136 Chapter 4. Elementary plasma waves
while the linearized continuity equation
∂ρ 1
∂t
+ρ∇·U 1 =0 (4.70)together with the equation of state
P 1
P
=γρ 1
ρ(4.71)
give
∂P 1
∂t
=−γP∇·U 1. (4.72)
To obtain an equation involvingU 1 only, we take the time derivative of Eq.(4.67) and use
Eqs.(4.69) and (4.72) to eliminate the time derivatives ofP 1 andB 1. This gives
ρ∂^2 U 1
∂t^2= −∇
(
−γP∇·U 1 +1
μ 0B·∇×(U 1 ×B)
)
+
1
μ 0(B·∇)∇×(U 1 ×B).
(4.73)
This can be simplified using the identity∇·(a×b)=b·∇×a−a·∇×bso that
B·∇×(U 1 ×B)=∇·[(U 1 ×B)×B]=−B^2 ∇·U 1 ⊥. (4.74)Furthermore,
B·∇=B∂
∂z=ikzB. (4.75)Using these relations Eq. (4.73) becomes
∂^2 U 1
∂t^2=∇
(
c^2 s∇·U 1 +v^2 A∇·U 1 ⊥)
+ikzvA^2 ∇×(U 1 ×zˆ). (4.76)To proceed further we take either the divergence or the curl of this equation to obtain
expressions for compressional or incompressible motions.
4.3.6 MHD compressional (fast) mode
Here we take the divergence of Eq. (4.76) to obtain
∂^2 ∇·U 1
∂t^2=∇^2
(
c^2 s∇·U 1 +vA^2 ∇·U 1 ⊥)
(4.77)
or
ω^2 ∇·U 1 =
(
k^2 ⊥+k^2 z)(
c^2 s∇·U 1 +vA^2 ∇·U 1 ⊥)
. (4.78)
On the other hand if Eq.(4.76) is operated on with∇⊥=∇−ikzzˆwe obtain∂^2 ∇⊥·U 1
∂t^2=∇^2 ⊥
(
c^2 s∇·U 1 +v^2 A∇·U 1 ⊥)
+k^2 zvA^2 ˆz·∇×(U 1 ⊥ ׈z). (4.79)Using
∇×(U 1 ×zˆ)=ˆz·∇U 1 ⊥−ˆz∇·U 1 =ikzU 1 ⊥−zˆ∇·U 1 (4.80)
Eq.(4.79) becomes
ω^2 ∇⊥·U 1 =k^2 ⊥(
c^2 s∇·U 1 +vA^2 ∇·U 1 ⊥)
+k^2 zv^2 A∇·U 1 ⊥. (4.81)