Geoscience Reference
In-Depth Information
Electron displacement distributions
⊥
to
B
τ
=0
.
0ms
τ
=2
.
0ms
τ
=4
.
0ms
τ
=6
.
0ms
τ
=8
.
0ms
τ
=10
.
0ms
Gaussian
0.5
Electron displacement distributions
⊥
to
B
0.4
0.4
0.3
0.2
0.2
0.1
0
−2
0
10
8
−3
−2
−1
0
1
2
3
0
6
4
Δ
r
⊥
(
τ
)
/σ
⊥
(
τ
)
2
2
0
Δ
r
⊥
(
τ
)
/σ
⊥
(
τ
)
Time delay [ms]
Electron displacement distributions
to
B
Electron displacement distributions
to
B
τ
=0
.
0ms
τ
=2
.
0ms
τ
=4
.
0ms
τ
=6
.
0ms
τ
=8
.
0ms
τ
=10
.
0ms
Gaussian
0.5
0.4
0.4
0.3
0.2
0.2
0
0.1
−2
10
8
0
6
0
4
2
−3
−2
−1
0
1
2
3
2
0
Δ
r
(
τ
)
/σ
(
τ
)
Δ
r
(
τ
)
/σ
(
τ
)
Time delay [ms]
Fig. 7. Same as Figure 5 but for the case of a test electron. All distributions at all time delays
are normalized to unit variance. Note that the distributions of the displacements parallel to
B
become narrower than a Gaussian distribution (from Milla & Kudeki, 2011).
between successive Spitzer collisions, and (b) the ions are unable to return to within
π
of their starting positions after a gyro-period as a consequence of Coulomb collisions. As an
upshot, we will be able to handle the ion terms analytically in spectral calculations.
λ
B
/2
7.2 Statistics of electron displacements
Next, we study the effects of Coulomb collisions on electron trajectories using procedures
similar to those applied to ions. In Figure 7, the displacement distributions resulting
from an electron moving in an O+ plasma are presented. The top and bottom panels in
Figure 7 correspond, respectively, to displacement distributions in perpendicular and parallel
directions.
, while on the the
right, sample cuts of the distributions are compared to a Gaussian pdf. As in the ion case, we
note that the normalized distributions for perpendicular direction to be invariant with
On the left, the distributions are displayed as functions of
τ
and
closely match a Gaussian. However, the distributions of parallel displacements change with
τ
τ
,
and the shapes are distinctly non-Gaussian for intermediate values of
. More specifically, at
very small time delays (lower than the inverse of a collision frequency), the distributions are
Gaussian, but then, in a few “collision” times, the distribution curves become more “spiky”
(positive kurtosis) than a Gaussian. Although, at even longer delays
τ
τ
the distributions once
again relax to a Gaussian shape, it is clear that the electron displacement in the direction
parallel to
B
is not a Gaussian random variable at all time delays
τ
.