Equatorial Plasma Instabilities and Mesospheric Turbulence - The Earth's Ionosphere: Plasma Physics and Electrodynamics - page 188

In this derivation we have chosen to follow tradition and, unlike in Chapter 2,
write all gyrofrequencies as positive numbers (i.e., m ). In the z compo-

nent of the momentum equation, the zero-order electric field and electron drift
terms cancel in the equilibrium state about which we are making our perturba-
tions. For the terms in the x direction we have

e =+ eB /

0 = ( − eik / m )δφ + e δ V e z + ik ( k B T e / m )(δ n / n ) − ν e δ V e z

or, equivalently,

ik ( ) − k B T e m (δ ) − δ V e z + ν δ V e x =− e δφ/ m / n / n (4.38c)

e e

where again the zero-order terms cancel. The corresponding ion equation derived
from continuity is

i ω(δ n / n ) − ik δ V i x = 0 (4.38d)

The ion momentum equation yields

ik ( ) + k B T i / M (δ )

( i ω + ν i ) δ V i x = e δφ/ M n / n (4.38e)

where we have used i ν i = ν in . Since the ions are assumed to be at rest to

zero order, the zero-order polarization field is not included in the ion equation of
motion. Equations (4.38a)-(4.38e) constitute five equations in the five unknowns
δ

V e x , δ V e z , δ V i x , δφ , and δ n / n . The resulting determinant of coefficients is then

set to zero.

V D − ω/

k

1 0 0 0

− e − ν e 0 0 0

ik k B T e / m

ν e − e 0 ike / m − = 0 (4.39a)

0 0 − ik 0 i ω

ik k B T e / M 0 0 ( i ω + ν i ) − ike / M −

This can be evaluated in a straightforward manner to find the relationship
between ω and k (e.g., Sudan et al., 1973),

ik 2 C s

ω −

kV D = ( − 0 /ν i )

ω ( i ω + ν i ) − (4.39b)

i and C s

where 0 = ν ν i / = k B ( T e + T i )/ M . If we now set ω = ω − i γ

e e r

and require γ ω r and ν i , we find the real and imaginary parts of ω to be

ω r = kV D /( 1 + 0 ) (4.40a)

k 2 C s

2
r

γ = ( 0 /ν i ) ω − ( 1 + 0 ) (4.40b)