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in the area north of Australia. With the exception
of convergence zones, there is poor correlation of
geoid undulations with plate boundaries.
However, in most cases subduction zones
are associated with geoid highs that can be
explained by the presence of dense slabs
below the arcs. Significant examples are the
Peru-Chile, Aleutine, and Tonga subduction
zones (Fig.
14.3
). More difficult is to explain
the negative undulations of N. America and
central Asia, although the geoid low over the
Hudson Bay in Canada may be associated
with postglacial rebound. In general, long-
wavelength (>4,000 km) anomalies of the
geoid are mainly associated with large-scale
density anomalies and flows in the lower mantle,
while short-wavelength features correlate with
lithospheric structure (Hager and Richards
1989
and references therein). Interestingly,
Dziewonski et al. (
1977
) observed that the long-
wavelength lows correlated with seismically
fast, presumably cold and dense, regions of
the lower mantle, while long-wavelength geoid
highs correlated with seismically slow, possibly
light, lower mantle features. Moreover, the geoid
anomaly results to be positive for the regions
surrounding many hot spots, although these
features are located on low-density mantle.
Clearly, this is the reverse of what would be
expected, because in principle the geoid should
exhibit a positive correlation with internal density
anomalies. The solution of this apparent paradox
can be found in a pioneer work of Pekeris (
1935
),
who focused on thermal convection but showed
that the geoid results from the combined effect
of density distribution in the Earth's interior and
a process today known as
dynamic topography
,
which will be the subject of the last section of
this chapter. It is responsible for topographic
uplift over hot upwelling currents and subsidence
over cold downwellings. Pekeris (
1935
)showed
that the density anomalies generated by dynamic
topography close to the Earth's surface give
a contribution to the geoid that opposes and
eventually overcomes the contribution of deep
sources, giving geoid highs over low-density
mantle upwellings and lows over high-density
downwellings.
14.4 MacCullagh's Formula
The potential of a planetary mass, in particular
of the Earth, can be calculated using the classic
Poisson integral, which can be obtained either
integrating (
14.1
) or simply extending Newton's
gravity law to a continuous body. Using the
geometry shown in Fig.
14.4
,wehaveforthe
potential generated by a density distribution in a
region
R
:
V.r/
D
G
Z
¡.r
0
/
j
r
r
0
j
dx
0
dy
0
d
z
0
(14.38)
R
The term
j
r
r
0
j
in the integral (
14.38
) can be
expanded in power series. In fact, if
z
is a real
variable such that
1
z
< 1, then McLaurin's
expansion of the function
f
(
z
)
D
(1
z
)
1/2
gives:
1
2
z
C
1
2Š
1
2
3
2
z
2
f.
z
/
D
.1
z
/
1=2
D
1
C
1
3Š
1
2
3
2
5
2
z
3
C
C
:::
I
1
z
<1
(14.39)
Therefore, setting
z
2
x
—
—
2
we obtain that
the function:
1
p
1
2x—
C
—
2
G.x;—/
D
(14.40)
Fig. 14.4
Geometry for the calculation of Poisson's inte-
gral at a point
P
