Geology Reference
In-Depth Information
was used in the non-axial form to approximate
the Earth's magnetic field (Bartels
1936
). It is
usually referred to as the
eccentric dipole field
.
Wilson's analysis of inclination data gave an
offset •
z
D
191
˙
38 km. Most interestingly, Wil-
son (
1970
) proved that the field produced by the
eccentric axial dipole is an axially symmetric
field formed by a GAD field plus a quadrupole
component. For an axially symmetric potential,
we have that the spherical harmonic expansion
At the Earth's surface, the vertical and hori-
zontal components of the corresponding geomag-
netic field are given by:
8
<
2
g
2
3cos
2
™
1
H.™/
D
g
1
sin ™
3g
2
sin ™ cos ™
3
Z.™/
D
2g
1
cos ™
:
(6.67)
Therefore, for an eccentric dipole field the
following more complex expression:
a
r
nC1
V.r;™/
D
a
X
n
g
n
P
n
.cos™/
I
r
a
2
g
2
3cos
2
™
1
g
1
sin ™
C
3g
2
sin ™ cos™
(6.68)
3
2g
1
cos™
C
D
1
Z.™/
H.™/
D
tan I
D
(6.63)
Earth's surface). The vertical and horizontal com-
ponents of the magnetic field associated with this
potential are given by:
8
<
A comparison of this equation with the Wilson
solution (
6.62
) gives an estimate of the relative
importance of the quadrupole term with respect
to the dipole strength:
@V
@r
Z.r;™/
D
B
r
.r;™/
D
.n
C
1/
a
r
nC2
X
g
2
2•
z
a
Š
0:06
˙
0:01
g
n
P
n
.cos™/
D
g
1
D
(6.69)
nD1
:
H.r;™/
D
B
.r;™/
a
r
nC1
X
1
1
a
@V
@™
D
dP
n
d™
where we have used
a
D
6,371 km and
•
z
D
191
˙
38 km. In a later study, Wilson
and McElhinny (
1974
) showed that the mean
paleomagnetic field for the last 25 Myrs was an
eccentric dipole field with •
z
D
325
˙
57 km.
These authors also found a slow rate of change
in the long-term structure of the geomagnetic
field, hence in the parameter •
z
, during this
time interval. However, the first comprehensive
study about the non-dipolar components of the
time-averaged geomagnetic field was published
by Coupland and Van der Voo in 1980. These
authors assumed an axial geometry for the
paleomagnetic field and searched the best-
fitting low-degree zonal Gauss coefficients,
g
n
, of a spherical harmonic expansion of the
potential. With the available paleomagnetic
data, they found that significant departures from
the GAD symmetry were associated with the
zonal
g
n
D
n
D
1
(6.64)
In the case of Wilson's field, only the dipole
and quadrupole components of (
6.63
)and(
6.64
)
survive, so that the potential
V
is given by:
V.r;™/
D
a
g
1
r
3
P
2
.cos ™/
(6.65)
a
2
r
2
P
1
.cos™/
C
g
2
a
3
The
Legendre
polynomials
appearing
in
(
6.65
)
are
given
by:
P
1
(cos™)
D
cos™ and
P
2
(cos™)
D
(3 cos
2
™-1)/2.
Therefore,
we
have:
V.r;™/
D
a
g
1
2r
3
3cos
2
™
1
(6.66)
a
2
r
2
cos ™
C
g
2
a
3
quadrupole
and
octupole
components,