Instabilities and Structure in the High-Latitude Ionosphere - The Earth's Ionosphere: Plasma Physics and Electrodynamics - page 501

Substituting for V jy in the second equation and substituting into the first yields
an expression involving only V jx :

1

q j B 2
M j ν

= −

k B T j
M j ν jn

1

n ∂

n q j

M j ν jn

V jx +

x +

E x

2
jn

∂

q j B 2
M j ν

q j Bk B T j
M j ν

1

n ∂

n
∂

−

y +

E y

2
jn

2
jn

Similarly, we can derive

1

q j B 2
M j ν

= −

k B T j
M j ν jn

1

n ∂

n q j

M j ν jn

V jy +

y +

E y

2
jn

∂

q j B 2
M j ν

q j Bk B T j
M j ν

1

n ∂

n

−

x +

E x

2
jn

2
jn

∂

Remembering that the E × B and the diamagnetic drift velocities can be written

k B T j
q j B 2 n ∇

B 2 V Ej = E × B / , V Dj =− n × B

the equations for the perpendicular velocity can be recombined, giving

1

q j B 2
M j ν

k B T j
M j ν jn n ∇ ⊥

q j
M j ν jn

V j ⊥ + =− n + E

⊥

2
jn

q j B 2
M j ν

q j B 2
M j ν

+ V D + V E

2
jn

2
jn

Defining the perpendicular diffusion coefficient for motion antiparallel to a
density gradient by an expression of the form V =− D ( ∇ n / n ) yields,

M j ν jn 1

jn

2
j

2

D j ⊥ = k B T j / + /ν (10.14a)

j >> ν jn , and this expression becomes

Now for F-region altitudes

M j

j

D j ⊥ = k B T j /

2

ν jn /