Physics 777Plasma Physics and

Magnetohydrodynamics (MHD)

Instructor: Gregory Fleishman

Lecture 4. Linear Waves in the Plasma

30 September 2008

Plan of the Lecture

• MHD Waves

• Waves in Isotropic Plasma

• Waves in Magnetized Plasma

• Emission of Waves by a Given Electric Current

• Emission by Rectilinearly Moving Charge, General Derivation

Section 1. MHD Waves (see Somov, Chapt. 15)

Fourier transform yields:

Section 2. Waves in Isotropic Plasma

Transverse (free-space) modes

Longitudinal Waves

Maxwellian Plasma

where

1)

Use iterations to solve

Ion Sound Waves

For

if

Section 3. Waves in Magnetized Plasma

Zeros and Resonances

Resonances

Show that this means quasilongitudinal wave E||k

if

Substitution of tensor components into coefficient A yields:

This Eq. has three roots

Neglecting ion contribution, we obtain two of three:

Asymptotic expressions

Zeros

Neglecting ion contribution, we find:

F wZ

X

X

Z

O

F-w

A

Normal waves for oblique propagation

Simplifications:

1

2

Whistler mode

Waves in Hot Plasma

Recall:

It is convenient to express this via Bessel functions

Maxwellian plasma

Bernstein Modes

Section 2. Macroscopic Maxwell Equations. Linear response

Introduce polarization vector; continuity Eqn. is fulfilled:

Form displacement vector: D=E+4P; the most general (non-local) linear relation for statistically uniform medium reads:

Section 4. Emission of Waves by a Given

Electric Current

Recall:

where

is the Maxwellian tensor, j is an external electric current (including nonlinear plasma current in a general case).Let’s solve this inhomogeneous algebraic equation for E

- energy loss of a given current (from electrodynamics)

where

In the basis of the eigen-vectors we obtain diagonal form

Substitution yields:

where

Section 5. Emission by Rectilinearly Moving Charge, General Derivation

Radiation field far away from the charge (from e/d)

Define and obtain:

Nonrelativistic case:

Ultrarelativistic case:

We are looking for

Section 6. Homework

• Derive formula for the energy emitted by a rectilinearly moving charge in a given field in the nonrelativistic case.