6. Wave Phenomena6.1 General Wave Properties(1)

Following Schunk’s notation, we use index 1 to indicate the electric and magnetic wave fields, E1 and B1, and the plasma charge variations, 1c, caused by the waves.The direction of the propagating wave is given by the propagation constant K.

To find the plasma waves we must solve Maxwell’s differential equations in the plasma environment.

121 1 0 0

1

11

711 0 1 0 0 0

1 s 1 1 1 s 1s s

1 , 8.85 10 ,

2 0

3

4 , 4 10 , .

where e ; e

c

s s c s

x SI units permittivity

xt

x x SI units permeabilityt

n u n

E

B

BE

EB J

J

6.1 General Wave Properties(2)We solve Maxwell’s equations by taking the curl of (3):

1 1

21 1 1

21 1

1 0 0 0 2

22 1 1

1 1 0 0 02

c1 1

But

and using(4): - results in

6.6

This quation can be easily solved for vacuum where = 0, and = 0. In th

xt

x

t t t

t t

E B

E E E

J EB

E JE E

J

c11 1 c1

0

22 1

1 0 0 2

0 0 20 0

22 1

1 2 2

is case

0 since according to 1 , and = 0. Then

0

1 1or by setting , . .,

10 wave equation in vacuum. 6.7

t

i e cc

c t

E E

EE

EE

6.1 General Wave Properties(3)

1 10 10

10

11 1

: , Re cos

where for simplicity I assumed is real. Recall: cos sin .

From Mawell equation 3 we can get .

Notice that when the exponential functions a

i t

ix

Solution t e t

e x i x

xt

K rE r E E K r

E

BE B

1 1 1 1

1 1

re used, application of the

operators and simply means multiplication with i and -i ,

respectively. Therefore: -i

This means that is to and to . With the help of Maxw

t

i

K

K E B K E B

B K E

1 1 0

1

1 1

ell equation

1

it is easy to show that for 0 the electric field is to :

0 . Note: only true for vacuum or isotropic medium.

c

c

i

E

K

K E K E

6.1 General Wave Properties(4)

1 1 1 1

21 1 1 1 1

1 1

Further from

: In vacuum, , , are orthogonal to each other following

the right-hand rule.

:

The flow of energy car

K,

r

or .KK K

Summary E B K

K E B K K E K B

KK K E E K B E B k

Poynting Vector

E B

1 1 0 1 1

1 1

ied by an electromagnetic wave in the direction

is given by the Poynting vector: / 6.13

Since are sinusoidal time varying functions, is a function of t.

In general we are not in

K

S E B E H

E H S

1 1

0

terested in the fast in and out energy fluxes, but

want to know the time-averaged flux:

1 2dt where is the wave period. It is easy to

verify that when using the exponential notation:

wT

ww

TT

S E H

S 1

*1

1Re time-avearged Poynting vector 6.14

2 E H

BE

Sk

=0 in vacuum

6.2 Plasma Dynamics(1)

The propagation of waves in a plasma is governed by Maxwell’s equations and the transport equations. We assume that the 5-moment simplified continuity, momentum, and energy equations (5.22a-c) can describe the plasma dynamics in the presence of waves. If we neglect gravity and collisions these equations become (Euler equations):

( ) 0 continuity eq. 6.21

[ ] [ ] 0 momentum eq. 6.22

0 energy eq. 6.23

1 16.21 0

Substitute in

ss s

ss s s s s s s s

ps ss s

v

s s s ss s s s s s s

s s s

nn

t

n m p n et

cD pp

Dt c

n n D nn n n

t n t n n Dt

u

uu u E u B

u

u u u u

6.23 :

0 6.25s s s s s

s

D p p D n

Dt n Dt

6.2 Plasma Dynamics (2)

1

This implies that

0, since 6.26a

1 1

1 10

. ., 0, same

s s

s

s s s s s s s s s ss s

s s s s s

s s s s s s s ss s s

s s s s s

s s s ss

s

D p

Dt

D p D p p D D p p D

Dt Dt Dt Dt Dt

D p p D D p p Dn m n

Dt n m Dt Dt n Dt

D p p Di e n

Dt n Dt

-1s

as 6.25 . And 6.26a implies that

. 6.26

Notice that this is the of a gas. The value for is

=3/5 for adiabatic flow, and =1 for isothermal flow. Since V ,

we can also w

s

s

pconst b

equation of state

rite

. sp V const

6.2 Plasma Dynamics (3)

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 const

n m

kTp

m

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 find

n , , and (10 unknowns).

ss s

nn

t

u

u E B

6.2 Plasma Dynamics (3a)

0 0 0 0

:

1. Solve for equilibrium conditions finding n , , , ( I dropped

index s on n and u) that satisfy the differential equations.

