A Chaos theory of ice ages

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Jamal Munshi, Sonoma State University, all rights reserved

There is no satisfactory explanation for ice ages. For at least two million years the size of the northern polar ice cap has followed a cyclical pattern; growing at times to cover most of the northern continents during an "ice age" and then receding to approximately where it is today during the "interglacial" periods.

The traditional theory of the ice age cycle was first forwarded by Milutin Milankovitch. The theory attempts to link the earth's precession to ice ages. The period of the earth's precession is 26,000 years; so we would expect ice formation to peak and to have warm interglacials every 26,000 years. But this is not the case. The evidence suggests that icy periods last from 20 to 100 thousand years and interglacials between 7 and 20 thousand years and not integer multiples of the precession period. The non-periodic nature of the phenomenon has not been adequately addressed.

Another mystery of the ice age is that within any icy period there are violent cycles of ice meltings. During the meltdown, large chunks of ice slide out to sea and the continental ice sheets get thinner. But within a few years it begins to get thicker again. The commonly held explanation for this behavior is due to Hartmut Heinrich. Heinrich postulates that as the ice gets thicker it acts as insulation and allows internal heat from the earth to melt the bottom of the ice and cause glacial flows. The problem with the Heinrich theory is that evidence suggests that glacial flows are not regional but global and at such a large scale that synchronization of localized hot spots is highly improbable.

Theories such as these subsume that for any given climactic condition there is a stable steady state ice level on the northern continents; and that any change from the steady state level can only be caused by a significant event with sufficient energy to cause the change. But this is not always the case in nature. Many natural systems exhibit non-linear dynamics and are metastable. In these systems many different "equilibirium" states are possible and even the slightest trigger (the proverbial butterfly) can bring about substantial changes in the equilibrium state.

A graphical model of metastability is shown below. The ball in the upper frame is in stable equilibrium. It will require a great deal of energy to shift the ball to another equilibrium state and if such a shift is observed a theory like that of Heinrich or Milankovitch might be required. The ball in the lower frame is in metastable equilibrium. Although it appears to be in steady state, many other steady state conditions are equally likely and minute random events can make wholesale changes to the position of the ball.


Ice formation in the northern continents is such a system. The time series of ice fractions is in chaotic equilibrium at wildly different levels of ice. The non-linearity in the system is imposed by the annual summer/winter heat cycles, by the reflective nature of ice, and by the nature of solubility of carbon dioxide in water. Such a non-linear model may be used to explain ice ages, interglacials, Heinrich glaciers, and even the non-periodicity of these events. The waxing and waning of the ice fraction is nonlinear because ice is melted by heat that the planet has absorbed from sunlight; and the heat absorbed by the planet is a function of the ice fraction because ice reflects sunlight. The heating of the planet is nonlinear because warming of the oceans releases absorbed CO2 into the atmosphere which causes warming to accelerate.

The reason that these non-linearities do not end in extremes is that ocean currents form a self-correcting system that forces regression to the mean. When warming exceeds certain limits the Arctic is no longer cold enough to suck warming currents from the tropics and that plunges those areas back into coldness. When coldness exceeds natural limits, the warming currents accelerate and warm the Arctic to cause a reversal of the cooling phase of the planetary temperature cycle.

The chaos model shown below demonstrates the surprising impact of this non linear behavior. In the model, a sine function is used to generate the annual incident solar radiation on the northern hemisphere of the tilted earth as it rotates on its axis and revolves around the sun. We begin the simulation with an assumed size of the polar cap which has a tendency to grow unless melted by solar radiation. A small perturbation (1%) is added to the solar radiation function to account for random effects.

We find that large swings in the ice fraction is possible under these conditions. "Ice ages" (at high ice fractions) form naturally and tend to persist. Just as naturally the ice recedes in brief interglacial periods. What's more surprising is the existence of the Heinrich events within these epochs. Both the ice age cycle and the Heinrich events are produced as a result of a nonlinearity in the heating function and without imposing an external causal force.

The more ice you have the less energy gets absorbed and even more ice can be formed. Conversely, the more ice you melt, the more energy you can absorb and more ice you can melt. This is the dynamic that can be set off in either direction by minute random effects; and it is this nonlinearity that is responsible for periods of otherwise inexplicable growth in ice formation and periods of melting and shrinking of the ice fraction.



References

an instability theory of air-ice interaction
a theory of ice ages