Introduction

Reconstructions of past ice-sheet volume, extent, and dynamics are often used to constrain possible future contributions to global sea level from the extant Greenland and Antarctic ice sheets under projected warmer-than-present climate states1,2,3. Such reconstructions are based on a variety of types of evidence, many of which are indirect proxies. Typically it is assumed that ice sheets oscillate out-of-phase with climate, that is, warming leads to retreat and cooling leads to growth. However, some proxy records appear to show ice-sheet growth during climate warming, with a range of explanations proposed to explain this4,5,6,7,8,9. The phasing of climate–ice-sheet relationships can, therefore, change, but why and how this happens has yet to be explored in a systematic and objective manner.

For periods of the relatively recent past, up to 800 kyr or so, ice core data are sufficiently direct and temporally well-constrained that robust correlations between orbital forcing, greenhouse gas fluctuations, and high latitude climate are possible10,11,12. In other cases the proxies are ‘tuned’ so that peaks in their respective signals are aligned with astronomical cycles (precession, obliquity, eccentricity), with insolation, or with model results intended to simulate astronomical forcing4,13. These tuning approaches imply a direct generative mechanism that translates the presumed forcing into the observed response, with a lag that is generally poorly constrained4,14,15. In such cases, the phasing alignment between ice-sheet volume, global sea level change, and climate variability is impossible to determine precisely, due to millennial-scale uncertainties in chronology16.

To aid in the interpretation of stable isotope variations as proxy records of ice-sheet variability older than c. 1 Ma, process-based models are usually employed as a way of assessing physically plausible climate and ice sheet responses to orbital and greenhouse gas changes17,18. However, model simulations of past warm periods often employ a timeslice approach with static climate fields, and the computational expense of epoch-scale coupled climate–ice-sheet modelling makes such endeavours prohibitive. Yet timeslice modelling experiments show that the imposition of an instantaneous step warming leads to ice sheet equilibrium approximately 10 kyr later19, implying at least a 10 kyr lag in ice sheet response with respect to the imposed climate forcing. Furthermore, natural climatic warming from orbital variability tends to approximate a gradual, sinusoidal change, rather than an instantaneous step change. Studies that have simulated ice sheets under such conditions also note lag times between peak forcing and peak response of 11–28%, where lags during growth are at the lower end of this range and the slower responses occur during decay17.

Chronological uncertainty of palaeoenvironmental proxies, combined with variable and unknown ice sheet response times, mean that climate–ice-sheet phasing relationships are difficult to determine with certainty for periods of the geological past whose warming levels are comparable to future conditions under current or higher greenhouse gas emissions pathways. Taken together, the limitations of both proxy-based reconstructions and ice-sheet-specific simulations mean that a comprehensive mechanistic evaluation of ice-climate relationships is currently lacking.

To help address these shortcomings, we present an ensemble of 198 simplified ice sheet simulations that each evolve under a range of background climate states (i.e., a change in time-invariant warming levels, Supplementary Fig. S1) that also oscillate at periods consistent with each of the three primary Milankovitch periodicities – 23, 41 and 100 kyr. The purpose of choosing each of these forcing periods is not to simulate ice sheet changes that can be directly attributed to each of the Milankovitch frequencies, but to illustrate how ice-sheet response changes at these characteristic timescales, regardless of the origin (e.g., direct orbital forcing vs. multiples of shorter periods) of these forcing cyclicities. Experiments in the main body of the ensemble are run for 500 kyr each, to allow multiple glacial–interglacial cycles to be captured. In addition, gradual secular cooling simulations spanning 1 Myr are also run in order to capture transitions in ice sheet response under oscillating climates whose background climate state also linearly evolves. Finally, we investigate the impact of solid earth responses related to mantle viscosity, which are particularly instructive in understanding glacial cycle asymmetry. Collectively, our experiments total 102 million years of modelled time.

