Fundamentals of Plasma Physics

146 Chapter 4. Elementary plasma waves

thenk⊥must be in theydirection while in cylindrical geometry, this means that ifuσ⊥ 1 is
in theθdirection, thenk⊥must be in therdirection. The inertial Alfvén wave is known as
a cold plasma wave because its dispersion relation does not depend on temperature (such
a mode would exist even in the limit of a cold plasma). The kinetic Alfvénwave depends
on the plasma having finite temperature and is therefore called a warm plasma wave. The
shear mode can be coupled to ion acoustic modes since both shear and ion acoustic modes
involve finiteEz 1.

4.4.2 Two-fluid compressional modes

The compressional mode involves assuming thatBz 1 is finite and thatEz 1 =0.Having
Ez 1 =0means that there is no parallel motion and, in particular, implies thatJz 1 =0.
Thus, for the compressional mode Faraday’s law has the form

∇⊥·(E⊥ 1 ×zˆ)=iωBz 1 (4.135)

−ikzE⊥ 1 ×zˆ =iωB⊥ 1. (4.136)

Using Eq.(4.136) to substitute forB⊥ 1 in Eq.(4.111) and then solving forE⊥ 1 gives

E⊥ 1 =

iωv^2 A

ω^2 −k^2 zv^2 A

∇⊥

(

Bz 1 +

μ 0 P⊥ 1

B

)

×z.ˆ (4.137)

Since

E⊥ 1 ׈z=−

iωvA^2

ω^2 −k^2 zvA^2

∇⊥

(

Bz 1 +

μ 0 P⊥ 1

B

)

(4.138)

Eq.(4.135) becomes

∇⊥·

(

v^2 A

ω^2 −k^2 zv^2 A

∇⊥

(

Bz 1 +

μ 0 P⊥ 1

B

))

+Bz 1 =0. (4.139)

If we assume that the perpendicular motion is adiabatic, then

P⊥ 1

P

=γ

n 1

n

=γ

Bz 1

B

. (4.140)

Substitution forP⊥ 1 in Eq.(4.139) gives

∇⊥·

((

vA^2 +c^2 s

)

ω^2 −kz^2 v^2 A

∇⊥Bz 1

)

+Bz 1 =0 (4.141)

where

c^2 s=γκ

Te+Ti

mi

. (4.142)

On replacing∇⊥→ik⊥,Eq.(4.141) becomes the dispersion relation

−k^2 ⊥

(

v^2 A+c^2 s

)

ω^2 −kz^2 v^2 A

+1=0 (4.143)

or
ω^2 =k^2 v^2 A+k^2 ⊥c^2 s (4.144)
wherek^2 =kz^2 +k^2 ⊥.Since∇·uσ⊥ 1 =ik⊥·uσ⊥ 1 =0, the perpendicular wave vector
k⊥is at least partially co-aligned with the perpendicular velocity.