Magnetohydrodynamic Waves - Hydromagnetic Waves in the Magnetosphere and the Ionosphere - page 79

System (4.17a)-(4.17d) can be rewritten in a convenient form in terms of
displacements ξ , given by

u = ∂ ξ

∂t .

(4.18)

Substituting this expression for ξ into (4.17a) and (4.17c) and integrating the
obtained ratios with respect to time, give density and magnetic field pertur-
bations in the form

ρ 1 = − ∇ · ( ρ 0 ξ ) , b = ∇ × [ ξ × B 0 ] . (4.19)

The first relation means that the plasma density at the certain point diminishes
proportionally to the quantity of liquid flowing out of the volume.
Transform the second equation in (4.19). Present the displacement vector
as the sum

ξ = ξ || + ξ ⊥ ,
where ξ || is the displacement component parallel to B 0 and ξ ⊥ is perpendicular
to B 0 . As the vector product of the collinear vectors ξ || and B 0 is zero, only
ξ ⊥ remains in the expression for b .Iftoapply

∇× to the equation for b in

(4.19), then we find

b = − B 0 ∇ · ξ

⊥ −

( ξ

⊥ ∇

) B 0 +( B 0 ∇ ) ξ , (4.20)

⊥

where it is accounted that B 0 = 0. The first term here is proportional

to the density perturbations ρ 1 . Equation (4.20) shows that as plasma is
compressed across the initial magnetic field B 0 , field-lines are frozen into the
plasma and are displaced so that the magnetic field in the direction of B 0 is
intensified proportionally to the plasma compression. As displacement ξ ⊥ is
changed in the direction of B 0 , the field-lines frozen into the plasma are bent.
The magnetic field acquires a transverse component ( B 0 ∇

∇ ·

as a result.
Substitution of (4.19) and (4.20) into (4.17b) yields one vector equation
for displacement

) ξ

⊥

ρ 0 ∂ 2 ξ
∂t 2

1
4 π B 0 ·

c s ∇ · = ∇ { ( ρ 0 ξ ) − [ ∇ × ( ξ × B 0 )] }

1
4 π {

+ [ ∇ × ( ξ × B 0 ) · ∇ ] B 0 +( B 0 · ∇ ) ∇ × ( ξ × B 0 ) } . (4.21)

This equation is rather cumbersome and requires special mathematical meth-
ods for investigating hydromagnetic wave propagation in suciently general
equilibrium configurations. However, many important effects can result by
studying hydromagnetic oscillations and waves in the simplest model. In
such model the field-lines of an equilibrium magnetic field are assumed to be
straight and embedded into the plasma with density dependent on one trans-
verse coordinate only. The rest of this chapter and Chapter 5 are devoted to
such 1D problems. The investigation of the general case when the curvature of