Magnetohydrodynamic Waves - Hydromagnetic Waves in the Magnetosphere and the Ionosphere - page 77

Equilibrium Configurations

The characteristics of hydromagnetic wave propagation are determined in lin-
ear approximation by equilibrium distributions of plasma and of the magnetic
field. The main objective of the present section is to expound only the ba-
sic concepts of equilibrium plasma configurations. We shall not consider many
important questions arising in the study of equilibrium plasma configurations,
referring the reader to numerous works where problems of plasma equilibrium
in a magnetic field are discussed (e.g. [14], where the mathematical treatment
of the magnetic equilibrium and its applications to the sun and planets with
a magnetic field are given).
According to (4.2), plasma can be in equilibrium only if its pressure gra-
dient is balanced by Ampere's force

P = 1

∇

c [ j

× B ] . (4.12)

Expressing j in terms of B with the help of (4.4) and using the vector

identity

1
2 ∇

( B · B )=( B ∇ ) B +[ B × [ ∇ × B ]] ,

the equilibrium equation (4.12) can be written as

1
4 π ( B

∇ ( P + P m ) − ∇ ) B =0 . (4.13)

Here

P m = B 2
8 π
can be interpreted as a magnetic pressure and

1
4 π ( B

∇ ) B

as a tension along field-lines.
A geometric presentation of equilibrium equations can be obtained if a
unit vector h = B /B , directed along the magnetic field, is introduced. Then
(4.13) becomes

B 2
8 π

= B 2

B 2
4 πR n ,

∇ P + ∇ ⊥

4 π ( h

∇ ) h = (4.14)

where n is normal to the field-line, R is its curvature and )

is the transverse gradient. It follows from (4.14) that the magnetic field exerts
pressure B 2 / 8 π in the transverse direction and creates an additional force in
the direction of the field line concavity as a result of their tension.
Consider a simple equilibrium configuration often used in modeling wave
processes in the magnetosphere as well as in other plasma objects. Let the

∇ ⊥ = ∇ − h ( h ∇