Fundamentals of Plasma Physics

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).