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exceeds (or is less than) some threshold level. Thus, according to this formulation:
dV
=
dt
¼
C
1
ð
S
S
0
Þ
if S
>
S
0
dV
=
dt
¼
C
2
ð
S
S
0
Þ
if S
<
S
0
where V is the ice volume, S is solar intensity, S
0
is the threshold solar intensity, t
is time, and C
1
and C
2
are (different) constants. These equations were integrated
over 1,000-year time steps, and the accumulation of ice volume vs. time was
tabulated. He added the constraint that dV
dt cannot be negative when V
0.
Calder (1974) admitted that his assumptions ''were almost frivolous'', which raises
the question of what he meant by ''almost''. Calder (1974) apparently adjusted the
constants in the above equations to fit long-term oxygen isotope data over 800,000
years, although he was not very clear about exactly what he did or what constants
he used. Nor is it clear what starting values he used for the integration. It is
known empirically that ice ages seem to require several tens of thousands of years
to build up, but deglaciation can take place in only a few thousand years. Hence,
it seems likely that one must choose C
2
C
1
to get a best fit to the data.
However, Calder did not divulge the necessary details.
Weertman (1976) developed a model to estimate whether solar variations over
many thousands of years are sucient to produce large variations in the size of ice
age ice sheets. He set up a model of an ice sheet and derived equations for the rate
of change of its volume (or dimensions). He considered a two-dimensional ice
sheet that rests on a land surface that was flat before the ice sheet formed on it.
Figure 9.13
shows the ice sheet in profile.
The lower ice surface is, of course, not flat because of isostatic depression of
the land surface. The northern edge of Weertman's ice sheet borders at the Arctic
Ocean and the southern edge is on land. The total width of the ice sheet is L.
Weertman (1976) hypothesized a snow line such that accumulation takes place
wherever the height of the ice sheet locally exceeds the snow line, whereas ablation
occurs wherever the snow line is higher than the ice sheet. Based on his rough
=
Figure 9.13. Weertman's ice sheet model.
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