0 APS-Sherwood Texas 2006-April 21-24 Study of nonlinear kinetic effects in Stimulated Raman...

1APS-Sherwood Texas 2006-April 21-24

Study of nonlinear kinetic effects in Stimulated Raman

Scattering using semi-Lagrangian Vlasov codesAlain Ghizzo1, P. Bertrand1, T.W. Johnston2,

M. Albrecht-Marc1,T. Reveillé1

1. LPMIA, CNRS-UMR7040, Université Henri Poincaré, Nancy, BP 239, F-54506 Vandoeuvre, France

2. I.N.R.S. Energie et Matériaux, Varennes, Québec

[email protected]


Topics

1. Vlasov plasmas

2. Vlasov codes and PIC codes

3. Application I: Resonant wave particle interaction

4. Application II: SRS-B in optical mixing

5. Conclusions


Topics

1. Vlasov plasmas




5. Conclusions


Introduction

. Vlasov models have long been used to study collisionless plasmas.

• Vlasov codes: powerful tool for studying in details the particle dynamics due to very fine resolution in phase space.

Questions for applications:• Need for a kinetic model?• PIC or Vlasov simulation?


Vlasov plasmas: collective effects

A dichotomy experiment: (e,m) -> 2(e/2, m/2) -> 4(e/4, m/4) -> etc…

= dimensionless parameter, - divided by 2 at each dichotomy- « graininess parameter »


Topics

1. Vlasov plasmas




5. Conclusions


Comparison PIC-Vlasov (1)

Vlasov Codes

: real space dimension

is the graininess due to particules

PIC Codes

: momentum space dimension

: sampling of momentumspace in each direction

Sampling the x-space needs

Real space X momentum space



Assume the same CPU time •to push a particle (PIC)•to move a phase space mesh point (Vlasov)

The ratio of the computationnal effort between Vlasov and PICdepends on•PIC graininess (must be as low as possible)•Sampling of momentum space (must be as high as possible)



Dv =1 Dv =2 Dv =3

gPIC =10-2 1 100 10 000

gPIC =10-4 0.01 1 100

gPIC =10-6 0.0001 0.01 1

Prefer PIC

Prefer Vlasov


Topics

1. Vlasov plasmas




5. Conclusions


Stimulated Raman Scattering

Using Coulomb gauge

with

Vacuum PLASMA

Scattered wave (1)

Plasma wave (2)

LASER

Pump wave (0)

Vlasov equation for electrons 1D momentum space


SRS : 3 mode coupling

Vacuum PLASMA

Scattered wave (1)

Plasma wave (2)

LASER

Pump wave (0)

Quasi particles (photons, plasmons)

Energy conservation

Momentum conservation

Electron plasma in a fixed ion homogeneous background

€

ω0k0( )=ω1k1( )+ω2k2( )

€

k0=k1+k2


Three mode coupling : a fluid description

Scalar potential (Plasma mode)

Multiple time-space scale expansion of fluid equations

€

A⊥=12εA0(x

1,t1)ei(k0x0−ω0t0)+12εA1(x

1,t1)ei(k1x0−ω1t0)+c.c.

€

Ex(x,t)=12εE2x

1,t1( )eik2x0−ω2t0( )+c.c.

€

∂∂t=∂∂t0+ε

∂∂t1+L and

∂∂x=∂∂x0+ε

∂∂x1

Vector potential (electromagnetic modes)



Hydrodynamic equations for electronsAssume slowly varying envelopes: i.e.

with

€

∂∂t+vg0

∂∂x

⎛ ⎝ ⎜ ⎞

⎠ ⎟a0=−Γa1a2∂∂t+vg1

∂∂x

⎛ ⎝ ⎜ ⎞

⎠ ⎟a1=Γa2*a0∂∂t+vg2

∂∂x

⎛ ⎝ ⎜ ⎞

⎠ ⎟a2=Γa0a1*

€

Γ=e2mkωp

2ε0ω0ω1ω2( )1/2



Envelope equations + periodic conditions Action conservation

Energy density of mode i

Action density of mode i

photon (0) photon (1) plasmon (2)

€

S0+S1=const=C1S0+S2=const=C2


Time evolution: pump + scattered

Pump wave action

Scattered wave action

Goodconservation

Check the fluid predictions against a fully kinetic Vlasov simulation


Time evolution : pump + plasma

pump

plasma

Poor Conservation !


Phase space portraits (1)

Color scale



Color scale


Accounting for « non fluid » particles

Good conservation

• Compute:Kinetic energy density above the lower separatrix:

• Divide by plasma wave frequency:


Topics

1. Vlasov plasmas




5. Conclusions


SRS-B in the « kinetic » regime (1)

• SRS-B reflectivity presents a bursting behavior

€

ω0=ωs+ωepw1 =2/3+1/3k0 = −ks+ kepw0.957=−0.60+1.559

• Nonlinear frequency shift - G.J. Morales and T.M. O’Neil, PRL 28, 417 (1972) -


SRS-B in the « kinetic » regime (2)

• Langmuir wave induced by SRS-B process

• Vortex-merging leading to weak turbulence

• BGK-like self-sustained structures (persisting over a long time)


Topics

1. Vlasov plasmas




5. Conclusions


Conclusions

Vlasov codes as compared to PIC codes

•lack of numerical noise •good resolution in phase space

provided the dimension of velocity space is as low as possible.

Kinetic effects in plasmas allow more phenomena than are found using only fluid theory with « ad hoc » kinetic damping.

0 APS-Sherwood Texas 2006-April 21-24 Study of nonlinear kinetic effects in Stimulated Raman...

Documents

Transcript of 0 APS-Sherwood Texas 2006-April 21-24 Study of nonlinear kinetic effects in Stimulated Raman...