MHD-Wave Exposure on the Ionosphere - Hydromagnetic Waves in the Magnetosphere and the Ionosphere - page 347

1
1+ σ i /σ e .

j e = j

But

m e
m i

σ i
σ e ∼

m e
m i

ν e
ν i ∼

1 ,

where m i being the average ion mass. Therefore

j ≈ j e .

This allows us to express j in terms of the magnetic pulsations components.
It follows from the Ampere's law that the vertical current density compo-
nent j z is related with the magnetic wave components as

∂b y
∂x −

.

c
4 π

∂b x
∂y

j z = (12.24)

From an equation analogous to (12.23), but written for the total current, we
find

j sin I = j z + σ P E x cot I + σ H E y cos I.
Taking into account that j e ≈ j , we rewrite (12.21)-(12.23)

j xe = j z cot I + σ Pe + σ P cot 2 I E x + σ He + σ H cot 2 I E y ,

σ He
sin I E x + σ Pe E y ,

j ye = −

j ze = j z + σ Pi E x cot I + σ Hi E y cos I. (12.25)

For the electron velocity, we obtain from (12.25)

N e e σ Pe +( σ Pe + σ Pi ]cot 2 I E x

j z cot I
N e e

1

v xe = − −

N e e σ He +( σ He + σ Hi )cot 2 I E y sin I

1

+

1
N e e

σ He
sin I E x −

1
N e e σ Pe E y ,

v ye =

j z
N e e −

1
N e e σ Pi E x cot I

1
N e e σ Hi E y cos I.

v ze = − − (12.26)

Above the E -layer, the electron and ion Pedersen conductivities are small.
Neglecting collisions, we get

c N e e
B 0

σ He = − σ Hi = − .