Reinisch_85.511_MHD1 Ch 4 Magnetohydrodynamics 2.1 Two-fluid plasma.
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Transcript of Reinisch_85.511_MHD1 Ch 4 Magnetohydrodynamics 2.1 Two-fluid plasma.
Reinisch_85.511_MHD 1
Ch 4 Magnetohydrodynamics2.1 Two-fluid plasma
We now use the macroscropic variables defined in section 2 to analysze a "fluid" consisting of two species: electrons and singly charged ions. The variables are , , and ( , for electrons and i
s s sn S s e iu
ons). (Actualy there is often a third species: neutrals)In Ch 2 the continuum an momentum equations where drerivedas the 0th and 1st moments of the Bolttzmann eqution (2.6).
continuitss s s
n n St
u
3s
y equations (see 2.40) 4.1, 2
f (the net production rate per unit volume) = ts
coll
S d v
Reinisch_85.511_MHD 2
The momentum equation was given in 2.61 :
where . These equations are too complicated to solve, andwe must simplify! First we assume the press
s sss s s s s s s s
coll
s s s
m St t
n m
u Mu u P a u
2
ure tensors are isotropic,
i.e., . Then we simplify . From 2.64 :
and neglecting the viscosity term :
ss s
coll
sst s s t s s s s n n s s s
t scoll
eet e e
coll
pt
Pm L mt
t
MP
M u u u u u
M u
; ,
; ,
t e n n e e et e
iit i i t i n n i i i
t ecoll
Pm L m t i n
Pm Lm t e nt
u u u
M u u u u
Reinisch_85.511_MHD 3
From 2.60 we get the average accelaration for ions and electrons :
i ii
e ee
eameam
E u B g
E u B g
Reinisch_85.511_MHD 4
The momentum equations now become:
; but
4.3
e e e e e e e e e et e e e tt e
e n n e e e e e e e
e e e e e et e e e t e n n e et e
i i i i i
n m p en n m n mt
Pm m L S L S P
p en n m n m P m m
nm pt
u u E u B g u u
u u
E u B g u u u u
u u
4
We assume sinusoidal wave-like distur
.4
Further approximations are nban
ecessary to solve the Dc
Es.e
i i i i it i i i tt e
i n n i i i i
i i i i i it i i i t i n n i it e
en n m n m
Pm m L S
p en n m n m P m m
E u B g u u
u u
E u B g u u u u
s.
Reinisch_85.511_MHD 5
4.2.2 Langmuir Plasma OscillationsWe start by looking at a "high frequency" disturbance in an unmagnetized (B = 0) plasma. This means the motions are too fast for the ions, so that we can consider the ions to just sit still. We also a
i e
ssume a "cold" plasma, i.e., T = T = 0; then 0, since . We also assume there is no electron production, i.e., 0,and no collisions. And we neglect gravity. Then:
0 4.16
i e s s B s
e
ee e
p p p n k TP
n nt
n
u
0 0 0
perturbation
4.17
To solve these simplified PDEs we use the . The field variables without the disturbance (background)
methodare
, , .
e e e e e
e e
m ent
n and
u u E
u E
Reinisch_85.511_MHD 6
0 1
0 1
0 1
When a disturbance occurs:, ( ) ,
, ( ) , 4.19
, ( ) ,
where the 1 variables are small deviations from the background values. Let's also assume now that the background
e e e
e e e
n t n n t
t t
t t
x x x
u x u x u x
E x E x E x
0 0
field variables for E=0 areuniform and neutral: . e in n n
0 00 0 0 0 0 0
0 0 0 0 0
0
For the background:
0 0 0
0 0.
0 is a solution.
ee e e e
e e e e e e
e
n nn n nt t
n mt
u u u
u u u u
u
Reinisch_85.511_MHD 7
0 1 1 1
Substitute (4.19) into continuity 4.16 and momentum 4,17 equation
0 4.16
4.17
; ;
ee e
e e e e e
e e e e
n nt
n m ent
n n n
u
u u E
u u E E
Reinisch_85.511_MHD 8
0 10 1 1
10 1 1
1 10 1 1 1 1 0 1
10 1
10 1 0 1 1 1 1
1
Continuity eqation:
0,
0
0. 4.24
Momentum equation:
; since 0
ee e
ee e
e ee e e e e
ee
ee e e e e
ee
n nn n
tn
n nt
n nn n n nt tn nt
n n m e n nt
m
u
u
u u u
u
u E u u
u 1 4.22
All second-order 1 terms have been neglected (linearization).
