An ice sheet that skips a beat of the tilt

The orbital side of the ice ages is solved. For any definition of a summer — the sunniest day, the whole half-year, the days above a melting threshold — the dependence on the Earth’s eccentricity, its tilt and the season of its closest approach to the Sun can be computed as precisely as anyone needs, and the energy of a summer above a melting threshold follows the tilt far more closely than it follows the precession, because a nearer Sun in summer also means a shorter summer. What is not solved is the other side: what an ice sheet does with the sunlight it is given.

The record of the last eight hundred thousand years makes the question sharp. The volume of ice on the continents, read from the oxygen isotopes in deep-sea shells and from the air in Antarctic ice, rises and falls in a sawtooth. It builds slowly for eighty or ninety thousand years, reaches a maximum with ice sheets three kilometres thick over North America and Scandinavia, and then collapses in ten thousand years or less — a termination. The terminations come roughly every hundred thousand years. The obliquity varies with a period of forty-one thousand; the precession of the equinoxes modulates the seasons with periods of nineteen and twenty-three thousand; the eccentricity does vary with periods near a hundred thousand years, but its effect on the sunlight averaged over a year is a change of a few tenths of a per cent, and its effect on the seasons appears only through the precession. No orbital forcing of any strength has a hundred-thousand-year period. The mismatch has been called the hundred-thousand-year problem for half a century.

Earlier still the record is different. Before about 1.2 million years ago the ice volume varied almost exactly with the forty-one-thousand-year cycle of the tilt — smaller ice sheets, more symmetric cycles, one per obliquity oscillation. Between 1.2 million and seven hundred thousand years ago the rhythm changed to the one seen since, and nothing in the orbit changed at the same time. The mid-Pleistocene transition is the second half of the problem, and the most economical answer to both is that the ice sheet counts the tilt’s beats and sometimes skips them.

Three lines of model #

The model drawn throughout is deliberately minimal, of the kind used to test the idea rather than to simulate an ice sheet. Ice volume grows by one unit per thousand years, with some noise. A termination begins whenever the volume plus a multiple of the obliquity anomaly exceeds a threshold, and the ice then decays to nothing over ten thousand years. The obliquity is the sum of the five largest terms of the standard series for the Earth’s tilt, which is dominated by the forty-one-thousand-year term and modulated by its neighbours at forty and thirty-nine, fifty-four and twenty-nine thousand years. The phases are fixed arbitrarily, so the model’s history is not the Earth’s; it is a demonstration of what the mechanism does.

The two halves of the figure use the same forcing and differ in one number. With a low threshold, the ice never grows large before an obliquity maximum is strong enough to tip it over, so it collapses on almost every beat: the ice volume follows the tilt, and its cycles are forty-one thousand years long and small. With a higher threshold, the ice needs more time to grow large enough to be vulnerable. A maximum that arrives while it is still too small passes without a termination; the ice survives it, keeps growing, and collapses on the next maximum or the one after. The cycles become long, large and asymmetric — a slow build and a sudden fall — and average a little over a hundred thousand years.

The two regimes look like the two halves of the Pleistocene record. That the sawtooth shape comes out of the model is not a coincidence: slow growth and fast collapse are built into its three lines. That the period comes out near a hundred thousand years is more interesting, because nothing in the model has that period.

The intervals are whole numbers of beats #

A hundred thousand years is the average of the intervals, and the average hides their structure.

The distribution of intervals is bimodal. Almost none are near a hundred thousand years; they are either near two obliquity cycles or near three. Each termination happens on an obliquity maximum, so the time between terminations is always the time between two maxima, which is a whole number of cycles — slightly irregular, because the maxima of the real obliquity series drift a few thousand years either side of a strict forty-one-thousand-year rhythm under the influence of its smaller terms. The average of a mixture of eighty-odd and a hundred-and-twenty-odd is the hundred thousand years the record shows.

That prediction can be checked against the record, and it has been. The terminations of the last million years are not spaced evenly: the intervals between them, as well as they can be dated, cluster near eighty and near a hundred and twenty thousand years, and the long one between the terminations about 430,000 and 340,000 years ago is especially clear. The spacing of glacial cycles is therefore not a hundred-thousand-year periodicity but an irregular sequence of two- and three-beat skips, as the model makes it.

Which beats count depends on how large the ice must grow #

The threshold in the model is the only thing that changes between the two regimes, and it is worth asking how the change works when the threshold moves gradually.

As the threshold rises, the mixture shifts through whole numbers: at first every interval is one cycle; then pairs appear and take over; then triples. The mean interval rises smoothly, from forty-one thousand years to more than a hundred and fifty, because the real obliquity series is not a single sine wave and the noise blurs every boundary — but underneath the smooth mean, each individual interval is still a whole number of beats. With a strictly periodic forcing and no noise the mean would climb in steps, with plateaus at one, two and three cycles; the irregular forcing turns the staircase into a ramp while keeping each step’s content.

