Fundamentals of Plasma Physics

144 Chapter 4. Elementary plasma waves

which gives the ion acoustic waveω^2 =k^2 zκTe/midiscussed in Sec.4.2.1. This shows that
the acoustic wave is associated with having finiteEz 1 and also requiresTe>>Tiin order
to exist.
Returning to shear waves, we now assume that the electric field is not electrostatic so
B⊥ 1 does not vanish and Eq.(4.118) has to be considered in its entirety. For shearwaves
the character of the parallel current changes depending on whether the waveparallel phase
velocity is faster or slower than the electron thermal velocity. Theω^2 /k^2 z>> κTe/me
case is called the inertial limit while theω^2 /k^2 z<< κTe/me case is called the kinetic
limit. The perpendicular component of Faraday’s law is

∇⊥Ez 1 ׈z−ikzE⊥ 1 ×zˆ=iωB⊥ 1. (4.120)

Substitution ofE⊥ 1 as determined from Eq.(4.111) into Eq.(4.120) gives

−

iω

v^2 A

∇⊥Ez 1 ׈z−ikz

(

μ 0 ∇P⊥ 1

B

×zˆ−ikzB⊥ 1 ׈z

)

×zˆ=

ω^2

vA^2

B⊥ 1 (4.121)

which may be solved forB⊥ 1 to give

B⊥ 1 =

1

ω^2 −k^2 zv^2 A

(

−iω∇⊥Ez 1 ×zˆ+ikzv^2 A

μ 0 ∇⊥P⊥ 1

B

)

(4.122)

and

B⊥ 1 ׈z=

1

ω^2 −k^2 zvA^2

(

iω∇⊥Ez 1 +ikzv^2 A

μ 0 ∇⊥P⊥ 1

B

×zˆ

)

. (4.123)

Substitution ofB⊥ 1 ×zˆinto Eq.(4.118) gives

∇⊥·

(

1

ω^2 −k^2 zvA^2

(

∇⊥Ez 1 +kzv^2 A

μ 0 ∇⊥P⊥ 1

ωB

×zˆ

))

=Ez 1

∑

σ

ω^2 pσ/c^2

ω^2 −γσk^2 zκTσ/mσ

.

(4.124)

However, because∇⊥·(∇⊥P⊥ 1 ׈z)=∇·(∇P⊥ 1 ×zˆ)=0the term involving pressure
vanishes, leaving an equation involvingEz 1 only, namely

∇⊥·

(

1

(ω^2 −kz^2 v^2 A)

∇⊥Ez 1

)

−Ez 1

∑

σ

ω^2 pσ/c^2

ω^2 −γσkz^2 κTσ/mσ

=0. (4.125)

This is the fundamental equation for shear waves. On replacing∇⊥→ik⊥, Eq.(4.125)
becomes

k^2 ⊥

ω^2 −k^2 zvA^2

+

ω^2 pe

c^2

1

ω^2 −γekz^2 κTe/me

+

ω^2 pi

c^2

1

ω^2 −γik^2 zκTi/mi

=0. (4.126)

In the situation whereω^2 /kz^2 >> κTe/me,the second term dominates the third term
sinceω^2 pe>>ω^2 piand so Eq.(4.126) can be recast as

ω^2 =

k^2 zvA^2

1+k^2 ⊥c^2 /ω^2 pe

(4.127)