2. Perturb the equilibrium state of the p

Using Perturbation Technique

u B E

0 0

0 1

0 1

0 1

0 1

lasma 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 E

r r

u r u u r

E r E E r

B r B B r

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

Carry out differenti

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

1 10 1 1 1 0 1 0 1 0 1

10

ations noting that all 0-index terms are constants:

0 6.33

where only first-order terms in 1-index functions were kept.

The momentum equation becomes

n nn n n n n

t t

n mt

u u u u u

u

0 0 1 1 0 0 0 1 1 0

0 0 0 1 0 0 0 0 1 0 1 0

s

0. 6.34

where e = e for ions/electrons.

s s s s s

s s s s

n m kT n n e n e n e

n e n e n e n e

u u E E E

u B u B u B u B

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

But

0 (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

uu u E u B u B

E u B

uu u E u B u

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:

one algebraic eq. 6.3

plane wave

7

i tn i i

i n n i i n

n n

e

K r

B

u E B K

K u u K

u K K u

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

6.36 to be solutions.

We a

s s s

s s s

n m i i kT i n n e

kT n ei i

n m m

u u K u K E u B u B

u K u K E u B u B

4 algebraic equations

1 1 1 1 1 1

1 1

22 1 1

1 1 0 0 02

lso must make use of Maxwell's equations to solve for the 10 unknowns

n , , , ; n , can have different values for the different species,

n , . From slide 6.1(2):

6

s s

t t

u E B u

u

E JE E

2 2

1 1 0 0 1 0 1 0 0 2

22

1 1 0 12

.6

1;

. 6.20

i i i i ic

K ic

K E K K E E J

E K K E J 3 more algebraic equations

6.2 Plasma Dynamics (5a)

1 1 0 1 s 1 1 s 1s 0

1 1 1

11 1 1

From slide 6.1 1 :

11 , e e

One more algebraic eq.

2 0 0. This eq. only tells that always .

3 . Three more algebraic eqs.

c c s ss

n i n

xt

E K E

B K B B K

BE K E B

Electrostatic Waves: B1= 06.3 Electron Plasma Waves (1)

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 0

the discussion we also assume 0, and 0,

then the algebraic transport equationelec s 6.37 and 6.38

0

become with

t

n

- :

roo

s s s

s

e e

n n

kT n ei i

n m m

e e

n n

u u E B

u K K u

u K u K E u B u B

K

11 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

c

e

kT n ei i a b

n m m

en c

u u K E

E

K E

6.3 Electron Plasma Waves (1a) (B1=0)

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 i

n m m

i n

n

K

K 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 en

n m im

kT e nn K

m m

6.3 Electron Plasma Waves (2) (B1=0)

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 s

e e e

e e

e e e

e e

p e e

e ep e

e e

kT e nK

m m

e n kTK

m m

V K

e n kTV

m m

2 2

peed.

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

6.4 Ion-Acoustic Waves (1) (B1=0)

1 0 1

11 1

0

We now consider the low frequency waves for which the ion motion must be

included. The ion transport equations are similar to 6.39 , with :

6.46

0 6.46

In Ga

s

i i i

i i ii

i i i

e e

n n a

kT n ei i b

n m m

K u

u K E

1 1 0 s 1s0

1 1 1 0

2 2 1

1uss's law ( e ) we must include positive and

negative charges:

i ( ) / . 6.46

Looking back at slide 6.3 (1), equation (6.40), for the electron motion:

c s

i e

e e e

e

n

e n n c

kT ni K

m

E

K E

10

2 2 01 1

0, or 6.40

0 6.48

e e

ee e e e

e

n m

enm K kT n

i

K E

K E

6.4 Ion-Acoustic Waves (2) (B1=0)

2 2 01 1

0 0

2 21 1 1 1

Similarly from the ion transport equations we get from 6.46b

0 6.49

Assuming = (neutral plasma), and adding 6.48 and 6.49 gives

( ) ( ) 0. Since

ii i i i

e i

i e e i i e e ei

enm K kT n

in n

mn m n K kTn kT n

K E

2 2 21 1

1 1

1 1 1 0

2 2 01 1 1

:

( ) 0 6.50

How do and relate? We can combine Gauss's law and the

momentum equation for the electrons

i ( ) / . 6.46

10

i e

i i i e e e

i e

i e

ee e e e

m m

m K kT n K kT n

n n

e n n c

enm K kT n i

i en

K E

K E K E

2 21

0

2 21 1 0 1

0

gives

1( ) / 6.51

e e e ee

i e e e e ee

m K kT n

e n n m K kT nen

6.4 Ion-Acoustic Waves (2a) (B1=0)

2

20

1 1 20

1 01 D2 2 2

0

2 2 21

If we neglect the electron inertia term again for low frequencies:

1 0

with , the Debye length 6.521

Substitute into 6.50

( )

e

e ei e

e

i ee

e D e

i i i e

m

K kTn n

e n

n kTn

K e n

m K kT n K

1

2 2

2 2

0,

gives the dispersion relation for ion plasma waves:

6.531

e e

i e e

i i e D

kT n

kT kTK

m m K

6.4 Ion-Acoustic Waves (2b) (B1=0)

22 2 D

2 2 2 2

2For very long waves: = 1.

