Physics 777 Plasma 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 w Z 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+4  P; 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-vectorswe 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 andobtain:

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.