Geoscience Reference
In-Depth Information
qB
m
is the particle gyrofrequency. The mean-square displacement (32) which is
periodic in
τ
i
s can be derived by invoking circular particle orbits with periods 2
π
/
Ω
and
mean radii
√
2
C
/
Ω
on the plane perpendicular to
B
. As a consequence of (31) and (32), ISR
spectra in a magnetized but collisionless ionosphere can be derived from the single particle
ACF
where
Ω
≡
2
k
2
⊥
C
2
1
2
k
2
C
2
2
sin
2
e
−
τ
e
−
(Ω
τ
/2
)
e
j
k
·
Δ
r
=
×
(33)
2
Ω
for electrons and ions.
Note that the ACF (33) becomes periodic and the associated Gordeyev integrals and spectra
become singular (expressed in terms of Dirac's deltas) in
k
→
0 limit. Spectral singularities
are of course not observed in practice since collisions in a real ionosphere end up limiting the
width of single particle ACF's in
tan
−
1
α
=
(
⊥
)
τ
in the limit of small “aspect angles”
k
/
k
.
Despite the singularities in (33), it turns out that for finite aspect angles
larger than a few
degrees, the collisionless result (33) leads us to the most frequently used ISR spectral model
at F-region heights. This is true because given a finite
k
α
1
2
k
2
C
2
2
, the term
e
−
τ
in (33) restricts
)
−
1
the width of the ACF to a finite value of
∼
(
k
C
even in the absence of collisions (or when
)
−
1
, as well
τ
(
collision frequencies are smaller than
k
C
). It can then be shown that for
k
C
Ω
τ
π
as
2
(easily satisfied by massive ions), the ACF (33) for ions recombines to a simplified
1
2
k
2
C
2
2
form
e
−
τ
as if the plasma were non-magnetized. Also with finite
k
, the ACF (33) for
1
2
k
2
sin
2
C
2
2
electrons simplifies to
e
−
α
τ
Ω
can be
easily invoked to ignore the rightmost exponential in (33) (or even more accurately, replace it
with its average value over
, since for the light electrons a condition
k
C
⊥
k
2
⊥
C
2
Ω
−
τ
, namely, 1
2
). These ion and electron ACF's exhibit similar
τ
dependencies and lead to similar shaped Gordeyev integrals. The resulting ISR spectra are
of the “double humped” type shown in Figure 1b.
6. Modeling the Coulomb collision effects in magnetized plasmas
As we have noted, the form (33) of the single particle ACF indicates that magnetic field effects
in ISR response are confined to small aspect angles, which is also the regime where collision
effects cannot be neglected (e.g., Farley, 1964; Sulzer & González, 1999; Woodman, 1967) given
the non-physical behavior of ACF (33) in
0
◦
limit.
α
→
Historical note:
The need to account for the effects of collisions in incoherent scatter theory of
ionospheric F-region returns was first pointed out by Farley (1964). Based on a qualitative
analysis, Farley recognized that ion Coulomb collisions would be responsible for the lack of
O
+
gyroresonance signatures on incoherent scatter observations carried out at 50 MHz at the
Jicamarca Radio Observatory located near Lima, Peru. This analysis was later verified by
the theoretical work of Woodman (1967) which was based on the simplified Fokker-Planck
collision model of Dougherty (1964). Many years later, after the application of modern
radar and signal processing techniques to the measurement and analysis of ISR signals (e.g.,
Kudeki et al., 1999), Sulzer & González (1999) noted that, in addition to ion collisions, electron
Coulomb collisions also have an influence on the shape of the ISR spectra at small magnetic
aspect angles. Based on a more complex Fokker-Planck Coulomb collision model, Sulzer &
González found that the collisional spectrum is narrower (just like the observations of Kudeki
et al., 1999) than what the collisionless theory predicts and that the effect of electron collisions
extends up to relatively large magnetic aspect angles. Recently, this work has been refined and
