Introducing Some Basic Concepts

4.14.09

Linear Theories of Waves

• (Vanishingly) small perturbations• Particle orbits are not affected by waves.• Dispersion relation is independent of wave

energy.• But linear theory may actually describe

some conceptually nonlinear processes. The best example is the mode coupling process.

• Nonlinear process and linear theory.

Nonlinear Process Described by Linear Theory

2 2p p

220 0

1 0e ebn

n kv

22 2 2 2p 0 pe ekv

Nonlinear or Quasilinear Theories of Waves

• Explicit appearance of wave energy in the theory.

• Physically nonlinear but mathematically linear• Both physically and mathematically nonlinear

Ensemble of Systems

A group of similar systems but suitably randomized so that statistical study is meaningful.

Concepts of Random Quantities

• Theoretically each physical quantity in a many-particle system consists of two parts: the ensemble averaged value and a fluctuating part.

• By definition the fluctuating part is random and its ensemble averaged value vanishes.

Statistic Approaches to Plasma Physics

• BBGKY hierarchy• Prigogine and Balecu scheme• Klimontovich formalism

which introduces a totally new approach in statistical theory of plasma physics.

Random Density Function

• A random density function is defined as follows

where and denote the position and momentum of a given particle.

1

( , , ) ( ) ( )si

s si sii N

N t t t

r p r r p p

( )si tr ( )si tp

3 3( , , )s sN t d d N r p r p

Phase-Space Orbit

( , , )tr p

'( '), '( '), 't t tr p

Phase-Space Continuity Equation

• The density function satisfies

• Here the microscopic fields yield

( ', ', ') d '' 0

' ' ' 's s s sdN t N N N

dt t dt

r p p

vr p

d 1( , ) ( , )M M

se t tdt c

pE r v B r

Field Equations

In addition to the kinetic equation we also need the Maxwell equations

31 4( , , )

MM

s ss

e d N tc t c

E

B vv r p

1 MM

c t

B

E 0M B

34 ( , , )Ms s

s

e d N t E v r p

So Far…

• The theory is completely formal.• Practically not useful

• A statistical treatment is needed.

Phase Space Probability Density

• The ensemble averaged value is what we know as the distribution function

• We may also define

( , , ) ( , , )s s sN t n F tr p r p

( , , ) ( , , ) ( , , )s s s sN t n F t N t r p r p r p

Microscopic Field and Fluctuations

• Ensemble averaged microscopic field

• We define

M

M

E E

B B

mM

Mm

E E E

B B B

Ensemble Averaging of the Klimontovich Equations

• If we neglect fluctuations completely, it is obtained

e 0s s ss

F F F

t c

v B

v Er p

31 4( , , )s s

s

e d F tc t c

E

B vv r p

1

c t

B

E 0 B

34 ( , , )s ss

e d F t E v r p

These are the Vlasov equations

• If electromagnetic fields are neglected, the equations reduce to

0s s s s

s

F F e F

t m

E

vr v

34 ( , , )s ss

e d F t E v r p

Linearization Scheme

• For practical reason we introduce

and assume so that the equations can be linearized. The result is

0( , ) ( ) ( , , )s s sF t F f t p p r p

0s sF f

0 0s s s s

s

f f e F

t m

E

vr v

34 ( , , )s ss

e d f t E v r p

We use linearized Vlasov equations for:

• Derivation of dispersion relations

• Discussion of propagating modes

• Study of plasma instabilities

Considering Density Fluctuation

• Since we know

• And

( , , ) ( , , )s s sN t n F tr p r p

( , , ) ( , , ) ( , , )s s s sN t n F t N t r p r p r p

First order fluctuating quantity

If we neglect the terms involving products of fluctuating quantities, we obtain

0s s s m s

s

N N e F

t m

E

vr v

34 ( , , )m s ss

e d N t E v r p

Comparison of Two Sets of Equations

0s s s m s

s

N N e F

t m

E

vr v

34 ( , , )m s ss

e d N t E v r p

0 0s s s s

s

f f e F

t m

E

vr v

34 ( , , )s ss

e d f t E v r p

An Important Conclusion

• When an unperturbed distribution function describes an unstable state, it means that both ensemble-averaged perturbation and microscopic fluctuating field would grow with time.

• In general an instability is more important for the latter because it leads to the origin of the turbulence.

Fluctuating Fields

• Consisting two components• One can propagate in plasmas• The other cannot

Significance of Kinetic Instabilities

• A kinetic instability usually excites a spectrum of fluctuating fields whereas a reactive instability often amplifies coherent waves.

• Therefore in general plasma turbulence is attributed to kinetic instabilities.