Geoscience Reference
In-Depth Information
And since
using Kramer-Kronig
relations (e.g., Yeh & Liu, 1972), a full blown solution of the modeling problem can be
formulated in terms of “single particle correlations”
σ
e
,
i
(
k
,
ω
)
can be uniquely obtained from Re
{
σ
e
,
i
(
k
,
ω
)
}
e
j
k
·
Δ
r
underlying the thermal density
2
2
spectra
|
n
te
(
k
,
ω
)
|
and
|
n
ti
(
k
,
ω
)
|
.
This general formulation is as follows (see Appendix 2 in Kudeki & Milla, 2011, for a detailed
derivation): In terms of a one-sided integral transformation
ˆ
∞
e
−
j
ωτ
e
j
k
·
Δ
r
s
J
s
(
ω
)
≡
d
τ
,
(16)
0
known as
Gordeyev integral
for species
s
(
e
or
i
for the single-ion case), we have
2
|
n
ts
(
k
,
ω
)
|
σ
s
(
k
,
ω
)
1
−
j
ω
s
J
s
(
ω
s
)
k
2
h
s
=
2Re
{
J
s
(
ω
s
)
}
and
=
,
(17)
N
o
j
ω
o
where
ω
s
≡
ω
−
k
·
V
s
is a Doppler
-shifted frequency in the radar frame due to mean velocity
V
s
of species
s
,
h
s
≡
o
KT
s
/
N
o
e
2
is the corresponding Debye length, and the
k
-
spectrum
of electron density fluctuations in the equilibrium plasma is given by (14) or its multi-ion
generalizations.
ω
The “general framework” of ISR spectral models represented by (16)-(17) and (14) (as well as
(10)) takes care of the macrophysics of the incoherent scatter process due to collective effects,
while microphysics details of the process remain to be addressed in the specification of single
particle ACF's
e
j
k
·
Δ
r
.
e
j
k
·
Δ
r
4. Single particle ACF's
for un-magnetized plasmas
We have just seen that ISR spectrum of ionospheric plasmas in thermal equilibrium can be
specified in terms of single particle ACF's
e
j
k
·
Δ
r
e
j
k
·
Δ
r
. In general, an ACF
can be explicitly
(Δ
)
computed if the probability distribution function (pdf)
f
r
, where
Δ
r
is the component of
e
j
k
·
Δ
r
Δ
r
along
k
, is known. Alternatively,
can also be computed directly given an ensemble
Δ
τ
(
Δ
)
Δ
of realizations of
r
data will reflect the dynamics of random particle motions taking place in ionospheric plasmas.
r
for a given time delay
. In either case, pdf's
f
r
or pertinent sets of
When
Δ
r
is a Gaussian random variable with a pdf
r
2
e
−
Δ
r
2
2
Δ
(Δ
)=
2
f
r
,
(18)
π
Δ
r
2
the single-particle ACF
ˆ
1
2
k
2
e
j
k
·
Δ
r
e
jk
Δ
r
f
e
−
Δ
r
2
=
(
Δ
r
)
d
(
Δ
r
)=
(19)
r
2
depends on the mean-square displacement
of the particles. In such cases incoherent
scatter modeling problem reduces to finding the appropriate variance expressions
Δ
r
2
Δ
.
In a non-magnetized and collisionless plasma the charge carriers will move along straight line
(unperturbed) trajectories with random velocities
v
. In that case the displacement vectors will
be
Δ
r
=
v
τ
(20)
