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


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


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


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.


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


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


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


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


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


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


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


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


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


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



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



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


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



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.


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):


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


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


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


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


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

reinisch_85.511_mhd1 ch 4 magnetohydrodynamics 2.1 two-fluid plasma

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