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(a) Electron ACF (
α
=0
◦
)
Simulation
Brownian
(b) Electron ACF (
α
=0
.
01
◦
)
Simulation
Brownian
No-collisions
(c) Electron ACF (
α
=0
.
05
◦
)
Simulation
Brownian
No-collisions
1
1
1
0.8
0.8
0.8
0.6
0.6
0.6
0.4
0.4
0.4
0.2
0.2
0.2
0
0
0
0
50
100
150
200
250
300
0
50
100
150
200
250
300
0
10
20
30
40
50
60
Time [ms]
Time [ms]
Time [ms]
(d) Electron ACF (
α
=0
.
1
◦
)
Simulation
Brownian
No-collisions
(e) Electron ACF (
α
=0
.
5
◦
)
Simulation
Brownian
No-collisions
(f) Electron ACF (
α
=1
◦
)
Simulation
Brownian
No-collisions
1
1
1
0.8
0.8
0.8
0.6
0.6
0.6
0.4
0.4
0.4
0.2
0.2
0.2
0
0
0
0
5
10
15
20
25
0
0.5
1
1.5
2
2.5
0
0.2
0.4
0.6
0.8
1
Time [ms]
Time [ms]
Time [ms]
0
◦
, (b)
λ
B
=
α
=
Fig. 8. Electron ACF's for
3 m at different magnetic aspect angles: (a)
0.01
◦
, (c)
0.05
◦
, (d)
0.1
◦
, (e)
0.5
◦
, and (f)
1
◦
. Note the different time
α
=
α
=
α
=
α
=
α
=
scales used in each plot (from Milla & Kudeki, 2011).
r
2
Δ
⊥
Fitting the simulated
to match (46) and (47) we find
,
ν
≈
ν
e
/
i
(52)
and
ν
⊥
≈ ν
e
/
i
+
ν
e
/
e
,
(53)
where
N
e
e
4
ln
Λ
e
ν
e
/
e
=
(54)
3/2
o
m
e
C
e
12
π
and
√
2
N
e
e
4
ν
e
/
i
=
√
2
ln
Λ
e
ν
e
/
e
=
(55)
o
m
e
C
e
3/2
12
π
are the Spitzer electron-electron and electron-ion collision frequencies.
However,
the
e
j
k
·
Δ
r
e
Brownian ACF model (45) fails to fit the electron ACF's
computed with simulated
trajectories as shown in Figure 8 for a range of magnetic aspect angles and
3 m. The blue
curves correspond to the ACF's calculated with the Fokker-Planck model (simulations), while
the green curves are the electron ACF's calculated using expression (45) together with our
approximations for
λ
B
=
. Additionally, the electron ACF's for a collisionless magnetized
plasma are also plotted (red curves).
ν
and
ν
⊥
We can see that the Fokker-Planck and the Brownian
0
◦
, and also that the agreement is still good at
very small magnetic aspect angles (see panels a, b, and c). However, substantial differences
between the Brownian and estimated ACF's become evident as the magnetic aspect angle
increases (see panels d, e, and f).
α
=
ACF's matched almost perfectly at
