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CHAPTER 9 • Insolation Control of Ice Sheets 161

and reacts slowly to changes in the heating applied, period of a cycle is the interval of time separating suc-
changes in water temperature lag behind changes in cessive peaks or successive valleys. In this example, the
intensity of the heat source. phase lag of ice volume behind summer insolation rep-
Ice sheets have the same lagging response to sum- resents one-quarter of the cycle length.
mer insolation (Figure 9–8 bottom) but on a much The same relationship can be applied to the separate
longer time scale. As summer insolation declines from a ice volume responses driven by the insolation cycles of
maximum value, ice begins to accumulate. The rate at orbital tilt and precession. At the orbital tilt cycle, the
which ice volume grows reaches a maximum when sum- ice volume response would have the same regular sine
mer insolation has fallen to its lowest value because wave shape as the summer insolation signal, but it
summer ablation is at a minimum at that time. Later, as would lag behind it by one-quarter of the 41,000-year
summer insolation begins to increase, the ice sheet con- wavelength or about 10,000 years (Figure 9–9A). At the
tinues to grow for a while longer because insolation val- precession cycle, the ice volume response would again
ues are still relatively low. The ice sheet does not reach lag insolation by one-quarter of its 23,000-year length,
its maximum size until insolation crosses the value that or just under 6000 years. The ice volume response at
initiates net ice ablation. As a result, the maximum in this cycle also shows the same amplitude modulation
ice volume lags thousands of years behind the minimum as the precession insolation signal (Figure 9–9B). As
in summer insolation. we will see later, the actual lags of the ice sheets behind
As summer insolation continues to increase, the rate the solar forcing are large but not quite a full quarter
of ablation increases (see Figure 9–8 bottom). When wavelength.
insolation reaches its maximum value, the rate of abla-
tion also reaches its maximum. Again, however, the 9-3 Delayed Bedrock Response beneath Ice Sheets
minimum in ice volume does not occur at the insolation
maximum because falling insolation values still remain As ice sheets grow, so does the pressure of their weight
high enough to melt even more ice for thousands of on the underlying bedrock. Although the density of
3
years. Later, about halfway through the drop in insola- solid ice (just under 1g/cm ) is much lower than that of
tion, ice volume reaches its maximum value and then the underlying rock (about 3.3 g/cm ), ice sheets can
3
starts to decrease in a regime of increasing ablation. reach thicknesses in excess of 3000 m, equivalent to the
This relationship can be described by the equation weight of 1000 m of rock. This load is enough to
shown in Box 9–1. depress the underlying bedrock far beneath its level
The persistent delay in ice volume relative to sum- when no ice sheets existed.
mer insolation shown in Figure 9–8 (bottom) is the One way to visualize this process is a thought experi-
phase lag between the two cycles. Remember that the ment in which an ice sheet 3.3 km thick is instantaneously

BOX 9-1 LOOKING DEEPER INTO CLIMATE SCIENCE
Ice Volume Response to Insolation

he dependence of ice volume on summer insolation resulting rate of ice volume change, which depends on the
Tcan be expressed by this equation: inverse of the value of T, or 1/T.
The second factor controlling the rate of ice volume
d(I) 1 (S–l) change is the term S – I, which is a measure of the degree of
——— = ——
d(t) T disequilibrium (offset) between the summer insolation forc-
d(I) ing signal (S) and the ice volume response (I). Conceptually,
where I is ice volume, ——— is the rate of change of ice
dt the disequilibrium can be thought of in this way: the ice vol-
volume per unit of time (t), T is the response time of ume response (I) is constantly chasing after the insolation
the ice sheet (in years), and S is the summer insolation forcing signal (S), but it never catches up to it. For example,
signal. when insolation is at a minimum, ice volume is growing at its
This equation specifies that the rate of change of ice fastest rate toward its maximum value, but has not yet got-
volume with time is a function of two factors. One factor ten to that value. When ice volume finally does reach its
is T, the response time of the ice sheets, measured in maximum size, the insolation curve has already turned and
thousands of years. The equation specifies that the larger risen halfway toward its next maximum, causing the ice vol-
the time constant T of ice response, the slower will be the ume curve to reverse direction and shrink.