Magnetohydrodynamic Waves - Hydromagnetic Waves in the Magnetosphere and the Ionosphere - page 81

Perturbations of hydrodynamic variables ρ 1 , u and electromagnetic field

components E E ⊥ ,E and b b ⊥ ,b are expressed in terms of displace-
ment ξ as

b B 0

ρ 1
ρ 0

b B 0

= ∂ ξ ⊥
∂z

− ∇ · ξ , , − ∇ · ξ ⊥ ,

= =

u = ∂ ξ

1
c u ∂t ,

E ⊥ = − × B 0 , E = E z =0 . (4.26)

Here b ⊥ , E ⊥ and b ,E are magnetic and electric wave components perpen-
dicular and parallel to the ambient magnetic field B 0 .

Alfven Waves

A wave with zero perturbations of density ρ 1 and field-aligned displacement ξ
can be excited in plasmas. As a consequence, the wave field-aligned magnetic
component b =

−

B 0 ∇ ·

ξ ⊥ also vanishes. In this case the wave should carry

with it a non-zero vortex

Ω =[ ∇× u ]

and a non-zero longitudinal component of current density

c
4 π [

j = ∇× b ] ,

where

= [

.

B 0
B 0

∇× ∇ × ·

[ b ] b ]

This wave mode is called the Alfven wave [1]. Equations for the Alfven waves
can be found from (4.24)-(4.25). Transverse displacement ξ in an Alfven

⊥

wave is found from

= ∂ 2 ξ ⊥

∂ 2 ξ ⊥
∂t 2

1
c 2 A

L ξ

∂z 2 −

=0 . (4.27)

⊥

In addition, since ρ 1 = 0, then it follows from (4.26) that ξ must be solenoidal

⊥

∇ · ξ ⊥ =0 . (4.28)

For an individual field-line, (4.27) coincides with the elastic string equa-
tion. According to (4.27), transverse displacement in an Alfven wave propa-
gating in the positive z direction is

ξ ( x, y, z, t )= ξ + ( x, y, ζ ) , ζ = z − c A t, (4.29)

⊥

where ξ + ( x, y, ζ )ateachfixed ζ is an arbitrary solenoidal 2D vector field
orthogonal to B 0 .From ∇ · ξ ⊥ = 0 it follows that ξ ⊥ canbeexpressedin