Geoscience Reference
In-Depth Information
x
−
x
+
0.12
∆ S = S
−
− S
+
0.1
0.08
0.06
S
−
S
+
0.04
0.02
0
−5
0
5
x
Fig. 5.5.
Poynting vector discontinuity near the resonance point. The difference
between Poynting vectors at
x
=
x
−
and
x
=
x
+
corresponds to the wave energy
dissipation in the FLR
The imaginary part in (5.42) is
π
sign(
δ
n
)=
sign(
Λ
) (see 5.38). Thus energy
losses on a unit area within the resonance region can be estimated as
−
2
k
y
ω
8
π
|
Λ
n
||
b
0
|
2
.
∆S
x
≈−
|
Q
n
(
z
)
|
(5.43)
k
A
(
x
n
)
The total energy dissipation
W
on the entire resonance magnetic shell is
found by integrating (5.43) over the field line and summing up on all the lines
for which the FLR-condition is satisfied for the set frequencies
ω
, i.e. on the
box cross-section
x
= const. With the normalization requirement (5.11) we
can define the total energy
W
-expending on the plasma heating within the
resonance region as
2
k
y
|
Λ
n
||
b
0
|
ω
8
π
W
≈
l
y
.
(5.44)
k
A
(
x
n
)
At numerical calculations it is convenient to use the expression
8
π
b
(
x
n
)
S
x
(
x
)=
i
ω
2
c
(
x
)
b
∗
(
x
)
|
2
,
|
Q
n
(
z
)
|
(5.45)
2
b
(
x
n
)
|
where
c
(
x
)and
b
(
x
) are obtained from (5.29)-(5.30).
Let us compare the energy loss in the resonance point calculated from
(5.43) and (5.45). Figure 5.5 shows the
x
-component of the time averaged
Poynting vector as a function of coordinate
x
near the resonance point. The
fields are normalized by the longitudinal magnetic field in the resonance point.
Search WWH ::
Custom Search