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
( 4.93 ) assumes the following simplified form:
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
dipole equation ( 4.49 ) will be substituted by 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)
which reduces to ( 4.97 ) for r D a (i.e., at the
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,
 
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