This gives the mid-Pleistocene transition a reading that does not need anything to happen to the orbit. If whatever sets the threshold — how large the ice must be before a warm summer can start it collapsing — rose gradually between 1.2 million and seven hundred thousand years ago, the climate would move from the one-cycle end of this figure to the two-and-three-cycle middle, the ice ages would lengthen, grow and become asymmetric, and the orbit would be the same throughout. Two proposals for what raised the threshold are commonly argued. One is a slow decline in atmospheric carbon dioxide over the Pleistocene, which makes it easier for ice to survive a warm summer. The other is the erosion of the soft sediment that the earlier ice sheets had rested on: an ice sheet on deformable sediment slides and stays thin, while one on hard bedrock grips and grows thick, and the earlier ice sheets would have removed their own soft beds over a few dozen cycles. Neither is established, and they are not exclusive.

Terminations start at high tilt #

If the terminations are paced by obliquity, each one should begin near a maximum of the tilt, whichever maximum it turns out to be.

The model’s terminations begin when the tilt is high: the distribution of the obliquity anomaly at their onset is shifted well above its distribution at random times. This is the property that can be tested against the record without knowing the threshold, the growth rate or anything else about the ice. Each dated termination has an obliquity phase — where in its forty-one-thousand-year cycle the tilt was at the time — and if terminations were unrelated to the obliquity, those phases would be spread uniformly. The test was carried out on the terminations of the last seven hundred thousand years with their dating uncertainties included, and the phases were found clustered near obliquity maxima, well beyond what chance allows. The same test for the precession found no significant clustering in that analysis, though later work with more terminations and different dating has found a role for precession in determining which obliquity maximum a termination falls on.

That division of labour is exactly what the summer-energy argument prepared. If ice sheets melt according to the total energy of a summer above a melting threshold, they respond mainly to the tilt, and the precession enters only secondarily. If they responded to the brightest day of summer, precession would dominate and terminations would cluster at precession maxima. The phase statistics favour the first.

A spectral peak made by counting #

The traditional evidence for a hundred-thousand-year cycle is a spectral peak: a Fourier analysis of the ice-volume record shows its largest power near that period.

The model produces the same peak from forcing that has no power there at all. At the low threshold the ice spectrum is a sharp line at forty-one thousand years, reproducing the forcing. At the high threshold it has a broad, ragged peak at a hundred thousand years or a little more, with the forty-one-thousand-year line still present but secondary. The broad peak is the signature of a sawtooth with irregular intervals between two and three beats: its fundamental period is their average, and its breadth is their irregularity.

The lesson is general. A peak in a periodogram says that a series has power at a frequency; it does not say that anything in the series’ cause has that frequency. The tallest peak in a spectrum of pure noise is not small, and a nonlinear system — one with a threshold, a memory, a sudden collapse — can do something subtler still: put power where its forcing has none, at combinations and sub-multiples of the forcing’s own periods. The ice-volume spectrum’s hundred-thousand-year peak is therefore evidence that the climate responds nonlinearly to the orbit, not that the orbit forces it at that period. The eccentricity’s hundred-thousand-year cycle is present in the forcing, weakly, through its modulation of the precession, and it may help decide which beats are skipped; but in this model it plays no part, and the peak appears anyway.

How the record is read, and why its clock matters #

The ice volume in the record is not measured directly. It is inferred from the ratio of oxygen-18 to oxygen-16 in the shells of foraminifera on the deep-sea floor. Water evaporated from the ocean is light, and when it falls as snow and is locked into ice sheets the ocean left behind is heavy; a shell growing in that ocean records its composition. The same ratio also depends on the temperature of the water the shell grew in, so the record is a blend of ice volume and deep-ocean temperature, which have to be separated with other measurements. Stacked from dozens of cores, it gives a curve of global ice volume with a resolution of a few thousand years over the last five million.

The curve has no clock of its own. Sediment accumulates at rates that vary, and the ages of most of the record come from correlating it with something else — and for decades that something was the orbit itself. The standard timescales were built by tuning the record’s wiggles to the calculated insolation, a procedure that guarantees the record will show the orbital periods and makes it impossible to use the same record to test whether they are there. The phase test on the terminations was possible only because an independent timescale had been built, from sediment depth alone, assuming steady accumulation between a few dated tie points, with its uncertainties carried through. On that untuned clock the obliquity pacing survives, which is what makes it evidence rather than an echo of the tuning.

