Geoscience Reference
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where
v
is ice volume, R is the current climate regime (R
¼
I ,
g
,orG),
v
R
are
reference ice volumes for different regimes, F is insolation forcing, and t
R
and t
F
are time constants. This model implies that the rate of addition of new ice
decreases as ice volume builds up. Physically, one might expect the rate of new ice
formation to increase at first as ice volume increases due to the cold trap for
moisture that is created but, eventually, the amount of prevalent moisture for
large ice volumes would presumably decrease and the rate of accumulation would
diminish with ice volume.
Paillard normalized ice volume to unity:
v
g
¼ v
G
¼ v
max
¼
1,
v
i
¼
0. The
forcing F is mathematically manipulated by a procedure that this writer was
unable to follow. F was said to depend on f , which in turn depended on x and a,
but neither x nor a were defined in the paper. Nor is it clear to this writer how the
above differential equation was integrated with time or how transitions between
regimes occurred within the model. Moreover, as in the case of the simpler model,
it is not clear to what extent this curve-fitting procedure has a physical basis.
Models that integrate periods of low insolation over time to produce ice formation
with some threshold and high insolation over a shorter period to produce
deglaciation seem to produce impressive results. While I do not begrudge Dr.
Paillard's self-satisfaction with his results, his conclusion
''In any case, and in contrast to recent claims, this conceptual model clearly
demonstrates that the geological record can easily be explained in the framework
of the classical astronomical theory''
seems a bit overly enthusiastic.
An interesting aspect of Paillard's results is that he finds very different ice
volume patterns vs. time for the first million years and the second million years,
despite the seemingly repetitive morphology of the insolation vs. time curve over
2 million years. How this comes about remains a mystery to this writer.
It does not seem necessary to define the three regimes of the climate system.
Paillard's equation for the rate of ice volume growth is actually very similar to
that used by the Imbries 18 years earlier. Unfortunately, like all the other models,
it is dicult to resurrect the model and explore it further because no units are
given.
9.6.2 Review of the Imbrie model
In attempting to check the Imbries' results and examine the model further, one
encounters the problem of making the units of x and y compatible. The descrip-
tion given by the Imbries is sparse. It seems likely that they may have used the
following procedure. Over any interval of time, the values of x (solar insolation)
are tabulated at regular intervals. The average value of x and the standard devia-
tion over the interval are calculated. The values of x are measured from the
average in units of multiples of the standard deviation (x could be positive or
negative). The average value and standard deviation of noon solar intensity at
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