Results

Phasing relationships in modelled ice sheet behaviour

In our first suite of experiments, we compare the response of our generic, but realistic, ice sheet (Fig. 1a; Supplementary Fig. S2) under a range of five oceanic and six atmospheric states, for the three primary orbital frequencies described above. From this ensemble of 90 experiments we calculate Pearson’s correlation coefficients to discern the phasing relationship of ice sheet response to applied sinusoidal climate forcing (Fig. 1d–f). Our metrics for comparison are grounded ice volume, grounded ice area, and calving flux from marine margins. What is immediately apparent is that the difference in response of ice sheets under different climate states is greatest at lower frequency forcings (Fig. 1d, e). Under a 100 kyr forcing, ice-sheet volume and area are positively correlated with applied forcings where the atmospheric conditions are warmest, and oceanic forcing is relatively weak. Under such conditions the ice sheet is primarily ‘terrestrial’ in the sense that floating ice at the margins is much reduced. Conversely, ice volume and forcing are negatively correlated under climate states characterised by a cold atmosphere coupled with strong oceanic control. Figure 1b, c (and Supplementary Fig. S3 for 41 and 23 kyr periodicities) illustrates these contrasting behaviours. Within our matrix of experiments, there is a clear division that separates positively correlated, in-phase, responses from negatively correlated, anti-phased, responses. This diagonal structure is evident under 100 kyr (Fig. 1d) and 41 kyr (Fig. 1e) forcings, and in both cases is most clearly expressed in terms of ice volume and area. Calving fluxes exhibit similar climate state dependencies, but the relationships are more complex. At 23 kyr timescales (Fig. 1f), however, the relationship in all three metrics is markedly different. The separation between in-phase and anti-phase relationships still appears to exist (though less clearly expressed than at longer timescales, with generally weaker correlations), yet the axis of the divide is now vertical, such that cooler atmospheres with weak ocean forcing can result in in-phase ice growth, and very warm atmospheres with strong ocean forcing can produce an anti-phased ice response.

Fig. 1: Ice sheet response to climate forcing.
Fig. 1: Ice sheet response to climate forcing.
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a Radially-symmetric ice sheet model configuration under a cold climate state, showing ice sheet dome and fringing ice shelves. b,c Ice sheet profiles during interglacial (orange) and glacial (blue) phases of warm and cold climate states. Black lines show bed topography and the effect of modelled isostasy. d–f Pearson’s correlation coefficients calculated from timeseries of climate forcing and modelled timeseries of grounded ice volume, grounded ice area, and calving flux, for 100, 41 and 23 kyr forcing frequencies. NaN values and hatched cells indicate two simulations where ice growth became so extensive that meaningful correlations could not be calculated.

Phasing relationships in empirical records and palaeo simulations

Reconstructions of former ice sheet behaviour rely on geographically sparse geological proxies (terrestrial outcrops, seismic stratigraphic profiles, or marine sediment cores) that are generally uncertain in age. There are limited data available to reconstruct ice elevation in ice sheet interiors, making empirical estimates of ice volume challenging. This is particularly true in places where tectonic, erosional, or depositional processes have substantially altered topography over timescales longer than c. 1 Myr20,21,22. Many of these limitations are exacerbated with increasing age, meaning that ice sheet reconstructions of warmer-than-present climate states, such as the mid-Miocene Climatic Optimum (MMCO, c. 17.5–14.5 Ma) or the Oligocene (c. 34–23 Ma) must draw on sea level proxies (benthic foraminifera δ18O, raised shorelines, or sequence stratigraphic inferences) from which ice volume can be indirectly approximated.

These shortcomings notwithstanding, ocean chemistry data suggest that Oligocene Antarctic ice volume may have been greater during the Mid-Oligocene Glacial Interval (MOGI, c. 28–26.3 Ma) than after the Oligocene–Miocene Transition (c. 23 Ma), due in part to higher topography in West Antarctica23. Somewhat counterintuitively, near field geological records indicate that this large ice sheet retreated as high latitude climate cooled through the late Oligocene24. Marine mudstone sequences from cores at the margin of East Antarctica also suggest that the ice sheet margin retreated during cool climates associated with eccentricity minima7,25,26, through the early to middle Miocene. Geological records show that the ice sheet then retreated to its terrestrial margin for most of the MMCO, but some data27 suggest that, during several intervals within this ‘warm’ period, it grew to a greater volume or extent than it did during early phases of the subsequent, and cooler, Miocene Climate Transition (MCT, c. 14 Ma). This relationship is consistent with the ‘snow-gun’ hypothesis, which posits that, during the Miocene, the main driver of ice sheet growth was increased moisture availability from upwelling warm waters, rather than atmospheric cooling5,28.

