4.3 Low frequency magnetized plasma: Alfvén waves 137Equations (4.78) and (4.81) constitute two coupled equations in the variables∇·U 1
and∇⊥·U 1 ⊥, namely
(
ω^2 −k^2 c^2 s)
∇·U 1 −k^2 v^2 A∇⊥·U 1 ⊥ =0
k⊥^2 c^2 s∇·U 1 +(
k^2 vA^2 −ω^2)
∇⊥·U 1 ⊥ =0. (4.82)
These coupled equations have the determinant
(
ω^2 −k^2 c^2 s)(
k^2 v^2 A−ω^2)
+k^2 v^2 Ak⊥^2 c^2 s=0 (4.83)which can be re-arranged as a fourth order polynomial inω,
ω^4 −ω^2 k^2(
vA^2 +c^2 s)
+k^2 k^2 zvA^2 c^2 s=0 (4.84)having roots
ω^2 =k^2(
vA^2 +c^2 s)
±
√
k^4 (vA^2 +c^2 s)
2
− 4 k^2 kz^2 v^2 Ac^2 s
2. (4.85)
Thus, according to the MHD model, the compressional mode dispersion relation has the
following limiting forms
ω^2 =k^2 ⊥(
v^2 A+c^2 s)
ifkz =0 (4.86)
ω^2 =k^2 zvA^2
or
ω^2 =k^2 zc^2 s
ifk^2 ⊥ =0. (4.87)4.3.7 MHD shear (slow) mode
It is now assumed that∇·U 1 =0and taking the curl of Eq.(4.76) gives
∂^2 ∇×U 1
∂t^2
= v^2 A∇×∇×(
∂U 1
∂z
×zˆ)
= v^2 A∇×
∂U 1
∂z
∇·︸︷︷ˆz︸
zero+ˆz·∇∂U 1
∂z
−zˆ∇·∂U 1
︸ ︷︷∂z︸
zero−
∂U 1
∂z
·∇︸︷︷︸zˆ
zero
= v^2 A∂^2
∂z^2∇×U 1 (4.88)
where the vector identity∇×(F×G)=F∇·G+G·∇F−G∇·F−F·∇Ghas been
used.
Equation (4.88) reduces to the slow wave dispersion relation Eq.(4.63). The associated
spatial behavior is such that∇×U 1 =0,and the mode is unaffected by existence of
finite pressure. The perturbed magnetic field is orthogonal to the equilibrium field, i.e.,
B 1 ·B=0,since it has been assumed that∇·U 1 =0and since finiteB 1 ·Bcorresponds
to finite∇·U 1.