = ********ω**

-  **β**

**I. Linear stability: ****Resonant effects**

- 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****

- As configuration size reduces,
- characteristic equilibrium
- frequencies grow, and particles
- spread out along  axis – _number_
- _of particles at resonance increases._

- Lines correspond to resonances:

- 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*,

**Two basic types of ion orbits (E>>1):**

- Betatron orbit (regular)

- Drift orbit (stochastic)

- For drift orbit                             at the FRC ends  stochasticity.

## Regularity condition

**Regularity condition:**

- 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*.*

- regular

- stochastic

- 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 )

**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:

- 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 1⁄2 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)

*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 in
- plane; 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

##       Nonlinear evolution of tilt mode in kinetic FRC is different from MHD:
     - instabilities saturate nonlinearly when S* is small [Belova et al.,2000].

**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 V0.3v .

-  A

-  A

-  A

-  R

- Z

- 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.

## Slide 22

## Slide 23