During the warm Pliocene, exposure ages from East Antarctica appear to show inland thickening, despite margin retreat in marine basins and the complete loss of WAIS29. Ice sheet modelling allows mechanistic insights into these climate–ice-sheet relationships, and studies generally highlight two key factors. First, precipitation under a warmer (but still cold) high-latitude climate enabled more positive surface mass balance and ice sheet growth17,30,31. Second, long dynamic response times meant that changes in ice extent could be substantially out-of-phase with climate forcing and precipitation-driven changes in interior thickness17,31. Such relationships may also apply to younger sequences, but to a lesser extent. This is because, under cooler climates (Plio-Pleistocene), the typically greater proportion of marine-based ice means that ice advance and retreat is governed to a greater degree by oceanic rather than atmospheric conditions, with margin oscillations forced directly (by melting at marine margins), rather than arising from slowly-evolving internal ice dynamics. Nonetheless, ice sheets such as the former Laurentide Ice Sheet may have exhibited millennial-scale oscillations that were in-phase with climate (ice growth during warming), primarily through ice sheet melt-induced ocean cooling8.

Comparison of modelled and palaeo inferred phasing relationships

Despite the simplifications of our model domain and climatic forcing, there are nonetheless notable similarities between the ice sheet–climate phasing relationships we simulate and those inferred from proxy records and simulations of former ice sheets. In both cases, relatively warmer climates that were associated with higher precipitation at high latitudes (‘snow-gun hypothesis’) appear to result in thicker ice sheets that had reduced or absent ice shelves. As a consequence of higher interior elevations and minimal marine margins, the phasing of ice sheet growth and decay was more tightly coupled to atmospheric than to oceanic temperatures. Conversely, models and observations suggest that colder and more arid climatic regimes favoured extensive ice shelf expansion that enhanced the response of the ice sheet to changes in oceanic, rather than atmospheric, conditions. This general relationship appears to be consistent between model results and observations, but no doubt oversimplifies real-world ice sheet–climate interactions. For example, under climate states that lie between the warm and cold end members, modelled and observed phasing relationships are more complex in character, are more ambiguous in phasing, and tend to be influenced to a greater degree by local topographic or environmental factors, such as bed topography or regional climatic conditions.

Discussion

Mechanistic analysis

The primary ensemble of experiments has revealed climate state-dependent behaviours that, at least at 41r and 100 kyr periodicities, exhibit opposing trends under warm climates with weak oceanic forcing and under cold climates with more vigorous ice-ocean interaction. These differing climate–ice-sheet relationships appear to be consistent with geological evidence and palaeo-ice sheet modelling studies of some Miocene and Oligocene warm periods (see earlier section).

To provide a mechanistic framework with which to understand why these contrasting behaviours arise, we first examine the relationship between ice volume and climate. In our experiments, we supply mass to our model domain through precipitation that under negative atmospheric temperatures falls as snow. Accumulated snow and ice is subsequently lost from the modelled ice sheet either through surface melting, if atmospheric temperatures are above zero centigrade, or through ocean-driven melting at submarine faces of simulated ice shelves. Because our imposed climatologies are deliberately simplified, mass gain from snowfall and mass loss through melting are both expected to increase as temperatures rise, and decrease as temperatures fall. This direct and in-phase coupling means that ice volume variability depends on the changing dominance of atmospheric over oceanic forcing. The diagonal structures evident in the ice volume (and area) relationships in Fig. 1 indicate that there is a clear switch between climate states where atmospheric forcing dominates ocean melt and those where ocean melt dominates surface accumulation.

To disentangle the relative importance of atmospheric and oceanic forcing we compare our primary ensemble of 90 simulations with a duplicate set of 90 that employ identical atmospheric forcing, but oceanic regimes in which only the magnitude, but not amplitude, of melt rate variability is changed (Supplementary Fig. S4). Intriguingly, comparing these simulations shows that, at 23 kyr periodicity, there is very little difference (at least in terms of time-integrated correlation coefficients). Differences in the pattern of correlations are more evident in the 41 kyr subset, and are most strongly expressed under 100 kyr fluctuations. In these latter experiments, changing oceanic melt magnitude – but not amplitude – leads to a clear and consistently linear relationship with atmospheric temperature, irrespective of ocean state. That is, the diagonal structure is absent, and although there are differences in absolute ice volume between scenarios, the phasing relationship between ice volume (and area) and climate variability is entirely controlled by atmospheric climate state.

