Geoscience Reference
In-Depth Information
and a system for
b
n
and
c
n
Q
n
Q
m
)
c
2
A
(
x, z
)
b
m
−
ω
2
m
(
x
))
Q
n
Q
m
c
m
,
b
n
=
(
ω
2
−
−
c
4
A
(
x, z
)
δ
mn
−
b
m
−
Q
n
Q
m
c
2
A
(
x, z
)
c
m
.
k
y
c
n
=
−
ω
n
(
x
)
Q
n
Q
m
(6.12)
ω
2
−
In a simple model of a one-dimensional plasma box (
c
A
=
c
A
(
x
)), the dif-
ferent harmonics of a field line (i.e., different Fourier components) do not inter-
act with each other. Therefore, the infinite system of coupled equations splits
into independent equations (5.26)-(5.28). In a 2D case, i.e.
c
A
=
c
A
(
x, z
), how-
ever, harmonics interact and an infinite system of linked equations is obtained
.
Let us assume that the condition of an Alfven resonance is satisfied at
x
=
x
0
, i.e., for some harmonic with number
s
, the function
ω
2
ω
s
(
x
)has
a simple zero at
x
0
. Then the point
x
0
is a regular singularity of the system
(6.12). Let us multiply (6.12) by
x
−
−
x
0
and rewrite the obtained equations in
the matrix form:
u
(
x
)=
u
1
(
x
)
u
2
(
x
)
,
x
0
)
u
(
x
)=
T
(
x
)
u
(
x
)
,
(
x
−
T
=
(
x
.
−
x
0
)
R
(
x
−
x
0
)
S
(6.13)
(
x
−
x
0
)
R
Q
u
1
(
x
)and
u
2
(
x
) are column vector functions with elements
b
1
(
x
)
,b
2
(
x
)
, ...
and
c
1
(
x
)
,c
2
(
x
)
,...
, respectively;
T
(
x
) denotes a matrix function of
x
which
is regular at
x
−
x
0
;and
Q
,
R
,
and
S
are matrices with elements
x
0
)
δ
nm
−
,
k
y
Q
nm
=(
x
−
ω
n
(
x
)
Q
n
Q
m
ω
2
−
Q
n
Q
m
c
2
A
,
R
nm
=
−
ω
2
m
(
x
))
Q
n
Q
m
.
(
ω
2
S
nm
=
−
−
c
4
A
We expand matrix
T
(
x
) into Taylor series in the vicinity of the point
x
0
x
0
)
2
T
2
+
T
(
x
)=
T
0
+(
x
−
x
0
)
T
1
+(
x
−
···
.
Here
T
0
=
T
(
x
0
)=
00
Q
0
0
,
T
1
=
R
0
S
0
Q
1
R
0
,...
;
0
is a zero matrix,
Q
0
=
Q
(
x
0
)
(
Q
0
)
nm
=
q
m
δ
ns
,
Search WWH ::
Custom Search