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Non-neutral Plasma Shock
工
HU Xiwei（胡希伟）
HE Yong （何勇）
Hu Yemin（胡业民）
Huazhong University of Science and Techonology
2006.10.25
Zhejiang University, Hangzhou, China
1
Outline
 Introduction
 Motivation
 One
dimensional case
 Two
dimensional case (with magnetic
field produced by shock current)
 The
electrostatic instabilities on the
non-neutral shock front
2
Introduction
to
non-neutral plasma shock
3
Plasma shock

Plasma shock can arise:
when a fluid velocity is larger than
the ion sound velocity (without magnetic field)
or the Alfven velocity (with magnetic field) .

A steady shock (or shock front, shock
profile) is the result of a balance between
compressive and dissipative effects.
4
Front, Upstream and Downstream



Shock front (or shock
profile): a steady wavefront, propagating at
supersonic speed through
the undisturbed fluid.
Upstream⑴: the
unshocked (undisturbed)
fluid before the shock front
Downstream⑵: the
shocked fluid (disturbed)
behind the front
front
5
The neutral plasma shock

In most cases, we describe the plasma
shock with the single (neutral) fluid or
magnetohydrodynamic (MHD) equations.
So the plasma shock is neutral essentially.

In neutral plasma shock there may be
plasma current and magnetic field, but
there is not charge separation and
electrostatic field.
6
Electric field in the front

The ion fluid elements will
run ahead of the electron
fluid elements due to its
larger inertial.

This causes a charge
separation, then a
electrostatic field E and a
potential difference Δφ
between the up- and down
stream.
7
Effects of electric field in front (I)

The electric field E will hold back the ion
fluid elements and draw the electron fluid
elements. This effect is similar the role of
dissipation in the formation of steady shock
(shock front or profile).
So, the E could play very important role
in the formation of plasma shock profile,
especially the collisionless (non-dissipative)
shock profile.
8
Effects of electric field in front (II)

The potential difference Δφ will modify the
jump (discontinuity) relations – the so called
Hugoniot relations and the critical Mach
number Mc (the least velocity for shock
emergency) .

The electric field will drive a new kind of
electrostatic instability in the shock front as
the role of density gradient in the RayleighTaylor instability.
9
Motivation
In our simulation for the imploding shock in
a single sonoluminescing bubble, we
found that there is a extreme strong
electrostatic field in the shock front.
 The electric field is caused by the charge
separation.
 The charge separation is due to the
difference of electron fluid and various ion
fluid velocities.

10
Profiles of the charge number density Δn,
electric field E, and electric current density J
striding on the shock front at the moment near
the emergence of the maximum electric field.
11
12
Electric field in strong laser and in imploding shock
Plasma produced by
extreme-short pulse laser
Imploding shock front
in SBSL
E (V/m)
5×1011
2×1010
Δt (s)
10-12
10-11
Δr (m)
10-6
10-8
ne (cm-3)
1021
1022
Polarization
Electromagnetic
Electrostatic
13
One dimensional case
 Plasma
shock carries low current.
 The
effects of magnetic field
produced by the shock current
can be neglected.
14
Static coupled equations
in a reference frame on the front
The double fluid equations and Poisson equation
na u a  a
ma na u a
dua d
d  du 
 naTa     a" a   qae na E  Ra
dx dx
dx 
dx 
2
3
dT
du
 du  dq
na u a a  naTa a   a"  a   a  Qaes
2
dx
dx
dx
 dx 
dE
 4eZni  ne 
dx
where q a is the equivalent heat flow
qa 
qca q fa
qca q fa
,
qca   a
dTa
,
dx
q fa  
f
3/ 2
naTa ,
ma
f  0.01 ~ 0.1.
15
Profiles: ━ M1=1.6
┉ M1=1.5
16
Jump conditions across the shock
 1  2  1  0
2
Particle fluxes
2

T 
(mi i  me e )u  ( i  e ) u   0
 1

Momentum fluxes
2
(mi i  me e )u  5( i   e )T   2 J 0 Energy fluxes
1
2

Where
  n u ,
  1(Upstream),2( Downstream)
[ F ]12  F2  F1
17
Critical upstream Mach number M

min
1
When J 0  0, from u2  u1min we can obtain
the minimum upstream velocity for the
non-neutral shock emergency
M
min
1
u1min

 1 Z
C1s
Z
1
2
3
4
M1min
1.414
1.732
2
2.236
18
Two dimensional case

Cylindrical Non-Neutral Plasma Shock
with Current, Electric and magnetic Field.