et
E
Reinisch_85.511_MHD 9
0 1
11 0 0 1 0
0 0 0
1 1
We must make sure that the solution for the density satisfies Maxwell's equations. Gauss's law:
. (Recall )
Assume a plane wave solution :( , ) e
e e e
c ee i
e e
n n n
ene n n n n n
n x t n
E
1 1
1 1
1 0 1
1 1
xp
( , ) exp
, exp
Note: , . Apply to 4.22 and 4.24 :
0
e e
e
e e
e e e
i kx t
x t i kx t
x t i kx t
i it
i n n i
i m e
u u
E E
k
k u
u E
Reinisch_85.511_MHD 10
1 1
1 11 1 1
0 0
2
1 0 10
22 0
0
20
0
Eliminate and :
, and
0
for Langmuir waveDispersion relati s in cold plasma.
de
on
finiti
e e
e ee e e
e e
e ee
e
pee
i ien ene ei i i
m m
ei n n i nm
e nm
e nm
k u k E
k E k u k E
20
0
electron plasma frequency.
12
See Figure 4
on of the
.1
pee
e nm
f
Reinisch_85.511_MHD 11
pe
For true dispersion relations, . But for Langmuir waves,
does not depend on k. This means that in a cold plasmathere is , but just an oscillation with the plasma frequency .
(The group
k
no wave
g
Discussion of Example
velocity is zero, i.
1
e., v 0.)k
Reinisch_85.511_MHD 12
2 2 2
2
2 2 2 2
We will show later that inn a warm plasma, i.e., 0 :
3 Dispersion relation for Langmuir waves,
One defines the electron thermal spe
Fig
ed as v . Then
3 v
Phase
. 4.2
e
B epe
e
B ete
e
pe te
T
k Tkm
k Tm
k
2 2
22
1velocity 3 v for small k.
3vGroup velocity 3v
peph pe te
teg te
ph
v kk k k
kvk v
Reinisch_85.511_MHD 13
6.2 Plasma Dynamics(1)(see Ionospheres (Cambridge), by Schunk and Nagy)
We now discuss a more complete treatment with .The propagation of waves in a plasma is governed by Maxwell’s equations and the transport equations.
( ) 0 continuity eq. 6.21
[ ] [ ] 0 momentum eq. 6.22
. (polytropic energy relation, see (2.81) 6.26
Notice that this is the of a gas. T
ss s
ss s s s s s s s
s
s
n nt
n m p n et
p const b
u
u u u E u B
equation of state
-1e s
he value for is =3/5 for adiabatic flow, and =1 for isothermal flow. For an electron
gas, the best value to use is =3. Since V ,
we can also write . sp V const
00, 0sT B
Reinisch_85.511_MHD 14
6.2 Plasma Dynamics (2)
1 1
From 6.26 :
6.27
Substitute in the momentum equation (6.22):
[ ] [ ] 0 6.28
The continuity equation
s s s ss s s s s s s
s s s s
ss s
s
ss s s s s s s s s s
b
p p n kTp constn m
kTpm
n m kT n n et
u u u E u B
s s
was
( ) 0 6.21
We must solve these equations together with Maxwell's equations to findn , , and (10 unknowns).
ss s
n nt
u
u E B
Reinisch_85.511_MHD 15
6.2 Plasma Dynamics (3)
0 0 0 0
:1. Assume n , , , ( the index s is dropped for convenience) satisfy the differential equations for equilibrium conditions.2. Perturb the equilibrium state of the pl
Using Perturbation Techniqueu B E
0 0
0 1
0 1
0 1
0 1
asma and assume that this will cause small changes in and (linearization).
, , 6.31
, , 6.31
, , 6.31
, , 6.31
n t n n t a
t t b
t t c
t t d
B Er r
u r u u r
E r E E r
B r B B rAssume allconst and uniform
Reinisch_85.511_MHD 16
6.2 Plasma Dynamics (4)
0 10 1 0 1
0 10 1 0 1 0 1 0 1
0 1 0 1 0 1 0 1
Substitute perturbed functions into the continuity and momentum equations:
6.21 ( ) 0
6.28 [ ]
[ ] 0
s s s
s
n nn n
t
n n m kT n nt
n n e
u u
u uu u u u
E E u u B B
10 0 0 1 1 0 0 0 1 1 0
0 0 0 1 0 0 0 0 1 0 1 0
s
The momentum equation becomes
0. 6.34
where e = e for ions/electrons.
s s s s s
s s s s
n m n m kT n n e n e n et
n e n e n e n e
u u u E E E
u B u B u B u B
1 10 1 1 1 0 1 0 1 0 1
Carry out differentiations noting that all o-index terms are constants:
0 6.33
where only first-order terms in 1-index functions were kept.
n nn n n n nt t
u u u u u
Reinisch_85.511_MHD 17
6.2 Plasma Dynamics (4a)
0 0 0 0
10 0 1 1 0 1 1 0 0 1
1 0 0 0
10 0 1 1 0 1 1 0
But0 (equilibrium condition), and (6.34) becomes
0.
