Magnetohydrodynamic Waves - Hydromagnetic Waves in the Magnetosphere and the Ionosphere - page 83

B 0

( a ) ( b )

y

ρ

B 0

x

ϕ

Fig. 4.1. Sketches of the magnetic field-lines disturbed by ( a ) linearly polarized
shear Alfven wave and ( b ) by a torsional shear Alfven wave

Arbitrary initial perturbation is carried along the field-lines at Alfven ve-
locity with constant shape and without spreading in the transverse direction.
Guiding by field-lines is probably the most important property of Alfven waves
which is unchanged in inhomogeneous plasmas and even under finite curvature
of field-lines.
An arbitrary Alfven perturbation in homogeneous plasma can be presented
as a superposition of disturbances of non-interacting field-lines. For a field line
with coordinates ( x 0 ,y 0 ) displacement can be written as

ξ + ( x, y )= ∇ × δ ( x − x 0 ,y − y 0 ) z ,

where δ ( x, y ) is the Dirac delta-function. However, in an inhomogeneous
plasma, interaction arises between perturbations propagating along adjacent
field-lines and a bundle of interacting field-lines is excited as a result. Guiding
along field-lines is conserved in this case, as well.

Magnetosonic Waves

Consider two other types of waves with vortex Ω and longitudinal current
j vanish. The longitudinal magnetic field b , transversal compression ξ ⊥

and longitudinal displacement ξ are non-zero. It is convenient to use the
perturbation of the longitudinal magnetic field and longitudinal displacement
in the description of these waves. Applying transverse operator

∇ ·

∇ ⊥ · to (4.24)

and (4.25), we obtain for b and ξ :

b B 0 b B 0 − ,

∂ 2
∂t 2

∂ξ
∂z

1
c 2 A

2 2

⊥

∇ − = − β ∇ (4.33)

∂ 2
∂z 2 −

ξ || =

∂ 2
∂t 2

1
c s

∂
∂z

b B 0 .

(4.34)