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(a) Ion ACF (
λ
B
=3m)
α
=0
.
0
◦
α
=0
.
5
◦
α
=1
.
0
◦
α
=90
.
0
◦
Brownian
(b) Ion ACF (
λ
B
=0
.
3m)
α
=0
.
0
◦
α
=0
.
5
◦
α
=1
.
0
◦
α
=90
.
0
◦
Brownian
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
0
0.5
1
1.5
2
2.5
3
0
0.05
0.1
0.15
0.2
0.25
0.3
Time [ms]
Time [ms]
Fig. 6. Simulated single-ion ACF's at different magnetic aspect angles
α
for two radar Bragg
wavelengths: (a)
0.3 m. The simulation results (color lines) are
compared to theoretical ACF's computed using expression (51) of the Brownian-motion
approximation (dashed lines). Note that there is effectively no dependence on aspect angle
λ
B
=
3 m and (b)
λ
B
=
α
(from Milla & Kudeki, 2011).
where, assuming a Brownian-motion process with distinct friction coefficients
ν
and
ν
⊥
in
the directions parallel and perpendicular to
B
, the mean square displacements will vary as
e
−ν
τ
2
C
2
ν
r
2
Δ
=
ν
τ
−
1
+
(46)
2
and
2
C
2
2
cos
2
γ
)
Δ
r
2
e
−ν
⊥
τ
cos
⊥
=
(
2
γ
)+
ν
⊥
τ
−
(
Ω
τ
−
(47)
2
ν
⊥
+
Ω
≡
√
KT
/
m
and
tan
−
1
in which
qB
/
m
are, respectively, the thermal
speed and gyrofrequency of the particles. Furthermore, simulated
γ
≡
(
ν
⊥
/
Ω
)
, and
C
Ω
≡
r
2
Δ
⊥
match (46) and (47)
,
with
ν
⊥
≈
ν
≈
ν
i
/
i
,
(48)
where
N
e
e
4
ln
Λ
i
ν
i
/
i
=
(49)
o
m
i
C
i
12
π
3/2
is the Spitzer ion-ion collision frequency given by Callen (2006) and Milla & Kudeki (2011).
The simulations also indicate, in the case of oxygen ions,
r
2
r
2
C
i
τ
2
Δ
≈
Δ
⊥
≈
(50)
ν
τ
ν
⊥
τ <
Ω
i
τ
for short time delays
1 and
1, in consistency with (46) and (47). Hence
(45) simplifies to
1
2
k
2
C
i
τ
e
j
k
·
Δ
r
i
e
−
2
. (51)
Evidently, the single-oxygen-ion ACF's are essentially the same as in collisionless and
non-magnetized plasmas because (a) the ions move by many Bragg wavelengths
≈
λ
B
=
2
π
/
k