Fundamentals of Plasma Physics

142 Chapter 4. Elementary plasma waves

Combining Eqs.(4.103) and (4.101) and then integrating in time gives

n 1

n

=

Bz 1

B

(4.104)

which shows that compression/rarefaction is associated with having finiteBz 1.
In summary, there are two general kinds of behavior:

1. Modes with incompressible behavior;these are theshear modesand haven 1 =0,
   ∇·uσ⊥ 1 =0,E 1 ⊥=−∇⊥φandBz 1 =0,


2. Modes with compressible behavior;these are thecompressible modesand haven 1 =
   0 ,∇·uσ⊥ 1 =0,∇×E 1 ⊥=0,andBz 1 =0.
   Equation (4.99) provides a relationship between the perpendicular electric field and the
   perpendicular current. A relationship between the parallel electric field and the parallel
   current is now required. To obtain this, all vectors are decomposed into components par-
   allel and perpendicular to the equilibrium magnetic field, i.e.,E 1 =E⊥ 1 +Ez 1 zˆetc. The
   ∇operator is similarly decomposed into components parallel to and perpendicular to the
   equilibrium magnetic field, i.e.,∇=∇⊥+ˆz∂/∂zand all quantities are assumed to be
   proportional tof(x,y)exp(ikzz−iωt).Thus, Faraday’s law can be written as

∇⊥×E⊥ 1 +∇⊥×Ez 1 zˆ+ˆz

∂

∂z

×E⊥ 1 =−

∂

∂t

(B⊥ 1 +Bz 1 ˆz) (4.105)

which has a parallel component

ˆz·∇⊥×E⊥ 1 =iωBz 1 (4.106)

and a perpendicular component

(∇⊥Ez 1 −ikzE⊥ 1 )×zˆ=iωB⊥ 1. (4.107)

Similarly Ampere’s law can be decomposed into

ˆz·∇⊥×B⊥ 1 =μ 0 Jz 1 (4.108)

and

(∇⊥Bz 1 −ikzB⊥ 1 )×zˆ=μ 0 J⊥ 1. (4.109)

Substituting Eq.(4.99) into Eq.(4.109) gives

(∇⊥Bz 1 −ikzB⊥ 1 )×zˆ=−

iω

vA^2

E⊥ 1 −

μ 0 ∇P 1 ×zˆ

B

(4.110)

or, after re-arrangement,

∇⊥

(

Bz 1 +

μ 0 P⊥ 1

B

)

׈z−ikzB⊥ 1 ׈z=−

iω

v^2 A

E⊥ 1. (4.111)

The slow (shear) and fast (compressional modes) are now considered separately.