Geoscience Reference
In-Depth Information
The corresponding single particle ACF is
k
2
C
2
ν
e
−ντ
)
,
e
j
k
·
Δ
r
e
−
(
ντ−
1
+
=
(24)
2
having the asymptotic limits (22) as well as
k
2
C
2
ν
e
j
k
·
Δ
r
e
−
τ
=
(25)
ν
ν
ν
for
kC
and
kC
, respectively.
Note that when (25) is applicable, with
kC
,
k
in between
successive collisions. In Coulomb interactions, the time
ν
−
1
between “effective collisions” (an
accumulated effect of interactions with many collision partners via their microscopic Coulomb
fields) can be interpreted as the time interval over which the particle velocity vector rotates by
about 90
◦
.
2
an average particle moves across only a small fraction of a wavelength
Binary collisions of charge carriers with neutral atoms and molecules — dominant in the
lower ionosphere — can be modeled as a Poisson process (Milla & Kudeki, 2009) and treated
kinetically using the BGK collision operator (e.g., Dougherty & Farley, 1963). As shown in
Milla & Kudeki (2009), in binary collisions with neutrals the mean-squared displacement of
charge carriers is still given by (23), but the relevant pdf
f
(Δ
)
r
is a Gaussian only for short
ντ
ντ
and long delays
's the
ACF of a collisional plasma dominated by binary collisions will then deviate from (24) and
as a result collisional spectra will in general exhibit minor differences between binary and
Coulomb collisions except in
τ
satisfying
1 and
1, respectively.
At intermediate
τ
ν
ν
kC
and
kC
limits (Hagfors & Brockelman, 1971; Milla
& Kudeki, 2009).
As the above discussion implies, the single particle ACF in the high collision limit (
kC
)
is insensitive to the distinctions between Coulomb and binary collisions and obeys a simple
relation (25). In that limit it is fairly straightforward to evaluate the corresponding Gordeyev
integrals analytically, and obtain (via the general framework equations) a Lorentzian shaped
electron density spectrum (mainly the “ion-line”),
ν
2
2
k
2
D
i
|
n
e
(
k
,
ω
)
|
≈
2
,
(26)
N
o
2
+(
2
k
2
D
i
)
ω
C
i
/
ν
i
=
valid for
kh
KT
i
/
m
i
ν
i
denotes the ion diffusion coefficient in the collisional plasma. This result is pertinent to
D-region incoherent scatter observations (see Figure 2) neglecting possible complications due
to the presence of negative ions (e.g., Mathews, 1984). Also, from (26) it follows that
1 (wavelength larger than Debye length), where
D
i
≡
ˆ
∞
d
2
N
o
2
2
2
|
(
k
)
|
≡
π
|
(
k
,
ω
)
|
=
n
e
n
e
,
(27)
−
∞
which is in fact true in general — i.e., for all types of plasmas with or without collisions and/or
DC magnetic field — so long as
T
e
=
1. In view of radar equation (10), this result
leads to a well-known volumetric radar cross-section (RCS) formula
T
i
and
kh
r
e
|
2
r
e
N
o
π
(
k
)
|
=
π
4
n
e
2
(28)
for ISR's that is valid under the same conditions as (27). Hence, RCS measurements with ISR's
can provide us with ionospheric mean densities
N
o
.
