Geology Reference
In-Depth Information
considerable variability when
m
varies from
0to
n
, ranging between [2/(2
n
C
1)]
1/2
and
[2(2
n
)!/(2
n
C
1)]
1/2
. Consequently, if we build
a general solution to Laplace's equation by
superposition of Legendre polynomials, the
coefficients of the series have a wide range of
values, depending on
n
and
m
, to compensate
the variability of the functions
P
nm
.However,
an expansion in series of Legendre polynomials
would be more informative if the magnitude of
the expansion coefficients reflected the relative
importance of the corresponding polynomial
terms. Therefore, in geomagnetism the Ferrers
normalization is substituted by a different
criterion, which is called
Schmidt quasi
-
normalization
. The new polynomials have the
form:
the reference sphere is conventionally chosen to
have a radius equal to the Earth's mean radius,
R
e
Š
6,371 km. Furthermore, the radial distances
are expressed in units of
a
, and the whole series
is multiplied by the radius
a
, in order to have the
coefficients expressed in tesla:
a
r
nC1
V.r;™;¥/
D
a
X
nD1
n
X
g
n
cosm¥
mD0
C
h
n
sin m¥
P
n
.cos™/
I
r
a
(4.93)
The terms within the square brackets, which
depend only from longitude ¥, resemble the usual
Fourier's harmonic series. Multiplied by the Leg-
endre polynomials, they are known as
surface
spherical harmonics
:
s
2
.n
m/Š
cos m¥
sin m¥
P
n
.cos™/ (4.94)
P
n
.x/
.n
C
m/Š
P
nm
.x/
(4.90)
Y
n
.™;¥/
D
To obtain the general solution to Laplace's
equation we must first combine particular solu-
tions to Eqs. (
4.78
)and(
4.82
) with the normal-
ized Legendre polynomials. Then, the general
solution will be a linear combination of particular
solutions for any value of
n
and
m
. A particular
solution for the external field is:
V
e
.r;™;¥/
D
r
n
sin m
¥
cosm¥
These functions are the spherical equivalent of
sines and cosines of the more familiar Fourier's
harmonic series. It is easy to prove that their rms
magnitude over a sphere of radius
r
is indepen-
dent from the order
m
.
Let
dS
be a surface element on the sphere of
radius
r
:
P
n
.cos™/ (4.91)
dS
D
r
2
sin ™d™d¥
(4.95)
Then, the surface spherical harmonics satisfy
the following orthogonality condition:
Similarly, for the internal field it results:
V
i
.r;™;¥/
D
r
.nC1/
sin m¥
cos m¥
P
n
.cos™/
(4.92)
Z
Z
2
1
4
Y
n
.™;¥/Y
s
.™;¥/sin ™d™
d¥
0
0
From here on, we shall focus on the potential
associated with internal sources, which gives the
main contribution to the observed field. This
potential will be called simply
V
.Whentakinga
linear combination of particular solutions (
4.92
),
we must take into account that the solution is
appropriate only where the potential is harmonic.
Therefore, usually the potential is represented
outside a reference sphere of minimum radius,
a
,
that encloses all the sources. In geomagnetism,
1
2n
C
1
•
ns
•
mr
D
(4.96)
Therefore, the magnitude of the coefficients
g
and
h
in the harmonic expansion (
4.93
) measures
the strength of the corresponding terms in the
series according to the degree. These coefficients
are called
Gauss coefficients
, in recognition of the
great contribution of this scientist to the devel-
opment of the spherical harmonic representation.
