Magnetohydrodynamic Waves - Hydromagnetic Waves in the Magnetosphere and the Ionosphere - page 84

In low-pressure plasma,

c s
c 2 A

β = << 1 .

Therefore, neglecting all terms proportional to β , (4.33) and (4.34) reduce to

b B 0

∂ 2
∂t 2

1
c 2 A

2

∇ − =0 , (4.35)

∂ 2
∂z 2 − ∂t 2 ξ || =

∂ 2

b B 0 .

1
c s

∂
∂z

(4.36)

Equation (4.35) describes the FMS-waves. Medium elasticity in a FMS-
wave is created by magnetic pressure P m = B 0 / 8 π . Basing on analogy with
adiabatic gas-dynamic sound velocity

c s = γ P 0
ρ 0

( γ is the adiabatic index) write the Alfven velocity as

B 0
4 πρ 0

= γ m P m

c 2 A =

ρ 0 ,
where γ m = 2. The condition γ m = 2 is linked to the field being frozen into the
plasma, owing to which the magnetic field at ξ || = 0 is changed in proportion
to plasma density, while its pressure is proportional to plasma density squared.
The longitudinal displacement ξ || is small in the FMS-wave in low β plasmas.
It follows from (4.36) that b propagating at Alfven velocity produces ξ
proportional to β , since perturbations b move with Alfven velocity c A c s .

Perturbations with

b =0 and ∇ ⊥ · ξ ⊥ =0

are described by the uniform equation (4.36). These are ion sound waves
propagating at velocity c s along the magnetic field. Plasma perturbations
occur only along the magnetic field.

Dispersion Equation

Three hydromagnetic wave modes can propagate in homogeneous plasma:

1. Alfven waves: plasma density is undisturbed and field-lines are twisted due
to plasma transverse displacements.
2. Fast magnetosonic (FMS) waves: compressions and rarefactions of field-
lines accompanied by plasma density variations. For low-temperature
plasma, when β<< 1, longitudinal displacements of plasma in FMS-waves
are small, ξ ∝ β .