Solid, yielding mantle - Mantle Convection for Geologists - page 32

d

D

Figure 4.3. Columns with a cross-sectional area of 1 m 2 .

In fact, the lower panel shows some of the data with an exponential curve fitted
through them. The exponential has a time constant of 4.6 kyr (kiloyears; this is the
time for the uplift to decrease by a factor e).
The inferred sequence of events is sketched in Figure 4.2(b). An initial flat
reference surface (i) is depressed a distance d by the weight of glacial ice during
the Ice Age (ii). (The ice load peaked about 18 kyr before present and ended
about 10 kyr.) After melting removed the ice load, the reference surface started
rising back towards its isostatically balanced level (iii), and that rising continues at
present with velocity v . In order for the surface to move vertically, the underlying
mantle must flow away, at first, and then towards the depression, as indicated by the
arrows.
Contrary to the common image of physics as being an 'exact' science (there is
no such thing), or of using sophisticated mathematics to reach conclusions of high
precision, a lot of good physics can be done with very rough and simple calcula-
tions, especially for the Earth, where our information often has large uncertainties.
The post-glacial rebound problem has been analysed using the partial differential
equations governing viscous fluid flow, and taking account of many details of the
situation, and from such analyses the viscosity of the mantle under Fennoscandia
has been estimated. On the other hand, a fair estimate of the viscosity of the mantle
can be obtained through the following rough approximation, using the definitions
of stress, strain and viscosity given in the previous section.
The depression was caused by the weight of ice sheets 2-3 km thick. If the ice
sheet were in place for long enough, an isostatic force balance would have been
established between the ice pushing down and the mantle 'pushing' up. When the
ice melted, an imbalance of forces was created, because the mantle was still pushing
up. A more rigorous way to say it is that there was a pressure deficit in the mantle
under the depression, because the depression was filled by water or air rather than
by heavier rock. We can calculate that pressure deficit by considering the weight
of two vertical columns of equal depth and with a horizontal cross-section of 1 m 2 ,
as is illustrated in Figure 4.3.
The left-hand column is outside the depression and it contains all rock. Its
volume is D D m 3 , so its mass is ρ m D and its weight is gρ m D ,where

ρ m is the density of the mantle (it turns out that the effect of the crust cancels

× 1 × 1 =