Instabilities and Structure in the High-Latitude Ionosphere - The Earth's Ionosphere: Plasma Physics and Electrodynamics - page 525

that arise at high latitudes. By “local” in this context we mean that we ignore
coupling along the magnetic field lines. In the following discussion, we follow
the presentation by Fejer and Providakes (1988) quite closely.
In the lower E region (between about 90 and 100 km), the ions are unmagne-
tized (

ν i << i ), and the electrons are collisionless to zero order ( e >> ν e ), so

B 2 . Linear theory then is identical to the result in Chapter 4, and
the oscillation frequency and growth rate, in the reference frame of the neutral
wind, are given by

V D = E × B /

ω = · (

V D + V Di ) / (

+ )

k 1 (10.20)

r

+ ) − 1

ω r −

k 2 C s V Di 2

γ = ( 1 (/ν i ) k · −

(10.21)

+ 1 Lk 2 ω r − V Di + (ν i / i ) k y −

/ k · 2 α n 0

where
= 0 k 2 k 2

e k 2

k 2

2
e

2

⊥ /

+ /ν

|| /

and

0 = ν ν i / i

e e

In this expression V Di is the ion drift velocity, which could be due either
to neutral winds or to the ambient electric field. The assumptions used are
ω r << ν i , or in other words that the wavelengths are much larger than the ion
mean free path, and | γ | << ω r . In addition, we consider k >> k 0 , where k 0 =

L N ] − 1 , so the linear waves are nondispersive. These approxima-
tions are valid for wavelengths between a few meters and a few hundred meters.
For shorter wavelengths, kinetic theories are needed. The first term in the right-
hand side of (10.21) is the two-stream term, which includes a diffusive damping
term of the form k 2 C s . The second term describes the gradient drift instability,
while the last term is due to recombinational damping. The two-stream term
is dominant at short wavelengths (1m

(ν i / i ) [ ( 1 + )

≤ λ ≤ 20m) and yields instability when

k ). For reference a set of high-latitude parameters was chosen

and the threshold drift velocity for instability evaluated as a function of wave-
length. The results are shown in Fig. 10.31 for k

· V D > kC s ( 1 + ψ

|| =

0 at a height of 105 km.

10 4 s − 1 , 10 3 s − 1 , 10 7 s − 1 ,

The parameters used are ν e = 4 × ν i = 2 . 5 × e =

06 s − 1 . Several different gradient
scale lengths were used. Large-scale waves are easily excited only if the electron
density gradient is destabilizing ( L N is negative). Note also that the two-stream
threshold drift ( L N =∞

180 s − 1 , C s i = = 360m/s, and 2 α n 0 = 0 .

20m.
Therefore, the two-stream instability mechanism is essentially restricted to short
wavelengths.

) increases rapidly with a wavelength for λ ≥