Geoscience Reference
In-Depth Information
where
Φ
0
(
x
) is the phase in the source at
t
=0
,z
=0,
ω
c
A
(
x
)
k
A
(
x
)=
is the Alfven wavenumber. At distance
x
from the source, the phase of the
Alfven wave is
Φ
(
x, y, t
)=
−
ωt
+
Φ
0
(
x
)+
k
A
(
x
)
z.
The transverse wavenumber
k
x
(
x
)=
∂Φ
d
dx
ln
c
A
(
x
)
∂x
=
k
(0)
−
k
A
(
x
)
z
(4.50)
x
varies in proportion to
z
. Here
k
(0)
=
∂Φ
0
(
x
)
/∂x
.
x
10
8
cm/s and the scale-size of Alfven velocity
change is about Earth's radius
R
E
≈
In the magnetosphere,
c
A
∼
10
8
cm. As is known, the finiteness of
the Larmour radius and the electron inertia cause the transversal dispersion of
Alfven waves. For the ion temperature
T
i
≈
6
×
0
.
1 eV, the ion thermal velocity is
10
5
cm/s. The ion cyclotron frequency
ω
ci
in the magnetosphere is
ω
ci
∼
10 Hz
.
Then the Alfven wavenumber for which the transversal dispersion is
essential, can be estimated as
k
A
=
ω
ci
/c
A
≈
v
Ti
≈
10
−
9
cm
−
1
. Then the transverse
wavenumber varies as
10
−
9
R
E
10
−
18
z.
∆k
x
≈
z
≈
2
×
10
9
cm be the length of a mid-latitudinal field-line. Then
∆k
x
∼
Let
z
=5
×
10
9
cm
−
1
=10
−
8
cm
−
1
=10
−
3
km
−
1
and the scale size of the
phase mixing is
L
⊥
≈
10
−
18
2
×
×
5
×
10
3
km.
The transversal dispersion near the equatorial plane of the magnetosphere
becomes significant at transversal scales of about the proton cyclotron radius,
i.e.
10
6
cm. The dispersion scale is determined by the electron inertial length
λ
e
which is estimated as a ratio of the light velocity
c
to the electron plasma
frequency
ω
pe
.
Near the ionosphere
λ
e
=
c/ω
0
∼
∼
10
4
cm. The dissipation
caused by the longitudinal conductivity becomes significant at scales less than
10
5
cm. Thus, the phase mixing is unimportant at middle latitudes. However,
at high latitudes, especially for field-lines going to the geomagnetic tail, the
phase mixing can become important.
If the properties of the medium vary in the direction of plasma displace-
ment, then eliminating
b
from (4.46) and (4.47), we have a 2D wave equation
for displacements
ξ
x
∂
2
ξ
x
∂x
2
+
∂
2
ξ
x
∂
2
ξ
x
∂t
2
1
c
2
A
∂z
2
−
=0
.
(4.51)
It is evident that a variation of
c
A
in the plane of polarization does not produce
phase mixing. Recall that dependence on
y
is determined by the choice of the
source and is not connected to the peculiarities of propagation.
Search WWH ::
Custom Search