The current and electric field is in axial (z)
direction.

The magnetic field produced by the shock
current is in poloidal (θ) direction.
19
Basic steady equations for shock front
  n u   0,
1
m n u  u  P      e n (E  u  B)  R  ,
c

5
 1


   m n u2  n T     u  q   e n u  E  u  R   Q .
2

 2

  E  4 e( Zni  ne ),
4
B 
J,
c
J  e  Zni ui  ne ue  .
20
Jump conditions of particle, momentum
in r and z direction, and energy fluxes
n u      const,
  H neTe 1   0
2
d

n   / u .
 Pi Pe 1
Te
 2u  
3  1 u
 J 0 B    ne
   
 2   dz  I mr ( r )
 
r  i ,e  r r
r  
 r r c
2
2
d



1

T
1

u

u 
2
2
1
e

   m n u  nT    neTe  2 Ez   0   H ne  r    r r  r 2  dz  I mz (r )
 i ,e


 i ,e
1

2


1 neTe e
1 niTi i
2
3 u
m
n
u

5
n
T
u




n
T
u

u


T


T

J

  

 e e e
i
H
e
H
i
0 
   
r
r
m
r
m


i
,
e
e
i

1
2


d
0
2
2


u  u  u  
1 neTe e Te
1 niTi i Ti 
1 u u


 
   
 dz  I e ( r )
  
2
r
r me r
r mi r 
 r  
 i ,e  r r
21
Profiles of velocity and temperature
22
Downstream profiles
━ 2D case, ┉ 1D case
23
Shock current changes the
M
min
1
24
Summary (I)
 The
jump conditions across the
shock will be changed that the
particle flux and energy flux are
no longer conservative.
 The profiles of plasma
parameters in downstream are no
longer as same as the profiles in
upstream.
25
Summary (II)
The critical Mach number ( the minimum
upstream velocity for the non-neutral
min
M
shock emergency) 1 is larger than its
value (1.0) in neutral shock.
M1min  1  Z
When J0=0,
 The positive shock current (from up- to
min
M
down-stream) will reduce the 1 , and
the negative current will increase the M 1min .

26
Electrostatic instabilities
in shock front
(1D case)
27
The linear instability analysis
n d (n u )

0
t
dx
(m n u ) d (m n u2 ) dp


 e n E  0
t
dx
dx
∂1
3
d 1
5
( m n u2  n T )  [( m n u2  n T )u ]  e n u E  0
t 2
2
dx 2
2
dE
 4 πe( Zni - ne )
dx
p  n T
A( r )  A0 ( x)  A1 ( r , t )
x

A1 ( r , t )  A1 exp i (t  k y y )  i  k x ( s )ds 


0
x -- the direction of the shock propagating,
y – the direction perpendicular to shock propagating.
28
A Example
Dispersion relation about kx
  k x ( x)   r  k x ( x)   i  k x ( x) 
Disturbance is in the direction of x
– the shock propagating direction
29
Profiles of equilibrium parameters
labeled x = 0.1, 0.3, 0.5, 0.7, 0.9
as
No.1, No.3, No.5, No.7, No.9
30
The numerical results

The high frequency (   pe ) approximation
vs. low frequency (   pi) approximation.

With electric field (E≠ 0 ) vs. E = 0.

With dissipation – viscous and friction -(α≠0 ) vs. α=0.

Adiabatic process approximation vs.
diabatic (energy equation) case.
31
The high frequency
approximation
   pe
32
   pe , E  0,  (dissipation)  0
33
  pe , E  0, (dissipation)  0
34
E  0,  0
vs.
E  0,  0
35
Adiabatic vs. Diabatic (E≠ 0)
36
Summary
The E can drive the electrostatic instability
in the shock front in both parallel (kx) and
perpendicular (ky) directions.
 The E and density gradient are the
destabilizing factors and dissipation (the
viscosity etc.) is the stabilizing factor.
6
 The increasing rate  /  p  10
 There is evidence of zero frequency instable
mode (k y )  0,  (k y )  0 , which is a kind of
absolute instability.

37
Prospect
The electrostatic instability in the shock
propagating direction is comparable with
the Rayleigh-Taylor instability in neutral
plasma shock.
 The electrostatic instability with zero
frequency in the perpendicular shock
direction is comparable with the
Richtmyer-Meshkov instability in neutral
plasma shock.
 These works are in process.

38
39