Transcript Ch 4 Hydrodynamics - University of Massachusetts Lowell

Ch 4 Magnetohydrodynamics
2.1 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 ns , u s , and S s (s  e, i for electrons
and ions). (Actualy there is often a third species: neutrals)
In Ch 2 the continuum an momentum equations where drerived
as the 0th and 1st moments of the Bolttzmann eqution (2.6).
ns
    ns u s   S s continuity equations (see 2.40)
t
  fs 
3
S s (the net production rate per unit volume) =  
d
v

  t coll
Reinisch_85.511_MHD
 4.1, 2 
1
The momentum equation was given in  2.61 :
 u s

  Ms 
s 
  u s   u s     P s   s a s  
  ms u s S s
 t

  t coll
where  s  ns ms . These equations are too complicated to solve, and
we must simplify! First we assume the pressure tensors are isotropic,
  Ms 
i.e.,   P s  ps . Then we simplify 
 . From  2.64  :
  t coll
  Ms 
2




u

u




u s  Ps mn u n  Ls ms u s



st s
s
t
s s


  t coll
ts
and neglecting the viscosity term :
  Me 

   et e  u e  ut   Pe mnu n  Le meu e ; t  i, n
  t coll
t e
  Mi 
t  e, n
i n u n  Li mi u i ;

   it i  ui  ut   Pm
  t coll
t e
Reinisch_85.511_MHD
2
From  2.60  we get the average accelaration for ions and electrons :
ai 
e
 E  ui  B   g
mi
e
ae    E  u e  B   g
me
Reinisch_85.511_MHD
3
The momentum equations now become:


ne me    u e    u e  pe  ene  E  u e  B   ne me g   et ne me  u e  ut 
 t

t e
 Pe mnu n  meu e  Le  Se  ; but Le  Se  Pe
 pe  ene  E  u e  B   ne me g   et ne me  u e  ut   Pe  mnu n  meu e 
t e
 4.3


ni mi    ui    ui  pi  eni  E  ui  B   ni mi g   it ni mi  ui  ut 
 t

t e
 Pm
i n u n  mi u i  Li  Si 
 pi  eni  E  ui  B   ni mi g   it ni mi  ui  ut   Pi  mnu n  mi ui 
t e
 4.4 
Further approximations are necessary to solve the DEs. We assume
sinusoidal wave-like disturbances.
Reinisch_85.511_MHD
4
4.2.2 Langmuir Plasma Oscillations
We 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 assume a "cold" plasma,
i.e., Ti = Te = 0; then pi  pe  0, since ps  ns k BTs . We also assume there
is no electron production, i.e., Pe  0, and no collisions.
And we neglect gravity. Then:
ne
   neu e  0
 4.16 
t


ne me    u e    u e  ene E
 4.17 
 t

To solve these simplified PDEs we use the perturbation method. The
field variables without the disturbance (background) are
ne 0 , u e 0 , and E0 .
Reinisch_85.511_MHD
5
When a disturbance occurs:
ne  x, t   ne 0 (x)  ne1  x, t 
u e  x, t   u e 0 (x)  u e1  x, t 
 4.19 
E  x, t   E0 (x)  E1  x, t 
where the 1 variables are small deviations from the background values.
Let's also assume now that the background field variables for E=0 are
uniform and neutral: ne 0  ni  n0 .
For the background:
ne 0
n0
   ne 0ue 0  0 
   n0u e 0  0  n0  u e 0  0
t
t


ne 0 me    ue 0    ue 0  0   u e 0   u e 0  0.
 t

u e 0  0 is a solution.
Reinisch_85.511_MHD
6
Substitute (4.19) into continuity  4.16  and momentum  4,17  equation
ne
   neu e  0
t


ne me    u e    u e  ene E
 t

ne  n0  ne1 ; u e  u e1 ; E  E1
Reinisch_85.511_MHD
 4.16 
 4.17 
7
Continuity eqation:
  n0  ne1 
    n0  ne1  u e1  0,
t
  ne1 
    n0  ne1  u e1   0
t
ne1
ne1
  n0  ne1    u e1  u e1ne1 
 n0  u e1
t
t
n
 e1  n0  u e1  0.
t
Momentum equation:
 4.24 
u e1
 e  n0  ne1  E1 ; since  u e1   u e1  0
 n0  ne1  me
t
u e1
 me
 eE1
 4.22 
t
All second-order 1 terms have been neglected (linearization).
Reinisch_85.511_MHD
8
We must make sure that the solution for the density ne  ne 0  ne1
satisfies Maxwell's equations. Gauss's law:
c e
ene1
  E1 
  n0   n0  ne1    
. (Recall ni  n0 )
0 0
0
Assume a plane wave solution :
ne1 ( x, t )  ne1 exp i  kx  t 
u e1 ( x, t )  u e1 exp i  kx  t 
E1  x, t   Ee1 exp i  kx  t 

