Magnetohydrodynamic Waves - Hydromagnetic Waves in the Magnetosphere and the Ionosphere - page 78

magnetic field-lines be straight. Axis z in the Cartesian coordinate system
{ is then directed parallel to the field-lines.

Denote by x , y , z the unit vectors of the coordinate system. Since the mag-
netic field has only one component parallel to axis z, B = B z , the solenoidality
∇ ·

x, y, z }

B = 0 yields

∂ B
∂z

=0 , B = B ( x, y ) .

Then the equilibrium equation (4.14) reduces to

P + B 2
8 π

=0 ,

∂P
∂z

∇ ⊥ =0 . (4.15)

It follows from the second equation in (4.15) that pressure P is constant along
the field-line, P = P ( x, y ), and the first equation in (4.15) yields

P ( x, y )+ B 2 ( x, y )
8 π

=const . (4.16)

Condition (4.16) is valid only if Ampere's force is balanced by the plasma
pressure gradient. If forces of a different nature, for instance, gravitational,
are taken into account, plasma density and pressure can change along the
field-lines.

Linear Approximation

Most effects linked to ULF-propagation in the magnetosphere can be under-
stood within the limits of linear approximation. Consider small perturbations
of plasma near the equilibrium. Denote by B 0 the magnetic field equilibrium
value, by P 0 - plasma pressure and by ρ 0 - plasma density. The undisturbed
electric field E 0 and plasma velocity u 0 will be taken as zero. Put

B = B 0 + b , P = P 0 + p, ρ = ρ 0 + ρ 1 .

b , E ,p,ρ 1 , u are small perturbations near equilibrium. Substituting these
expressions into (4.1)-(4.4), (4.6), (4.8), taking into account equilibrium con-
ditions and neglecting all the degrees of perturbations except the first, we
obtain
∂ρ 1
∂t

+ ∇ · ( ρ 0 u )=0 , (4.17a)

− ∇ c s ρ 1 +

ρ 0 ∂ u
∂t

1
4 π [[

1
4 π [[

= ∇ × B 0 ] × b ]+ ∇ × b ] × B 0 ] , (4.17b)

∂ b
∂t

= ∇ × [ u × B 0 ] , ∇ · b =0 , (4.17c)

1
c [ u

E = − × B 0 ] , (4.17d)

where c s = c s ( ρ 0 ) is sound velocity in an equilibrium configuration.