MHD-Waves in Layered Media - Hydromagnetic Waves in the Magnetosphere and the Ionosphere - page 198

( x 1 ,x 2 ,x 3 ), the electric components will be found with the help of formulae
for transformation vector components from one coordinate system to another.
Let us express E ( r )and b ( r ) in terms of the scalar ϕ and vector A potentials

E ( r )= − ∇ ϕ + ik 0 A , (7.55)

b ( r )= ∇ × A . (7.56)

However, the E ( r )and b ( r ) fields are conserved for transform of kind: A →

A + ϕ + ik 0 Ψ . A particular choice of ϕ and A potentials is a gauge,

and a Ψ scalar function used to change a gauge is called a gauge function ([7],
[14]).
In Alfven waves b Az = b = 0, so the condition

∇ Ψ, ϕ →

∇ · b = 0 becomes

∂b Ax
∂x

+ ∂b Ay
∂y

=0 . (7.57)

Therefore, a gauge function can be chosen so that only one z -component of
the vector potential A is nonvanishing: b A ( r )= ∇ × A , where A = A z . Then

the magnetic field in the Alfven wave is given by

∂A
∂y

∂A
∂x

b Ax = , b Ay = − . (7.58)

Components of the electric field E A = − ∇ ϕ + ik 0 A and longitudinal current

j are

∂ϕ
∂x

∂ϕ
∂y

∂ϕ
∂z

E Ax = − , Ay = − , E Az = − + ik 0 A, (7.59)

∂ 2 A

∂y 2 .

∂x 2 + ∂ 2 A

c
4 π

j = j Az = − (7.60)

Substituting (7.58), (7.59) into (7.5) and taking into account vanishing of the
longitudinal electric component E Az = 0, we obtain

∂A
∂z

= ik 0 ε m ϕ, (7.61)

∂ϕ
∂z

= ik 0 A. (7.62)

Eliminating A from this system we get the 1D-wave equation for ϕ :

∂z 2 + ω 2

∂ 2 ϕ

ϕ =0 .

c 2 A

In the FMS-wave the field-aligned component of the electric current
vanishes

j Sz = j =0 .