Partially Ionized Plasma - Hydromagnetic Waves in the Magnetosphere and the Ionosphere - page 13

with

ν ne ν ni

a =

iω + ν n ,

b =

iω + ν n ,

n = ν ne + ν ni ,

− −

m e
m n

N
N n ν en ,

m i
m n

N
N n ν in .

ν ne = ni =

Substituting (1.60) into (1.44) and (1.45), we obtain

e
m e E =(

e
m e c [ v e ×

− − iω + ν e − aν en ) v e + B 0 ] − ( ν ei + bν en ) v i , (1.61)

e
m i E =

e
m i c [ v i ×

−

( ν ie + aν in ) v e −

− iω + ν i − bν in ) v i ,

B 0 ]+( (1.62)

where ν i = ν ie + ν in ,ν ie =( m e /m i ) ν ei . Equations (1.61) and (1.62) in
Cartesian coordinates with the z -axis pointing along B 0 ( B 0 = B 0 z ,

for definiteness, B 0 > 0), become

{ x, y, z }

e
m e E =(

− − iω + ν e − aν en ) v e + ω ce Sv e −

( ν ei + bν en ) v i , (1.63)

e
m i E =

− ( ν ie + aν in ) v e − ω ci Sv i +( − iω + ν i − bν in ) v i , (1.64)

where the electron and ion cyclotron frequencies are

eB 0
m e c ,

ω ci = eB 0

ω ce =

m i c ,

and

⎛ ⎞ ⎛ ⎞

v e,i = v e,i⊥
v e,i

=

010 v e,i x

v e,i y
v e,i z

⎝ ⎠ , ⎝ ⎠ .

S = − 100

001

The easiest way to solve (1.63) and (1.64) is turn to the basis of circle
polarizations

v ( ± )
e

= v ex ± iv ey , e = v ez ,

v ( ± )
i

= v ix ±

iv iy , i = v iz ,

E ( ± ) = E x ±

iE y , E = E z .

Then (1.63) and (1.64) for the transversal components yield

e
m e E ( ± ) ,

( ν ei + bν en ) v ( ± )

aν en ] v ( ± ) [ − i ( ω ± ω ce )+ ν e − − = − (1.65)

e i

e
m i E ( ± ) .

bν in ] v ( ± )

( ν ie + aν in ) v ( ± )

− +[ − i ( ω ∓ ω ci )+ ν i − = (1.66)

e i