Geology Reference
In-Depth Information
In the case of a time-averaged geomagnetic field
(see Sect.
4.3
), the series (
4.93
) is assumed to be
symmetric with respect to the spin axis, so that
the potential does not depend anymore from lon-
gitude. This is made by setting
m
D
0in(
4.93
),
so that at the Earth's surface we have simply:
V.™/
D
a
X
nD1
g
n
P
n
.cos ™/
(4.97)
Fig. 4.23
Positive (
gray
) and negative (
white
) patches for
some surface harmonics
A Legendre polynomial
P
n
has
n
zeroes be-
tween ™
D
0and™
D
. These special surface
harmonics, which depend only from colatitude,
are called
zonal harmonics
. In the general case,
a surface harmonic
Y
n
(™,
¥
) vanishes along
n
-
m
parallels of latitude, corresponding to the zeroes
of the
m
-th order derivative of
P
n
(™), and it also
has
m
zeroes at the poles, where the terms (1 -
x
2
)
m
/2
than the lower ones. Therefore, we can remove
the high-degree terms of a spherical harmonic ex-
pansion to emphasize long-wavelength features
of the field associated with core processes or, al-
ternatively, we could cut off low-degree terms to
enhance short-wavelength features of the crustal
field. To find a precise relationship between har-
monic degree and wavelength, let us consider a
small patch of a sphere with radius
a
and a local
Cartesian coordinate system (Ÿ,§,—)orientedas
in Fig.
4.21
. On the sphere, the surface harmonic
Y
n
(™, ¥) satisfies the equation:
D
sin
m
™ vanish. Regarding the dependency
from longitude, both cos
m
¥ and sin
m
¥ have 2
m
zeroes between 0 and 2 . Finally, for
n
D
m
the function
Y
n
(™, ¥) has only
n
zeroes at the
poles and 2
n
zeroes in longitude. The parallels of
latitude and the meridians along which a surface
harmonic
Y
n
(™,
¥
)
D
0 divide the spherical sur-
face of radius
r
into a series of
tesserae
where
the values of the function have alternate signs.
Therefore, when
m
>0and
n
>
m
these functions
are also called
tesseral harmonics
. Finally, for
n
D
m
the function
Y
n
(™, ¥) divides the spherical
surface of radius
r
into a series of sectors bounded
by meridians, thereby, we say that
Y
n
(™, ¥)isa
sectoral harmonic
. Figure
4.23
shows some ex-
amples of zonal, sectoral, and tesseral harmonics.
A key concept in spherical harmonic analy-
sisisthatof
harmonic wavelength
. In standard
Fourier's analysis, the relative contribution of
the sines and cosines to the series is determined
by their amplitude and wavelength, and a given
set of terms in the series can be related to a
specific physical phenomenon. A similar feature
characterizes spherical harmonics.
In particular, if we consider the surface har-
monics as waves on the surface of a sphere of
radius
a
, it is possible to define a wavelength œ
as the distance between two successive peaks or
zeroes of
Y
n
(™, ¥) and it is quite intuitive that
higher degree harmonics have shorter wavelength
sin ™
@Y
n
@™
1
sin ™
@
@™
1
sin
2
™
@
2
Y
n
@¥
2
C
D
n.n
C
1/Y
n
.™;¥/
(4.98)
We want to transform this equation to local
Cartesian coordinates (Ÿ,§,—). To this purpose, we
note that for any function
f
D
f
(™,¥):
@
2
f
@™
2
D
a
2
@
2
f
@f
@™
D
@f
@Ÿ
@Ÿ
@™
D
a
@f
@Ÿ
I
@Ÿ
2
@
2
f
@¥
2
D
a
2
sin
2
™
@
2
f
@f
@¥
D
@f
@§
@§
@™
D
a sin™
@f
@§
I
@§
2
Furthermore, we observe that the variations of
Y
n
(™, ¥) with colatitude occur on a much shorter
scale than sin™. Therefore, the term sin™ at the
left-hand side of (
4.98
) can be considered as
approximately constant. Consequently, Eq. (
4.98
)
can be simplified as follows:
@
2
Y
n
@
2
Y
n
1
sin
2
™
@¥
2
Š
n.n
C
1/Y
n
.™;¥/
(4.99)
@™
2
C
