Gravity - The Solid Earth - page 215

Likewise, the geoid height anomaly for an Airy-compensated ocean basin of
depth d (Fig. 5.6(a))is

( ρ s − ρ u ) z d z

d t

2 π G
g

h =− ( ρ w − ρ u ) z d z +

0 t − r 3

( ρ w − ρ u ) d 2

+ ( ρ s − ρ u ) 2 tr 3 − r 3 =− π G

g

(5.53)

On substituting for r 3 from Eq. (5.25) and rearranging terms, we obtain
h =− π G
g

( ρ u − ρ w ) d 2 t − d ρ s − ρ w
ρ s − ρ u

(5.54)

Thus, with the numerical values given for Eq. (5.52), the geoid height anomaly
is

∆ h 3 . 85 d (0 . 7 − 0 . 046 d )m (5.55)

where d is the ocean depth in kilometres. A compensated ocean basin 5 km deep
would result in a negative geoid height anomaly of about 9 m.
Geoid height anomalies calculated for the Pratt compensation model are larger
than those for the Airy model. The difference is particularly significant for seabed
topography: the Pratt model gives a geoid height anomaly about twice that of
the Airy model. This difference can be used to estimate the type and depth of
compensation for major features.
In Chapter 8, the topic of convection in the mantle is discussed in some detail.
Here, we assume that this process occurs and see what can be discovered about
it by using measurements of the geoid and the Earth's gravity field.
The complex flow of material within the mantle gives rise to gravity and geoid
height anomalies just as mountains and oceanic trenches do. Figure 5.11 illus-
trates the two contributions to the gravity anomaly due to a region of hot rising
mantle. The surface is elevated over the hot material and depressed over cold
regions. There is a negative contribution to the gravity anomaly from the hot
mantle due to its reduced density. However, there is also a positive contribution
to gravity due to the elevation of the surface. Since this positive contribution is
larger than the negative contribution due to the reduced density, overall there is
a positive gravity anomaly over the hot region. Figure 5.12 shows a numerical
simulation of convection in the mantle (refer to Section 8.2 for details of these
models). The columns of rising material are hotter and therefore less dense than
the columns of sinking mantle material. Calculations indicate that there are small,
positive gravity seabed and geoid height anomalies above the rising hot regions.
These anomalies occur because the deflection of the surface by the rising cur-
rent produces a larger anomaly than the negative anomaly which results from
the density deficit. Similarly, there are negative anomalies above the sinking
columns. Detailed studies of long-wavelength gravity, geoid and bathymetric