A Water Bag Model of Phase Space Holes in Nonneutral Plasma

Ido Barth and Lazar Friedland

I. Barth, L. Friedland, and A.G. Shagalov, Phys. Plasmas, 15, 082110 (2008)

BGK (Berstein, Greene and Kruskal) modes

Electrostatic, dissipationless (no Landau damping) waves

Fluid-type description is not sufficient (resonant particles).

Need of kinetic theory (Vlasov-Poisson)

Observed: plasma experiments , but difficult to control

Magnetosphere

Solar wind

Supernova remnant

Phase space holes BGK mode

How to excite and control BGK modes ?

Take a Penning trapped pure-ion plasma.

Apply an oscillating external potential.

Slowly reduce the driving frequency.

Phase space hole and the associated BGK mode locked to the drive is formed.

Autoresonance !

Our approach:

The System

Penning trapped plasma:

Vlasov-Poisson system:

Reflecting boundary conditions:

Driving potential

B

• L. Friedland, F. Peinetti, W. Bertsche, J. Fajans, and J. Wurtele , Phys. Plasmas 11, 4305 (2004)

2 2

0

1 , ,

drive

t x x x u

xx

f uf f

f u x t du

,0, ,0, ,

,1, ,1,

f u t f u t

f u t f u t

0d t

Simulation Results

Holes outside Holes inside Collision and resonance overlap

16.1du 7.0du 21.0du

Electrostatic wave amplitude: (a) (b) (c)

Phase space holes formation:

Phase space holes BGK mode

The Water Bag model

I. Plasma between limiting trajectories.

II. Perfect phase locking.

Assumptions:

Dynamics of the limiting trajectories

Hamiltonian system with slow parameters.

Each limiting trajectory has its own conserved action.

Add Fourier transformed Poisson equation

Sufficient number of algebraic equations to describe evolution of the limiting trajectories and the associate BGK mode.

Differential equations Algebraic equations

Simulation vs. Theory

Flat top Distribution

Mexwellian Distribution

Summary

We found a way to excite and control BGK modes in trapped single species plasmas.

Our water bag theory fully agrees with simulations.

Thank you !

Stability

Vlasov-Poisson System

2 2

0

1 , ,

drive

t x x x u

xx

f uf f

f u x t du

radial screening parameter.

dencity parameter.

,0, ,0, ,Reflecting boundary conditions:

,1, ,1,

f u t f u t

f u t f u t

0

2 cos cosdrive

d

d

kx t dt

t

Differential equations Algebraic equations

Limiting trajectory Hamiltonian:

Poisson equation: Holes outside

Holes inside

Adiabatic Invariants:

2 free holes = 1 trapped hole

txtx

txuftxuf

,,

,,,,

2 2 2

1 2 0A k F F F

Symmetry condition:

21

, ' 'cos '2

dH u x u u A kx

2 2 2

2 1A k F F

0,1,2 0,1,2

1cos ' '

2F u kx dx

0,1,2 0,1,2

1'

4J u dx

'cos'2 kxAHuu d

Adiabatic Driven Water Bag model

Adiabatic transformation between stationary driven water bag equilibria.

Each limiting trajectory posses an adiabatic invariant

We get a complete set of Algebraic equations for the self field and the limiting trajectories.