These comparisons reveal two important things. Firstly, that it is the amplitude of oceanic variability, rather than the absolute value (magnitude) of sub-ice shelf melt rate, that is of greatest importance in the switching behaviour observed in the model results shown in Fig. 1. Secondly, since this effect is primarily evident only at longer forcing frequencies, it suggests that there is a certain timescale of modelled ice sheet response below which atmospherically-controlled surface accumulation becomes less important, and oceanic melt dominates, regardless of its magnitude or amplitude of variation. This is because the buttressing effects of ocean-forced melt are instantaneous, whereas volume increase from surface accumulation is much slower because of the drying effect of increasing elevation and the fact that a proportion of any additional mass is lost through enhanced discharge32.

Time-integrated correlations reveal key insights into the way that our modelled ice-sheet volume resonates with our oscillating climate forcing, but this single metric masks some of the complexity apparent in the time evolution of our modelled domains. One intriguing behaviour that is evident in the timeseries data is that of harmonic resonance. Under certain conditions, modelled ice sheet responses resonate with the applied climate forcing but with a frequency that is an integer multiple of the forcing period. For example, in our 100 kyr experiments using a cool (c. +1.5 °C), but not cold, climate, and a medium to strong ocean forcing, ice volume peaks every 50 kyr, rather than every 100 kyr, reaching maxima both during peak warm and peak cold climates (Supplementary Fig. S5).

Empirical evidence lends support for such resonant effects. For example, a 50 kyr-period climate oscillation can be detected in sedimentary evidence of East Antarctic Ice Sheet fluctuations33 (Fig. 2e, f), and also in a previously-published compilation of benthic δ18O data34 (Fig. 2a–d). In the latter case, our new analysis of 50 kyr and 40 kyr variance utilises multitaper method power spectral analysis35 with three 2π prolate data tapers, and a 1-million year moving window. Variance in the 40 kyr band (Fig. 2b) is determined by integrating power between 0.023–0.027 kyr-1, while variance in the 50 kyr band (Fig. 2c) integrates power between approximately 0.0182–0.0222 kyr-1. Intervals of enhanced 50 kyr variance are identified by dividing the 50 kyr integrated power by the 40 kyr integrated power (Fig. 2d). This approach is advantageous for two reasons. First, there is a very weak 52–54 kyr obliquity signal in the theoretical astronomical solution36. Although the expected signal is small in amplitude, for a linear response it should increase with overall obliquity forcing. In normalising by the strongest expected obliquity component (40 kyr), we are accounting for increases in overall obliquity forcing and response. Second, if 40 kyr obliquity forcing is absent in the proxy record, it is useful to account for secular changes in background power within the δ18O record over time. There are a range of methodologies to do so, but normalising by the power in a nearby bandwidth is a reasonable approach. All analyses were conducted using the Astrochron package for R (ref. 37).

Fig. 2: Evidence of 50 kyr oscillations in climate records.
Fig. 2: Evidence of 50 kyr oscillations in climate records.
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a The global δ18O megasplice record of ref. 34. Evaluation of the megasplice using evolutive power spectral analysis with a 1 million year moving window, which reveals b 40 kyr and c 50 kyr components. d 50 kyr response normalised by the 40 kyr response, highlighting times when the 50 kyr signal dominates (yellow shading). High-resolution sediment counts from close to the East Antarctic Ice Sheet margin e also reveal a 50 kyr signal in a periodogram of ice sheet behaviour (f, yellow shading) during Miocene warming, consistent with the global δ18O record (figure modified from Ref. 33). Solid and dotted red lines are conventional AR1 noise model confidence levels.

In our modelled scenarios exhibiting this 50 kyr harmonic period, the close balance between warm-climate precipitation and moderate to high-amplitude melt rate variability means that both during glacial and interglacial periods there is sufficient surface accumulation to offset the changing rate of sub-ice shelf melt, and thereby encourage net growth. This delicate balance, however, is only seen in a few scenarios representing ‘intermediate’ climate states. In the benthic δ18O data34 (Fig. 2a–d), the highest 50/40 kyr values (panel d) are all associated with reductions in 40 kyr response (panel b), and increased 50 kyr response (panel c), supporting a non-obliquity origin (for example at 16200–16900 ka, 29000–29300 ka). These episodes also appear to be associated with reductions in obliquity sensitivity38, which would suggest a connection to periods of transition towards a more terrestrial, rather than marine-based, ice sheet that is dominated by enhanced ‘local’ precession/eccentricity forcing, including some ‘local’ obliquity (not ocean/gradient sourced). Although other explanations may exist, we suggest that one plausible interpretation of both modelled and observed periods of 50 kyr resonance is that they reflect transitional states of the ice sheet–climate system that are intermediate between warm and cold states.