We get the dispersion relation for ion acoustic waves,

or ion sound waves:

ion acoustic waves 6.55

ion acoust

D

i e es

i

i e es

i

K

kT kTK K V

m

kT kTV

m

ic speed 6.56

6.5 Upper Hybrid Oscillations

0 0

e1 0 1

1 1

Assume 0, and 0.Upper hybrid oscillations are

directed .We start again with the electron continuity 6.37 and momentum 6.38

equations:

n 657e e

e e

n a

ei

m

0

B E high frequency oscillations

B

K u

u E u

1 0

1 1 0

1 1 1

1 1

0 6.57

We also use two Maxwell equations:

i / Gauss's Law 6.57

i where 0 Faraday's Law 6.57

0 6.57

The dispersion relation for the upper hybrid oscillation becomes

e

b

en c

i d

e

B

K E

K E B B

K E K E

2 2 2

2 22 20 0

0

:

6.62

,

and K are not related, so there is no wave velocity defined. .

pe ce

ce pee e

eB n ewhere

m m

There is no wave

6.6 Lower Hybrid Oscillations

0 0Assume again 0, and 0. Lower hybrid oscillations are

directed . We must use

the electron and ion continuity 6.37 and momentum 6.38 equations

together wit

0

B E

low frequency electrostatic oscillations B

0 0

2

h the two Maxwell equations. This gives the algebraic equation:

6.68

where

and

are the electron-cyclotron and ion-cyclotron frequencies. Again, .

ce cie i

ce ci

eB eB

m m

no waves

6.7 Ion-Cyclotron Waves

0 0

0

2 2 2

Assume again 0, and 0.The ion-cyclotron waves are

that propagate in a direction

to . The algebraic dispersion relation becomes:

ci K

B E low

frequency electrostatic waves almost

perpendicular B

2

2

6.74

where

s

e es

i

V

kTV

m

6.8 Electromagnetic Waves in a Plasma (1)Now we consider the case where E1 and B1are non-zero. We start with the

general wave equation (6.20) assuming again a plane wave solution:

22

1 1 0 12

1 1

22

1 0 12

6.20

Let's look first for transverse waves, i.e., solutions for which 0 :

6.75

In a two-component plasma (electrons and one ion species) the c

K ic

K ic

E K K E J

K E K E

E J

1

0 1 0 1 0 0 0 0

0 1 0 0 0 1 0 0

urrent density

6.76

Perturbation and linearization:

/ for ions/electrons. Charge neutrality:

i i e e

s so s io eo o

s s s i e

i i e e i i s i e e

en en

n n n s i e n n n

en

en en e n n e n n

en

J u u

u u u J J J J u u

J u u u u u u

0 0 0 0 1 1 1 0 1 0

0 0 1 1 1 0 1 0 0 1

1 0 1 1 1 0 1 0

, . .,

6.80

i e i e i i e e

i e i i e e

i e i i e e

en en en

en en en i e

en en en

u u u u u u

J J u u u u J J

J u u u u

6.8 Electromagnetic Waves in a Plasma (2)

0 0 0 0

1 0 1 1

1

1 0 1

10 1 1 1 0 0 1

0

To simplify the notation, we assume 0.Then

High frequency approximation: 0

6.81

The electron momentum equation 6.38

i e i e

i e

i

e

s s s

T T

en

en

kT n ei i

n m m

E B u u

J u u

u

J u

u K u K E u B u B

0 1

1 1

20

1 1

0 becomes

0, therefore 6.82e

ee

e

en

ei

m

n eim

J u

u E

E

6.8 Electromagnetic Waves in a Plasma (2a)

1

222 0

1 0 12

1

2 2 222 20 0 0

1 0 0 020 0

2 2 2 2

Substitute this into (6.75):

6.83

Since 0, this gives the dispersion relation:

, or with

high-frequency dispers

e

pee e e

pe

n eK

c m

n e n e n eK

c m m m

c K

J

E E

E

ion relation 6.84

6.8 Electromagnetic Waves in a Plasma (3)

0 0

2 2 2 2

22

2

The phase velocity of high frequency EM waves is obtained from

cos cos

For a point of constant phase

d0 . Therefore:

dt

. Since

or 1

ph

ph pe

peph ph

E x E t Kx E

dxK Kvdt

v c KK

v c v cK c

2

.

