Geoscience Reference
In-Depth Information
6
x
10
−4
Δ
r
=
v
τ
5
4
3
r
(
t
)
r
(
t
+
τ
)
2
1
0
−4
−2
0
2
4
ω
/2
π
(KHz)
(a)
(b)
Fig. 1. (a) A cartoon depicting particle displacements
r
in a plasma with straight line charge
carrier trajectories, and (b) a sample ISR spectrum for a non-magnetized and collisionless
plasma in thermal equilibrium.
Δ
over intervals
. Assuming Maxwellian distributed velocity components
v
along wavevector
k
, we then have Gaussian distributed displacements
τ
Δ
r
=
v
τ
with variances
r
2
v
2
2
C
2
2
,
Δ
=
τ
=
τ
(21)
=
√
KT
/
m
is the thermal speed of the charge carrier.
where
C
The corresponding single
particle ACF is in that case
1
2
k
2
C
2
e
j
k
·
Δ
r
e
−
τ
2
, (22)
which leads (via the general framework equations) to the most basic incoherent scatter
spectral model exhibiting double humped shapes as depicted in Figure 6b when (22) is applied
to both electrons and ions (with
C
=
C
e
and
C
i
, respectively).
The ACF (22) is also applicable in collisional plasmas so long as the relevant “collision
frequency”
=
ν
is small compared to the product
kC
, i.e.,
ν
kC
, so that an average particle
k
in between successive collisions. Otherwise, (22)
will only be valid until the “first collisions” take place at
2
moves a distance of many wavelengths
τ
∼
ν
−
1
. At larger
τ
, the mean-square
r
2
displacement
Δ
as well as the pdf
f
(
Δ
r
)
will in general depend on the details of the
dominant collision process.
Long range Coulomb collisions between charged particles (e.g., electrons and ions) are
frequently modeled as a “Brownian motion” process
8
, a procedure which leads (e.g., Kudeki
& Milla, 2011) to a Gaussian
f
(
Δ
r
)
with a variance
2
C
2
ν
Δ
r
2
e
−ντ
)
=
2
(
ντ
−
1
+
.
(23)
8
As discussed in Kudeki & Milla (2011) and here in Section 7, in Brownian motion the position
and velocity increments are Gaussian random variables and correspond to stochastic solutions of a
first-order Langevin update equation with constant coefficients.
