Waves, Fluctuations and Turbulence

• General concept of waves• Landau damping• Alfven waves• Wave energy• Collisions and fluctuations

Definition• Any perturbation that propagates in

plasmas.• In general a wave is either damped or grow

in time while it propagates• Certain dispersion relation is satisfied.• Coherent waves• Fluctuations• Disturbance induced by large scale motion• Weak turbulence

A Simple Explanation of Landau Damping• Considering

0e ee

f e Ff

t m

v E

v

ˆ( , , ) ( , , ) i t ie ef t f e k rr v k v

0( )

ee

ie Ff

m

E

v k v

224 2 k

EE

t

j E

2 22

2( ) 2pe e

k

E Fd E

t k

v v k kv

4t

E

j

ˆ( , , ) ( , ) i t it t t e k rE r E k

Some Conventional Usage

• In plan wave solutions we usually consider

with . (Landau prescription)

exp i t i k r

0i

Conventional Definition of Wave Vector

0B

x

y

z

k

zk

General Dispersion Equation (cold)

2 2

2

2 2

2

0

0 0

0 0

zxx xy

yx yy

zz

k c

k c

Alfven and Magnetosonic Waves

• Alfven waves

• Magnetosonic waves

22 2 2

2 2 2

pizxx

p A

k c c

v

22 2 2

2 2 2

piyy

p A

k c c

v

Conventional Definition of Wave Vector

0B

x

y

z

k

zk

Wave-energy spectral density

• General expression

• Unmagnetized plasma

2

2 *1Re ( , )

8k

k k i ij k jk k

EW a a

k

2

Re ( , )8k

k k kk

EW

k

Wave-energy spectral density

• Alternative general expression

• Spectral density associated with kinetic energy

2 2

*( , )8 8k k

k k i ij k jk

E BW a a

k

2

( , ) 18k

k k kk

EW

k

Three special wave modes• Alfven and magnetosonic wave

• Langmuir wave

• Physical implication is important.

2 2 2

8 8 4k k k

k k xxk

E B BW

2 2

8 4k k

k kk

E EW

In all three cases

• Particle motion induced by a spectrum of turbulent waves can be stochastic, rather than fluid like.

• The kinetic energy density is equal to the energy density of the wave field.

• To the best of my knowledge no other wave mode has this special property.

“Collisions” and Fluctuations

General Considerations

• Traditional concept of “collision” is based on binary particle-particle interactions

• The issue of Debye sphere

• Physical picture of “collisions” is not very clear based on intuition.

3 3 910 10Dn

Landau’s Collision Integral (1937)

3'

( ')( )' ( ') ( )q qsqs sij q s

q q i j j

n FF Fd v Q F F

t m v v v

vvv v

2 2 34

( ')2 i jsq

ij s q

k kQ e e d

k

k v k vk

Balescu-Lenard Collision Integral

• It was derived in 1960 independently by R. Balescu and A. Lenard.

3'

( ')( )' ( ') ( )q qsqs sij q s

q q i j j

n FF Fd v Q F F

t m v v v

vvv v

22

2 3

4

( ')2

( , ')

i jsqij s q

k kQ e e d

k

k v

k k v

k vk

Vlasov EquationsKinetic equation neglecting fluctuations

The neglected term is

0s s ss

s

F e FF

t m c

v

v E Bv

ss

s s

eN

m n

Ev

Klimontovich Formalism

• According to Klimontovich (1967) the “collision integral” may be simply written as

( , ) ( , , )s ss

s s

F et N t

t n m

E r r vv

Implications

• So-called “collisions” are closely related to the correlation function

• For plasma that is unstable according to linearized Vlasov theory the fluctuations may be enhanced so that “collisions” become more important than we have assumed.

( , , ) ( , )sN t t r v E r

Thermal Fluctuations

• The physical origin • Spontaneous emission• Induced emission

Particles can emit waves via resonance

3 2

322,

32

,s s

sks

n ed v F

k

k vE E v

k

22 3

2

22 3

2

Im , 4

8

s s s

s s ss

n e Fd v

k

n m ed v F

k T

k k v kv

k v v

2,

Im ,8 8,

,k

T T

k

E E kk

,8d T

k kE E E E