The group velocity is

./

ph

g

g gph

v cK

vK

c cv c c v c

K v

6.8 Electromagnetic Waves in a Plasma (4)

0

0 00 0

ii

The phase index of refraction is defined as

1 since

where K = is the free space wave number2 2

The pase in the plane wave can then also be written as

e e

phph

nKt

c cKn v c

v

K n nK nc

K rK r

2 2

2 2

2 2

where is the wave normal.

Solve the dispersion relation for K:

K is real as long as 0 wave can propagate for .

For , i.e., K is purely imaginative, and

t

pe

pe pe

pe

pe

i t

Kc

K ic

e

K r

K

2

2 2

This is an evanescent wave with skin depth

Cutoff frequency1

where K = n = 0

K i t

pe

pe

e e

c

K

K r

6.9 Ordinary and Extraordinary Waves (1)

0

0 0 0 0 0

We now look for s in a plasma with 0. We first investigate waves perpendicular

to , i.e., . Again we assume E 0, , and we

neglect ion motion. We

high frequency EM wave soluti

wil

n

l

o

e e iT n n

B

B K B

1 0 1 0 1 0

0

1 0

ordinar

get different solutions depending on whether or . The wave with is called

the wave since has no effect on the wave propa-

gation. The wave w

y

eith is called the xtraordin

E B E B E B

B

E B

0

wave.

: In ionospheric radio science the terms ordinary and extraordinary waves are often used for waves in any direction with left-hand and right hand elliptical polarizations r

ary

e .

Note

B

K

Bo

6.9 Ordinary and Extraordinary Waves (2)We can use the following equations:

22

1 1 0 12

1 0 1

1 1 1 0

1 0 1

0

wave equation 6.20

continuity eq. 6.39

0 from 6.38 , momentum eq

ord

.

6.81

For the wave,ar in y

e e e

e ee

e

K ic

n n a

ei

m

en

E K K E J

K u

u E u B

J u

B E

1

0 1 1

1 0 1 0 1 0

0

Since always or 0 for ordinary wave.

Since accelarates the electron along , then =0

The equation are therefore the same as for the case =0, and the

dispersion relati

e e

K B E K K E

E B u B u B

B

2 2 2 2

on relation is again given by 6.84 :

for the waveordinarypec K

K

Bo

6.9 Ordinary and Extraordinary Waves (2a)

1 0 1

1 1 1

2 22 2 2 2

2 2 2

For the wave 0.

Set = and start cranking. The result is

dispersion relation

extraordinary

6.96pepe

pe ce

c K

E B K E

E E E

K

Bo

6.10 L and R Waves (1)

0

22

1 1 0 12

1 0 1

1 1 1 0

Consider high frequency transverse EM waves propagating parallel to .

Using the equations

wave equation 6.20

Gauss's law 6.39

0 momentum equ

e e e

e ee

K ic

n n a

ei

m

B

E K K E J

K u

u E u B

1 0 1

1

ation 6.38'

6.81

0.

This system of equations has two solutions.

een

J u

K E

z

xyK

B0

EL ER

6.10 L and R Waves (2)

0

( )1 10 0

For convenience we orient the coordinate system such that and

point in the direction of the the z-axis, then

right-hand circularly polarized wave re

or R wave, also calle

i Kz tR E x i y e

B K

E B

1 10 10 1 10 10

( )1 10

d e-wave . At z = 0:

cos( ) cos( ); sin( ) sin( )

left-hand circularly polarized wave

(or L wave, also called i-wave)

And the respective dispersion rela

Rx Ry

i Kz tL

E E t E t E E t E t

E x i y e

E

22 2 2

22 2 2

tions are:

R wave 6.1021 /

L wave 6.1021 /

pe

ce

pe

ce

c K a

c K b

xyK

B0

EL ER

6.11 Alfvén and Magnetosonic Waves

Low frequency transverse (i.e. ) electromagnetic waves are called:Alfvén waves, if

magnetosonic waves, if

The dispersion relations are, respectively:

0K B

0K B

1 E K

2 2 2

2 22 2

2 2

2 20

0

dispersion relation for Alfven waves 6.103

magnetosonic waves 6.1041 /

where the Alfven velocity and sonic thermal velocities are

and

A

s A

A

e eA s

io i i

K V

V VK

V c

B kTV V

n m m

EM waves in arbitrary direction

2

22 2 4 4 2 2

2

The index of refraction for an EM wave in arbitrary direction in a

magnetized plasma is given by the Appleton-Lassen equation:

11

1 11 sin sin cos 1

2 4

, , ,pe ce

X Xn

X Y Y Y X

cn K X Y

0

2

2

22

between and

0 : 11

190 : 1

11

Xn

Y

n X

Xn X

X Y

K B

K B0