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case, solar intensity in midsummer at some high northern latitude), and T is a
time constant that produces a lag between change in solar input and resultant
change in climate. Unfortunately, the Imbries do not seem to have defined the
units of x and y or their ranges. Since x is a measure of solar intensity and y is a
measure of ice volume, it is not immediately clear how they are related. Although
never defined, it seems likely that the quantity x is the deviation of solar irradiance
from some long-term average value; thus, it can either be positive or negative. The
Imbries wrote this equation without a minus sign in front of x, but that does not
make sense to this writer. When solar irradiance is lower than average (x
<
0) ice
volume increases, so the relationship between dy
dt and x must contain a minus
sign and I have arbitrarily inserted one. It is not clear why the y term was
included on the right-hand side of the equation. Under average solar conditions
(x
0) the presence of the y term forces the ice volume to shrink with time, so
this equation has a bias toward interglacial conditions and only sustained values
of x
=
y can build up an ice sheet. This might have been based on the assumption
that reduced humidity could slow down the growth of the ice sheet when it gets
large, so that dy
<
dt must diminish as y builds up. As it turned out, the Imbries
found the above equation did not correlate well with long-term isotope data no
matter how they varied T.
Having failed with the simplest model, the Imbries added an embellishment to
it. As mentioned previously, there is considerable evidence in isotope profiles that
suggest that ice ages build up slowly but decay relatively rapidly. Therefore, the
Imbries modified the model by choosing the effective time constant to be greater
during ice buildup—it takes longer to add ice when
=
ð
x
þ
y
Þ <
0 than it does
to remove ice when
0. However, as before, in reporting their model
I changed (x)to(
x) because increasing x should reduce the ice volume. Thus,
their modified model can be written (with my change):
ð
x
þ
y
Þ >
dy
dt
¼
1
þ
B
T
ð
x
þ
y
Þ
if
ð
x
þ
y
Þ >
0
dy
dt
¼
1
B
T
ð
x
þ
y
Þ
if
ð
x
þ
y
Þ <
0
Insertion of the constant B assures that ice buildup will take place more
slowly than ice sheet decay. Note that as B varies from 1/3 to 2/3 the ratio of
effective time constants varies from 2 to 5.
It is not immediately clear why ice volume should enter this equation on the
right-hand side. Previous modelers always inserted a
y term on the right-hand
side to reduce the rate of ice volume growth as ice sheet volume increased, but no
explanations for this were offered. The growth of ice sheets requires a source of
moisture and evidence suggests that as an ice sheet grows such sources become
more distant; therefore, with constant solar intensity the rate of ice sheet growth
should be lower as ice sheet volume grows. That might justify inclusion of
y on
the right-hand side, although the argument is rather subjective.
The Imbries reported arriving at ''optimum model values'' by tuning this
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