Magnetohydrodynamic Waves - Hydromagnetic Waves in the Magnetosphere and the Ionosphere - page 80

the field-lines as well as the 2D or 3D inhomogeneity of the plasma are taken
into account is performed in Chapter 6.
Suppose that the magnetic field B 0 = B 0 z . Then, if no external forces are
acting on the plasma, the pressure is P 0 = P 0 ( x, y ) and the magnetic field is
B 0 = B 0 ( x, y ) . Using the vector equality

B 0 ]= B 0 ∂ ξ

⊥
∂z −

∇ × [ ξ ⊥ × z [ B 0 ∇ · ξ ⊥ +( ξ ⊥ · ∇ ⊥ ) B 0 ]

we reduce (4.21) to

ρ 0 c 2 A ∇ ⊥ ρ 0 c 2 A + c s ∇·

1

L ξ ⊥ = − ξ ⊥

c s ∇ ⊥ ρ 0 + + ρ 0 c s ,

∂ ξ

ρ 0 c 2 A
2 ∂z

+ ξ ⊥ ·

∇ ⊥

(4.22)

c s ,

ρ 0 c 2 A
2

1
ρ 0 c s

∂
∂z

L s ξ = − ∇ · ( ρ 0 ξ ⊥ )+ ξ ⊥ · ∇ ⊥ (4.23)

where c 2 A = B 0 / 4 πρ 0 is the Alfven velocity,

∂ 2
∂z 2 −

∂ 2
∂t 2 ,

1
c 2 A

L =

∂ 2
∂z 2 −

∂ 2
∂t 2 .

1
c s

L s =

4.2 Homogeneous Plasma

Basic Equations

Propagation of small-amplitude hydromagnetic waves in homogeneous plasma
is discussed in most textbooks on plasma physics and magnetohydrodynamics
([5], [6], [15], [18]), and even more so in works specially devoted to electro-
magnetic wave propagation in plasma ([9], [19]). The basic properties of these
waves are briefly described in this section. Hydromagnetic waves in the ho-
mogeneous case will be analyzed in a way convenient for generalization on
inhomogeneous media.
In a homogeneous medium ∇ ρ 0 , ∇ c A , and ∇ c s vanish and (4.22) - (4.23)

reduce to

(1 + β ) ,

ξ ⊥ + β ∂ ξ ||
∂z

L ξ ⊥ = − ∇ ⊥ ∇ · (4.24)

∂
∂z ∇ ·

L s ξ || = − ξ ⊥ , (4.25)

where β = c s /c 2 A .