The same problem runs through every argument about the phase of the ice ages. A statement that terminations fall at obliquity maxima is only as good as the dates of the terminations, and dates good to a few thousand years are needed to tell an obliquity maximum from the neighbouring minimum. Radiometric dates from cave deposits and from coral reefs have pinned the last few terminations that well; the older ones rely on the depth-derived clocks. A timescale built by fitting the thing being tested is a hazard throughout astronomy’s own time-keeping as well, and the ice-age record is one of its clearest cases.

The eccentricity’s honest share #

None of this makes the eccentricity irrelevant; it makes its role specific. The precession changes the seasons only in proportion to the eccentricity: with a circular orbit the season of closest approach would not matter at all. So the eccentricity modulates the amplitude of the precession signal in summer sunlight, and in periods of high eccentricity the precession swings summer insolation at high latitudes by as much as the obliquity does. In the model drawn here that modulation is left out, and the rhythm appears without it. In the real record it probably helps decide which obliquity maximum a termination falls on — a maximum that coincides with a strongly precession-enhanced summer is a better candidate than one that does not — and that would link the timing of terminations weakly to the eccentricity’s hundred-thousand-year cycle without making it the cause.

The eccentricity also has a four-hundred-thousand-year cycle, the longest and most regular of its periods, and around four hundred thousand years ago it was at a minimum, with the precession signal nearly absent. The interglacial that followed was one of the longest and warmest of the last million years, lasting perhaps thirty thousand years rather than the usual ten. A model paced by obliquity alone has no reason to produce an unusually long interglacial there; a model in which precession helps trigger the next glacial inception, and was too weak to do so, does. That interval is the one place in the record where the eccentricity’s share can be read most clearly, and it is read as a delay in the end of a warm period rather than as a cause of the cycles.

What the model does not claim #

A model of three lines cannot claim to describe an ice sheet, and this one does not. It has no geography, no bedrock sinking under the load, no ocean, no carbon cycle, no dust darkening the ice’s surface as the climate dries, and no distinction between the North American and Scandinavian sheets, whose histories differed. Its threshold is a single number standing in for all of these; its forcing is obliquity alone, when the real summer energy has a precession component too. Real ice-sheet models couple a surface energy balance to ice flow and to the solid Earth, and they reproduce the late-Pleistocene cycles with a similar logic but through physical mechanisms: an ice sheet that has grown large sits on bedrock depressed by its weight, so its surface is low and warm; its southern margin is exposed to summer sun and to dust-darkened melting; and once it starts retreating, the retreat accelerates. Those mechanisms are what the threshold summarises.

Its phases are arbitrary, so it does not predict the dates of any real termination, and its parameters were chosen to show the two regimes clearly rather than fitted. What it demonstrates is a logical possibility with testable consequences: that the hundred-thousand-year rhythm can be produced by the forty-one-thousand-year tilt alone, that its intervals should be whole numbers of beats, that terminations should begin at high tilt, and that a spectral peak near a hundred thousand years does not require a forcing there. The first two have been checked against the dated record and hold up. None of them rules out a role for the eccentricity; they show that none is needed to produce the rhythm.

Why the tilt, and not the sunniest day #

The deepest reason the tilt should dominate lies in the arithmetic of a summer that is hotter because it is nearer the Sun but correspondingly shorter. An ice sheet does not melt on its brightest day; it melts on every day warm enough, and what matters is the sum. Kepler’s second law makes the precession’s effect on that sum small, while the tilt changes it directly, by raising the Sun higher in the high-latitude summer sky. An ice sheet large enough to have a long memory is precisely the kind of system that integrates, and an integrating system reads the tilt.

The same integrating property is what lets it skip beats. A system with no memory would respond to every maximum identically; a system whose response depends on its own accumulated state responds only when both the state and the forcing permit. The ice sheet’s size is its memory. The tilt itself is held steady by the Moon, varying by only a degree or so, and that small, regular variation is all the pacing the model needs — which is a reminder of how different the Earth’s climate would be on a planet whose tilt wanders chaotically over tens of degrees, as Mars’s does.

Still open: what raised the threshold #

If the threshold picture is right, the mid-Pleistocene transition is a change in the ice sheets or in the carbon cycle, not in the sky, and the question moves from astronomy to the Earth. The regolith hypothesis predicts that the earliest ice sheets left sediment-rich deposits and the later ones bedrock-scoured landscapes, which the glacial geology of North America broadly shows, but dating the change well enough to compare with the marine record is difficult. The carbon-dioxide hypothesis predicts a decline across the transition, and the oldest ice cores — now reaching back more than a million years in the first sections recovered from blue-ice areas and new deep drilling sites in Antarctica — are beginning to measure the atmosphere across it directly. Whether the carbon dioxide fell enough, early enough, to raise the threshold by the factor the model needs is the measurement that will decide between the two, and it is being made now.