# Presentation: 567463

## 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

- Z

- φ

- R

**Ψ**

##  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

## Numerical Studies of FRC stability

- 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**

- 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:**

- 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*/E*1.

- To isolate Hall effects
-  **Hall-MHD simulations**

- 1/S*

- Recent analytic results:  stability of the *n*=1 tilt mode at S*/E1 [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

- 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**

- 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

##  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.

-  

- 

- 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**

- 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

## 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

-  -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*,

**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:**

- 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

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

-   -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.

- Simulations with small *S**  show that small fraction of resonant ions (<5%)
- contributes more than 1⁄2 into energy balance – which proves the resonant
- nature of instability.

##       Nonlinear evolution of tilt mode in kinetic FRC is different from MHD:
     - instabilities saturate nonlinearly when s is small.

**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.

- _

-  _

## I. Non-linear effects: Large Larmor radius FRC

**I. Non-linear effects: ****Large Larmor radius FRC**

- R

- 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=1

- n=2

- n=3

- n=4

- LSX [Slough, Hoffman, 93]

- 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

- n=1

- n=0

- n=2

-  R

- Z

- 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 V0.3v .

-  A

-  A

-  A

-  R

- Z

- 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.