waves in plasma very short course

Waves in Plasma

Very short course

Outline

• Introduction – some definitions

• Self-consistency, linear theory

• Transverse waves in unmagnetized plasma

• Longitudinal waves, Landau damping

• Waves in magnetized plasma

• Beam-plasma instabilities, plasma masers

• Introduction – some definitions• Self-consistency, linear theory

• Transverse waves in unmagnetized plasma




What is a wave?A wave is a perturbation propagating in space

Gerald B. Whitham, ``Linear and Nonlinear Waves''

a lot of physical phenomena

Shock wave

Solitary wave

Solitary wave in a lab

Nonlinear waves

Out of scope of the course

We restrict ourselves by considering only linear waves

Linear waves

Mathematically, a process described by linear equations

Wave equation 012

2

2

tc

Partial solution in Cartesian coordinates – monochromatic plain wave

222)( , zyxtrki kkkcckAe

Particularly, Maxwell equations in vacuum are linear equations and can be reduced to the wave equation

What about plasma?

is a linear equation

Plasma is a ``fourth state of a matter'‘ as defined by Russian physicist Frank-Kamenetsky)

Temperature

Solid state Liquid Gas Plasma

Most general definition: plasma is an ensemble of charged particles

(less general one includes the condition of quasi-neutrality)

In the ensemble of charged particles, the long-range forces appear.

The motion of each charged particle is determined by electric and magnetic fields, which, in their turn, are created by a large number of other charged particles, i.e., depend on their velocities and locations.

The motion of ions and electrons occurs in self-consistent fields.

The self-consistent long-range e/m fields allow passing information about particles motion from one region to another without collisions. This is the main difference between plasma waves and, for instance, sound waves and waves at a water surface.

to be described by Maxwell equations

• Introduction – some definitions

• Self-consistency, linear theory• Transverse waves in unmagnetized plasma




Maxwell equations in vacuum

external current and charge densities do not depend on fields

In plasma, assuming 0 ,0 extextj

Only currents and charges remain that depend on fields

40

1

41

E

Ht

Hc

E

jct

Ec

H

),(4

0

1

),(41

trE

Ht

Hc

E

trjct

Ec

H

ext

ext

How to find this dependence?

Vnejne

,

0

1

Vnt

n

VHVc

Eme

dtVd

means a sort of particles

Equation of motion

Continuity equation

Simplest model for the self-consistent set of equations

Kinetic equation in the self-consistent fields

),,( trVf

Distribution function

Number density VdtrVftrn 3),,(),(

;),,( ;),,( 33 VdtrVfVejVdtrVfe

thermal motion is accounted

natural normalizing

Liuville equation: 0

dtVd

Vf

dtrd

rf

tf

dtdf

No particles are born

No particles are died

No collisions (free motion in e/m fields)01

fHVc

EmefV

tf

V

Vlasov equation:

Vlasov + Maxwell = Vlasov-Maxwell set of equations

One more example is the magnetohydrodynamic model

0

0

1

1

D

BtB

cE

tD

cH

Maxwell equations in medium

Plasma as a medium

40

1

41

E

Ht

Hc

E

jct

Ec

H

jtE

tD

HB

4

ED

Dielectric permittivity

For linear waves, only linear terms should be kept for j

and

Accordingly, the permittivity tensor does not depend on fields

Permittivity tensor contains all information about linear properties of a medium and depends on parameters of the medium steady-state: density, temperature, external fields, etc.

Linear theory

tH

cE

tE

cH

1

)ˆ(1 )(0

)(0 , trkitrki eHHeEE

HkiHEkiE

,

EkcH

Ec

Ekkc

ˆ E

ckkEEkk

ˆ)()( 2

2

0ˆ)( 2

22 E

cEkkEk

it

02

22

jjij

jjjii E

cEkkEk 02

22

jjijjiij E

ckkk

0det 2

22 ijjiij c

kkk Dispersion relation

),( kijij

Transverse waves and longitudinal waves

0ˆ)( 2

22 E

cEkkEk

k

E

kE

0Ek

EkEkk

2)(

0det 2

22 ijij c

k

ijij k ),(

0det ijED

For isotropic medium

0),( k0),(2

22 k

ck

Dispersion relation

Dispersion relation

In vacuumno way!ck

waves in plasma very short course

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