Fundamentals of Plasma Physics

132 Chapter 4. Elementary plasma waves

an inductive character (i.e.,∇×E= 0 ). Faraday’s law∇×E=−∂B/∂tshows that
if the electric field is electrostatic the magnetic field must be constant, whereas inductive
electric fields must have an associated time-dependent magnetic field.
We now further restrict attention to a specific category of theseω <<ωcimodes. This
category, called Alfvén waves are the normal modes of MHD, involve magnetic perturba-
tions and have characteristic velocities of the order of the Alfvén velocityvA=B/

√

μ 0 ρ.
The existence of such modes is not surprising if one considers that ordinary sound waves
have a velocitycs =

√

γP/ρand the magnetic stress tensor scales as∼B^2 /μ 0 ρso
that Alfvén-type velocities will result ifP is replaced byB^2 / 2 μ 0 .Two distinct kinds of
Alfvén modes exist and to complicate matters these are called a variety of names by dif-
ferent authors. One mode, variously called the fast mode, the compressional mode, or
the magnetosonic mode resembles a sound wave and involves compression and rarefac-
tion of magnetic field lines;this mode has a finiteBz 1. The other mode, variously called
the Alfvén mode, the shear mode, the torsional mode, or the slow mode, involves twisting,
shearing, or plucking motions;this mode hasBz 1 =0. This latter mode appears in two
distinct versions when modeled using two-fluid or Vlasov theory depending on the plasma
β;these are respectively called the inertial Alfvén wave and the kinetic Alfvén wave.

4.3.2 Zero-pressure MHD model

In order to understand the basic structure of these modes, the pressure will temporarily
assumed to be zero so that all MHD forces are magnetic. The fundamental dynamics
of both MHD modes comes from the polarization drift associated with a time-dependent
perpendicular electric field, namely

uσ,polarization=

mσ

qσB^2

dE⊥

dt

; (4.47)

this was discussed in the derivation of Eq.(3.77). The polarizationdrift results in a polar-
ization current

J⊥ =

∑

nσqσuσ,polarization

=

ρ

B^2

dE⊥

dt

(4.48)

whereρ=

∑

nσmσis the mass density. This can be recast as

dE⊥

dt

=

B^2

μ 0 ρ

μ 0 J

= vA^2 (∇×B 1 )⊥ (4.49)

where

vA^2 =

B^2

μ 0 ρ

(4.50)

is the Alfvén velocity. Linearization and combining with Faraday’s lawgives the two basic
coupled equations underlying these modes,

∂E⊥ 1

∂t

= v^2 A(∇×B 1 )⊥

∂B 1

∂t

= −∇×E 1. (4.51)