Geoscience Reference
In-Depth Information
Figure 9 shows a surface plot constructed from full IS spectrum calculations for
3m
(e.g., for the 50 MHz Jicamarca ISR system located near Lima, Peru) using the ACF library
constructed with the Monte Carlo procedure for electrons. The underlying electron Gordeyev
integral
J
e
(
ω
)
λ
B
=
2
is presented in Figure 10 where only Re
{
J
e
(
ω
)
}
∝
|
n
te
(
k
B
,
ω
)
|
is displayed.
The plots are displayed as a functions of aspect angle
. In both
figures it can be observed how these spectral functions sharpen significantly at small aspect
angles. In particular in the case of the IS spectrum, we can see that, just in the range between
0.1
◦
and 0
◦
, the amplitude of the spectrum becomes ten times larger while its bandwidth is
reduced by the same factor.
α
and Doppler frequency
ω
/2
π
Some interesting features of the IS spectrum caused by collisions can be pointed out. As
discussed by Milla & Kudeki (2011), in the absence of collisions, the magnetic field restricts
the motion of electrons in the plane perpendicular to
B
, forcing them to gyrate perpetually
around the same magnetic field lines — this would generate infinite correlation time of the IS
signal. With collisions, the electrons manage to diffuse
across
the field lines, and consequently
the correlation time of the IS signal becomes finite. As a result, in the limit of
0
◦
, the
width of the spectrum becomes proportional to the collision frequency. However, at other
magnetic aspect angles, the effects are slightly different. In a few hundredths of a degree from
perpendicular to
B
(
α →
0.01
◦
), the shape of the IS spectrum is dominated by electron diffusion
along
the magnetic field lines. As collisions impede the motion of particles, electrons diffuse
slower in a collisional plasma than in a collisionless one (where electrons move freely), which
implies that the electrons stay closer to their original locations for longer periods of time. As a
result, the correlation time of the signal scattered by the electrons also becomes longer, causing
the broadening of the IS signal ACF and the associated narrowing of the signal spectrum in
this aspect angle regime, as first explained by Sulzer & González (1999).
α
>
Spectrum dependence on electron density
N
e
and temperatures
T
e
and
T
i
has been studied
by Milla & Kudeki (2011). Since at very small aspect angles the electron Gordeyev integral
dominates the shape of the overall incoherent scatter spectrum, Milla & Kudeki (2011) found
that in the limit of
0
◦
the bandwidth of Re
α →
{
(
ω
)
}
J
e
, and therefore the IS spectrum, varies
according to
ν
⊥
k
2
C
e
.
(56)
2
e
⊥
+ Ω
ν
ν
⊥
≈ ν
e
/
i
+
ν
e
/
e
from the last section and taking
ν
⊥
Furthermore, using
Ω
e
, we can verify
that the bandwidth dependence (56) is proportional to
N
e
√
T
e
.
(57)
However, as
increases, in a few hundredths of a degree, the dependance of the IS spectral
width on
N
e
and
T
e
is exchanged, i.e., the bandwidth increases as either the density decreases
or the temperature increases. The reason for this is the exchange of roles between particle
diffusion in the directions across and along the magnetic field lines. It should be mentioned
that collision effects become less significant at even larger aspect angles where the spectrum
is shaped by ion dynamics. In that regime, the spectral shapes become independent of
N
e
as
long as
kh
e
α
1.
The volumetric radar cross section (RCS) pertinent in ISR applications is given by (e.g., Farley,
1966; Milla & Kudeki, 2006)
r
e
N
e
η
(
k
)
≡
σ
v
4
π
(58)