Again, the last 0, therefore:
s
s s s
s
s s s
n e
n m kT n n et
n e
n m kT n n et
E u B
u u u E u B u B
E u B
u u u E u B
0 1
1 1 1 1
1 0 1 0 1
0
(
1 0
)
1
0 6.35
We try solutions for all functions :
, , , . Remember , . Then 6.33 :t
0, or: (Schunk uses instead of
pla
)
6.37
ne wave
i tn i i
i n n i i n
n n
e
K r
u B
u E B K
K u u K K k
u K K u
Reinisch_85.511_MHD 18
6.2 Plasma Dynamics (5)
0 1 0 1 1 0 1 1 0 0 1
10 1 1 1 0 0 1
0
And (6.35):
0
0 6.38
6.37 and 6.38 are that must be satisfied for
the plane waves to be a s
s s s
s s s
n m i i kT i n n e
kT n ei in m m
u u K u K E u B u B
u K u K E u B u B
4 algebraic equationsolution.
Reinisch_85.511_MHD 19
121 1 0 0
1
11
711 0 1 0 0 0
1 s 1 1 1 s 1s s
The perturbations must satisfy Maxwell's equations
1 , 8.85 10 ,permittivity
2 0
3
4 , 4 10 ,permeability.
where e ; e . C
c
s s c s
x SI units
t
x SI unitst
n u n
E
BBE
EB J
J ombining (3) and (4):
Reinisch_85.511_MHD 20
111
22 1 1
1 1 0 0 0 2
2 21 1 0 0 1 0 1 0 0 2
22
1 1 0 12
Apply to 3 :
6.6
1;
. 6.20
t t
t t
i i i i ic
K ic
BBE
J EE E
K E K K E E J
E K K E J 3 more algebraic equations
Reinisch_85.511_MHD 21
1 1 0 1 s 1 1 s 1s 0
1 1 1
Check other Maxwell equations:11 , e e
One more algebraic eq.
2 0 0. This eq. only tells that always .
c c s ss
n i n
E K E
B K B B K
Reinisch_85.511_MHD 22
Electrostatic Waves: B1= 06.3 Electron Plasma Waves (Langmuir waves)
1
i1
We start the discussion by looking for high frequency electron plasma wave
solutions for which B 0.The wave frequency is high enough so that the ions cannot follow the motion, i.e., 0.To simplify
u
i0 e0 0
0 1 0 1
10 1 1 1 0 0 1
0
1
the discussion here we now assume 0, and 0. Then the algebraic transport equaelec tions 6.37 and 6.38 become
0.
With
t n
:
o
-
ro
s s s
s
e
n nkT n ei in m m
e e
n
u u E B
u K K u
u K u K E u B u B
10 1 1 1
0
1 1 0
1 1 0
0 6.39 ,
From Gauss's law :i / 6.39
e e ee e e
e e e
c
e
kT n en i i a bn m m
en c
K u u K E
EK E
Reinisch_85.511_MHD 23
1 1
11 1
0
1
Our immediate goal is to find the dispersion relation that relates K and .
Muliply 6.39 with and use 6.39 and 6.39 to substitute
for and :
0 6.40
e
e e ee
e e e
e
b a b
kT n ei in m m
i nn
KK u K E
K u K K K E
2 11 0
0 0
22 2 0
10
/ 0
0 6.41
e e ee
e e e e
s s ee
e e
kT n eiK enn m im
kT e nn Km m
Reinisch_85.511_MHD 24
22 2 0
0
22 20
0
2 2 2 2e
20
0
This gives the dispersion relation
0, or
, or
usually is set equal to 3. 6.42
plasma frequency; electron thermal
e e e
e e
e e e
e e
p e te
e ep te
e e
kT e nKm m
e n kT Km m
V K
e n kTVm m
2 2
speed.
The dispersion relation 6.42 relates with the wavelength (=2 /K).
Notice there is no propagating wave in a cold plasma where 0. In the cold plasma
plasma oscillation 6.45
e
p
T