Note:
 i ,   ik. Apply to  4.22 and 4.24  :
t
i ne1  n0ik  u e1  0
i meu e1  eEe1
Reinisch_85.511_MHD
9
Eliminate  ik  u e1 and ik  Ee1 :
ene1
e
e ene1
ik  Ee1  
, and ik  u e1 
k  Ee1  i
0
 me
 me  0
e2
i ne1  n0i
ne1  0
 me 0
e 2 n0
 
 0 me
2
Dispersion relation for Langmuir waves in cold plasma.
e 2 n0
 pe 
definition of the electron plasma frequency.
 0 me
f pe
1

2
e 2 n0
 0 me
See Figure 4.1
Reinisch_85.511_MHD
10
For true dispersion relations,     k  . But for Langmuir waves,
 does not depend on k. This means that in a cold plasmathere
is no wave, but just an oscillation with the plasma frequency pe .

(The group velocity is zero, i.e., v g 
 0.)
k
Discussion of Example 1
Reinisch_85.511_MHD
11
We will show later that inn a warm plasma, i.e., Te  0 :
k BTe
    3k
Dispersion relation for Langmuir waves, Fig. 4.2
me
2
2
pe
2
k BTe
One defines the electron thermal speed as v 
. Then
me
2
te
2
 2   pe
 3k 2 vte2
 pe
1
2
2
Phase velocity  v ph  
 pe  3kvte 
for small k.
k k
k
 k 2 3vte2
Group velocity  vg 
 3vte 
k 
v ph

Reinisch_85.511_MHD
12
6.2 Plasma Dynamics(1)
(see Ionospheres (Cambridge), by Schunk and Nagy)
We now discuss a more complete treatment with Ts  0, B0  0.
The propagation of waves in a plasma is governed by Maxwell’s equations
and the transport equations.
ns
   (ns u s )  0 continuity eq.
 6.21
t
u s
ns ms [
  u s   u s ]  ps  ns es [E  u s  B]  0 momentum eq.  6.22 
t
ps
 const. (polytropic energy relation, see (2.81)
 6.26b 

s
Notice that this is the equation of state of a gas. The value for  is
 =3/5 for adiabatic flow, and  =1 for isothermal flow. For an electron
gas, the best value to use is  e =3. Since s  V -1 ,
we can also write psV   const .
Reinisch_85.511_MHD
13
From  6.26b  :
6.2 Plasma Dynamics (2)
ps    const   s 1 s   
 ps 
 kTs
ms
 s
ps
 s
  s 1 s 
 ps
 n kT
 s  s s  s
s
ns ms
 6.27 
Substitute in the momentum equation (6.22):
u s
ns ms [
  u s   u s ]   s kTs ns  ns es [E  u s  B]  0
t
The continuity equation was
 6.28
ns
   (ns u s )  0
 6.21
t
We must solve these equations together with Maxwell's equations to find
n s , us , E and B (10 unknowns).
Reinisch_85.511_MHD
14
6.2 Plasma Dynamics (3)
Using Perturbation Technique :
1. Assume n 0 , u 0 , B 0 , E0 ( the index s is dropped for convenience)
satisfy the differential equations for equilibrium conditions.
2. Perturb the equilibrium state of the plasma and assume that this will
cause small changes in B 0 and E0 (linearization).
n  r, t   n0  n1  r, t 
 6.31a 
 6.31b 
 6.31c 
 6.31d 
u  r, t   u 0  u1  r, t 
E  r, t   E0  E1  r, t 
B  r, t   B 0  B1  r, t 
Assume all
const and uniform
Reinisch_85.511_MHD
15
6.2 Plasma Dynamics (4)
Substitute perturbed functions into the continuity and momentum equations:
 6.21 
  n0  n1 
t
   ( n0  n1 u 0  u1 )  0
 6.28    n0  n1  ms [
 u 0  u1 
 u 0  u1    u 0  u1 ]   s kTs  n0  n1 
t
  n0  n1  es [ E0  E1   u 0  u1    B 0  B1 ]  0
Carry out differentiations noting that all o-index terms are constants:
n1
n
 n0  u1  n1  u1  u0n1  1  n0  u1  u0 n1  0  6.33
t
t
where only first-order terms in 1-index functions were kept.
The momentum equation becomes
u1
n0 m
 n0 m  u0   u1   s kTs n1  n0es E0  n0es E1  n1es E0
t
n0es u0  B0  n1es u 0  B0  n0esu 0  B1  n0 esu1  B0  0.
 6.34 
where es =  e for ions/electrons.Reinisch_85.511_MHD
16
But
6.2 Plasma Dynamics (4a)
n0 es  E0  u 0  B 0   0 (equilibrium condition), and (6.34) becomes
 u1

