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Reduced Vlasov-Poisson model and it's instabilities. Denis Silantyev 1 Harvey Rose 2 Pavel Lushnikov 1 Mathematics & Statistics Department, University of New Mexico Los Alamos National Laboratory. 8 Jun 2012. Vlasov-Poisson collisionless plasma. - electron density distribution function. - PowerPoint PPT Presentation
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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