8 Jun 2012

Reduced Vlasov-Poisson model and it's instabilities

Denis Silantyev1 Harvey Rose2 Pavel Lushnikov1

1. Mathematics & Statistics Department, University of New Mexico2. Los Alamos National Laboratory

2

0gt

vEv

E zyx dddg vvv

t,,,z,y,x,gg zyx )vvv(

(1)

(2)

Units: electron charge and mass =1; length, λD ; time, 1/ωpe; electrostatic potential, Te/e, with Te the initial electron temperature.

Vlasov-Poisson collisionless plasma

- electron density distribution function

3

Framework

One preferred direction - laser propagation direction Laser intensity is high enough (near the instability threshold) →

sparse array of laser intensity speckles LW energy density is small compared to thermal

→ the most probable transverse electron speed is thermal

Effects to observe: electron trapping in LW electrostatic potential wells parallel to the

laser beam LW self-focusing ( sparse array of intense laser intensity speckles)

4

N

1i|| ),v,()),((v),,( tftδtg ii xxuvx

0v

Ev

)f(fxt iii

||||

|||| u

Euu iit

(0')

(1')

(2')ΦE ||

N

ii

N

ii df v

11

Reduced Vlasov-Poisson Model (Vlasov Multi-Dimensional Model)

iu

z

5

2D case. 2 streams

,fδδ,g z0xx0 )(vu)(vu)(v2

1)( vx 1uu i

1)(v 2 x1vv0 zx ddg

.

,e

)(f

z

z 2v

2

v

0

2

0vvvv zxxx dgd

1u

2u

n

θ

φ

4u

3u

2u

1u

v

n

,)(v)(N

1),(

N

1izi

00 fδxg uvv 2u iu

1vvv zyx0 dddg 1)( 2 nv0)( nv

Isotropy in transverse direction

3D case. N streams (uniformly distributed over angle φ)

6

2D Vlasov model. 2 streams. u1=u2=u=1

2D VMD model. 2 streams. u1=u2=u=1

2

2Φ2

Φ

2 11k

uvuv2

where

При

vv

dg

20

)(1

k

)(Z)(Zcosk 224cosθ

sinθζ Φ

2

vu

)(Z)(Z

tan)(Z)(Zk 224

θ

kz

x1u

2u

Dispersion relation (2D)

-well-known cold plasma two stream dispersion relation

dtt

eZ

t

2

1)( - plasma dispersion function

k

ωΦ v - phase velocity

10

Unstable

Stable

2D case. 2 transverse streams

11

2

u2i

ei

N

ii

1

1

N

iii

1

2 1usuch that

v

ρ

i=1

i=2

i=N

0u1 u2 uN

2D case. N transverse streams

12

N

iii )(Zcosk

1

222 Vlasov:

VMD:

N

i i

iii

)(Ztan)(Zk

1

222

cos

sinii

2

vu dt

t

eZ

t

2

1)(

2D case. N→ transverse streams

130k),(

)Im( .,N

C~

13

N

i i

ii

)(Ztan)(ZNk

1

222

2

N

1i2

0iΦ

2

Nπi2

cos

1Nk

)(v uWhen

vv

dg

20

)(1

k

θ

φ

kz

y

k┴x0iuN

i2

20iu

20iu

where

N

iiZcosNk

1

22 )(2 cosθ

Nπi2

cossinθζ

Φ0i

i2

v)(

u

Dispersion relation (3D)3D Vlasov model. N streams.

3D VMD model. N streams.

14

Unstable

Stable

15

16

20

Unstable

Stable

21

Measure of

anisotropy

22

23

24

θ1 and θ2 match up to 16 digits for N≥24

N21 5.942

76.708θθ

Convergence of anisotropy in 3D as N→

Llinear fit:

25

Envelope curve of different cross-sections w.r.t. φ N=4,6,8,10,12

28

N

Cmax

π

k),()Im(

][0,2

30

k=0.3θ→0

Anisotropy (in φ) of Langmuir branch with θ→0

)(ωRe

31

32

33

34

35

k=0.3θ→0

)(ωRe

Isotropy (in φ) of Langmuir branch with θ=0

36

Langmuir branch with θ=0. N=2,4,6,8

37

)(ωRe

Anisotropy (in φ) of Langmuir branch with θ=0.3

38

Langmuir branch with θ=0.3 N=2

39

Langmuir branch with θ=0.3 N=4

40

Langmuir branch with θ=0.3 N=6

41

Langmuir branch with θ=0.3 N=8

42

Anisotropy (in φ) of Langmuir branch with θ=0.5

43

Langmuir branch with θ=0.5 N=2

44

Langmuir branch with θ=0.5 N=4

45

Langmuir branch with θ=0.5 N=6

~1%

46

Langmuir branch with θ=0.5 N=8

47

Anisotropy (in φ) of Langmuir branch with θ=1

48

Langmuir branch with θ=1 N=2

49

Langmuir branch with θ=1 N=4

50

Langmuir branch with θ=1 N=6

51

Langmuir branch with θ=1 N=8

52

Conclusions

VMD model can be used for 3D simulations in regimes when plasma waves are confined to a narrow (θ<0.6) cone with quite good precision with only 6 or 8 transverse streams

Using VMD model we can drastically reduce necessary computing power since we only have to compute 6 or 8 equations that are effectively 4D instead of one 6D equation.

Thank you!

8 Jun 2012

Denis Silantyev1 Harvey Rose2 Pavel Lushnikov1

1. Mathematics & Statistics Department, University of New Mexico2. Los Alamos National Laboratory