n0 m 
  u 0   u1    s kTs n1  n0 es  E1  u1  B 0  u 0  B1 
 t

 n1es  E0  u 0  B 0   0.
Again, the last 
  0, therefore:
 u1

n0 m 
  u 0   u1    s kTs n1  n0 es  E1  u1  B 0  u 0  B1   0  6.35 
 t

We try plane wave solutions for all functions :

n1 , u1 , E1 , B1  e
. Remember   iK ,
 i . Then  6.33 :
t
i n1  n0iK  u1  u 0  iKn1  0, or:
(Schunk uses K instead of k )
i ( K r  t )
  u0  K  n1  n0K  u1
Reinisch_85.511_MHD
 6.37 
17
6.2 Plasma Dynamics (5)
And (6.35):
n0 m  i u1   u 0  iK  u1    s kTs iKn1  n0es  E1  u1  B 0  u 0  B1   0
 s kTs n1
es
i   u 0  K  u1  iK
  E1  u1  B 0  u 0  B1   0
n0 m
m
 6.38
 6.37  and  6.38 are 4 algebraic equations that must be satisfied for
the plane waves to be a solution.
Reinisch_85.511_MHD
18
The perturbations must satisfy Maxwell's equations
1   E1  1c  0 ,
 2    B1  0
 3   E1  
 0  8.85 x1012 SI units, permittivity
B1
t
 4    B1  0 J1   0 0
where
E1
, 0  4 x107 SI units, permeability.
t
J1   es ns1us1 ; 1c 
s
e n
s s1
. Combining (3) and (4):
s
Reinisch_85.511_MHD
19
Apply   to  3 :
    B1 
 B1 
    E1     

t
 t 
2

J

E
 2 E1      E1    0 1   0 0 2 1
t
t
 6.6 
1
 iK  E1  iK  iK  E1    0 0  i  E1  i0 J1 ;  0 0  2
c
 2
2
 2  K  E1  K  K  E1   i 0 J1 3 more algebraic equations.  6.20 
c

2
2
Reinisch_85.511_MHD
20
Check other Maxwell equations:
1
1   E1  1c  0 , 1c   es ns1  iK  E1   es ns1

s
0
s
One more algebraic eq.
 2    B1  0  K  B1  0. This eq. only tells that always B1  K.
Reinisch_85.511_MHD
21
Electrostatic Waves: B1= 0
6.3 Electron Plasma Waves (Langmuir waves)
We start the discussion by looking for high frequency electron plasma wave
solutions for which B1  0.The wave frequency is high enough so that the
ions cannot follow the motion, i.e., ui1  0.
To simplify the discussion here we now assume ui0  u e0  0, and Eo  B 0  0.
Then the algebraic electron transport equations  6.37  and  6.38  become
  u0  K  n1  n0K  u1
 s kTs n1
es
i   u 0  K  u1  iK
  E1  u1  B 0  u 0  B1   0.
n0 m
m
With es  -e :
 ne1  ne 0 K  u e1
iu e1  iK
 e kTe ne1
ne 0 me
e
 E1  0
me
From Gauss's law   E1  1c  0 :
iK  E1  ene1 /  0
Reinisch_85.511_MHD
 6.39a, b 
 6.39c 
22
Our immediate goal is to find the dispersion relation that relates K and  .
Muliply  6.39b  with K  and use  6.39 a  and  6.39b  to substitute
for K  u e1 and K  E1 :
i K  u e1  iK  K
 e kTe ne1
ne 0 me
e
K  E1  0

me
 6.40 
 kT n
i ne1
e
 iK 2 e e e1 
 ene1 /  0   0
ime
ne 0 me
ne 0
2
 2
ne 0 
e
kT

2 s
s

ne1    K
0
me 0 
me

Reinisch_85.511_MHD
 6.41
23
This gives the dispersion relation
 2  K
2  e kTe
me
e 2 ne 0

 0, or
me 0
e 2 ne 0  e kTe 2
 

K , or
me 0
me
2
 2   p2   eVte2 K 2 usually  e is set equal to 3.
e 2 ne 0
p 
me 0
kTe
plasma frequency; Vte 
me
 6.42 
electron thermal speed.
The dispersion relation  6.42  relates  with the wavelength  (=2 /K).
Notice there is no propagating wave in a cold plasma where Te  0.
In the cold plasma
 2   p2
plasma oscillation
Reinisch_85.511_MHD
 6.45 
24