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Waves, Fluctuations and Turbulence
• General concept of waves• Landau damping• Alfven waves• Wave energy• Collisions and fluctuations
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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
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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
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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
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Some Conventional Usage
• In plan wave solutions we usually consider
with . (Landau prescription)
exp i t i k r
0i
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Conventional Definition of Wave Vector
0B
x
y
z
k
zk
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General Dispersion Equation (cold)
2 2
2
2 2
2
0
0 0
0 0
zxx xy
yx yy
zz
k c
k c
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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
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Conventional Definition of Wave Vector
0B
x
y
z
k
zk
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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
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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
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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
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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.
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“Collisions” and Fluctuations
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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
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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
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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
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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
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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
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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
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Thermal Fluctuations
• The physical origin • Spontaneous emission• Induced emission
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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
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2,
Im ,8 8,
,k
T T
k
E E kk
,8d T
k kE E E E