Geology Reference
In-Depth Information
Fig. 4.25
Examples of zonal, sectoral, and tesseral surface harmonics, generated using the software utility
sph
by Phil
McFadden, which can be downloaded at:
http://www.ngdc.noaa.gov/geomag/geom_util/sph.shtml
the
average
magnetic energy
density
over
a
They are (Lowes
1966
):
spherical surface of radius
a
. This is given by:
I
1
4
r
Y
n
.™;¥/
r
Y
s
.™;¥/dS
I
2
0
ǝ
B
2
Ǜ
S.a/
D
1
1
8 a
2
0
B .r/
B .r/dS
S.1/
S.a/
n.n
C
1/
2n
C
1
•
ns
•
mr
I
D
(4.110)
1
8 a
2
0
D
r
V.r/
r
V.r/dS
We shall use these relations to evaluate the
integral (
4.106
). The objective is to find an ex-
pression for the average magnetic energy density
at distance
a
as a function of the harmonic coef-
ficients. Let us first evaluate the derivatives of the
gradient on a sphere of radius
a
. We can rewrite
(
4.73
) as follows:
S.a/
(4.106)
To calculate the integral at the right-hand side
of (
4.106
), we must rewrite the general solution
(
4.93
) in terms of surface harmonics. We first set:
(
P
n
.cos ™/cos m¥
I
m
0
P
jm
n
.cos ™/sin
j
m
j
¥
I
m<0
(4.107)
Y
n
.™;¥/
D
@V
@r
r
C
1
r
@V
@™
™
C
1
r sin ™
@V
@¥
¥
r
V
D
(
g
n
I
m
0
h
jm
n
I
m<0
@V
@r
r
Cr
s
V
(4.111)
v
n
.™;¥/
D
(4.108)
Then, using (
4.109
)wehave:
LJ
LJ
LJ
LJ
rDa
D
a.n
C
1/
X
n;m
LJ
LJ
LJ
LJ
rDa
Then, the potential can be rewritten as follows:
@V
@r
a
nC1
1
r
nC2
V.r;™;¥/
v
n
Y
n
D
.n
C
1/
X
n;m
a
r
nC1
D
a
X
nD1
X
Cn
v
n
Y
n
v
n
Y
n
.™;¥/
I
r
a
(4.112)
mDn
LJ
LJ
LJ
LJ
rDa
D
X
LJ
LJ
LJ
LJ
rDa
@Y
n
@™
1
r
@V
@™
1
r
nC1
(4.109)
a
nC1
v
n
n;m
The functions
Y
n
(™, ¥) satisfy not only the
conditions (
4.96
) but are also subject to additional
conditions of orthogonality for the gradients on
the unit sphere.
D
X
n;m
@Y
n
@™
v
n
(4.113)
