4.3 Low frequency magnetized plasma: Alfvén waves 133
The fields and gradient operator can be written as
E 1 = E⊥ 1
B 1 = B⊥ 1 +Bz 1 zˆ
∇ =ˆz
∂
∂z
+∇⊥ (4.52)
sinceEz 1 =0in the MHD limit as obtained from the linearized ideal Ohm’s law
E 1 +U 1 ×B=0. (4.53)
The curl operators can be expanded as
∇×E 1 =
(
ˆz
∂
∂z
+∇⊥
)
×E⊥ 1
=ˆz×
∂E⊥ 1
∂z
+∇⊥×E⊥ 1 (4.54)
and
(∇×B 1 )⊥ =
((
ˆz
∂
∂z
+∇⊥
)
×(B⊥ 1 +Bz 1 ˆz)
)
⊥
=ˆz×
∂B⊥ 1
∂z
+∇⊥Bz 1 ׈z (4.55)
where it should be noted that both∇⊥×E⊥ 1 and∇⊥×B⊥ 1 are in thezdirection.
Slow or Alfvén mode (mode whereBz 1 =0) In this caseB 1 =B⊥ 1 and Eqs.(4.51)
become
∂E⊥ 1
∂t
= v^2 Azˆ×
∂B⊥ 1
∂z
∂B⊥ 1
∂t
= −zˆ×
∂E⊥ 1
∂z
. (4.56)
This can be re-written as
∂
∂t
(ˆz×E⊥ 1 )= −vA^2
∂B⊥ 1
∂z
∂B⊥ 1
∂t
= −
∂
∂z
(ˆz×E⊥ 1 ) (4.57)
which gives a wave equation in the coupled variablesˆz×E⊥ 1 andB⊥ 1 .Taking a second
time derivative of the bottom equation and then substituting the top equation gives the wave
equation for the slow mode (Alfvén mode),
∂^2 B⊥ 1
∂t^2
=vA^2
∂^2 B⊥ 1
∂z^2
. (4.58)
This is the mode originally derived by Alfven (1943).