Given that this state of delicate balance would have to be encountered during long-term Cenozoic cooling (from warmer-than-present to the current, cold, state), we run three 1 Myr-long experiments in which we either i) linearly cool the climate by 4 °C over the length of the simulation, keeping forcing amplitudes unchanged, ii) maintain atmospheric temperature oscillations at a constant mean but linearly increase the amplitude of melt rate variability, or iii) linearly cool the atmosphere and also increase sub-shelf melt rates. The results of these experiments are shown in Fig. 3. Under an atmospheric cooling scenario in which the amplitude of melt rate variability is high for the duration of the run, modelled ice volume fluctuations (Fig. 3a, blue line) exhibit peaks at 50 kyr cyclicity during the first third (warmest part) of the simulation, with a weaker 100 kyr signal at this time (Fig. 3b, red line). As atmospheric cooling progresses, the 100 kyr signal becomes increasingly apparent (Fig. 3b, yellow line), and by the final third of the experiment the 100 kyr signal is by far the most dominant (Fig. 3b, blue line). Fitting linear trends to the time evolution of ice volume we see increasing skewness (Fig. 3c, blue line), a measure of asymmetry, and to a lesser extent a decrease in kurtosis (Fig. 3c, brown line), perhaps suggesting a more ‘peaky’ signal. In contrast, neither of the other two scenarios reveal the same dominant 50 kyr harmonic resonance, or the same degree or sign of changing asymmetry and peakiness. On this basis, we infer that although strong ocean forcing is necessary to produce changes in the phase and asymmetry of our modelled ice volume oscillations (Supplementary Fig. S5), the emergence of strong half-period ice sheet resonance in our model depends additionally on an evolving atmospheric temperature and associated moisture supply.

Fig. 3: Climatic controls on ice sheet resonance.
Fig. 3: Climatic controls on ice sheet resonance.
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a Ice volume change under a climate cooling of 4 °C and uniformly strong ocean forcing, both oscillating at 100 kyr periodicity. b Periodogram showing the dominant frequencies in ice volume change during initial third (red), middle third (gold), and final third (blue) of the one million-year long simulation. c Changes in skewness and kurtosis of ice volume change based on a 100 kyr moving window, reflecting increasing asymmetry of ice volume oscillations under increasingly cooler atmosphere. d–f Same as (a–c) but for a simulation in which mean atmospheric temperatures remain constant but ocean forcing increases in strength. g–i Same as (a–c) but for a simulation in which the cooling climate of a–c is combined with the strengthening ocean forcing of d–f. The strongest evidence for both increasing asymmetry as well as harmonic (50 kyr) resonance is seen in the first scenario, suggesting that atmospheric, rather than oceanic, forcing plays the most substantial role in these changes.

Cycle asymmetry

In the 500 kyr and 1 Myr experiments described above, asymmetrical ice volume oscillations arise within a small climatic state-space (primarily cold or cool atmosphere and strong ocean forcing), and only under 100 kyr periodicity forcing (Supplementary Figs. S5–S7). Previous studies have explained 100 kyr asymmetric oscillations of, for example, the Laurentide Ice Sheet as a function of solid-Earth feedbacks or calving of lake-terminating margins39,40,41. In both cases, the asymmetry reflects a non-linear ice sheet response to an imposed forcing, and a hysteresis in ice volume42,43. In our experiments the non-linearity arises only under certain climatic conditions, but despite this behavioural pattern being relatively rare (at least within the ensemble described here), the close similarity of this waveform to ice volumes inferred from benthic δ18O records of the Late Pleistocene13, and during climatic cooling of the Oligocene–Miocene transition23, suggest that closer inspection of the mechanism behind this pattern is warranted.

We ran a new ensemble of eighteen 500 kyr experiments that use identical climatological and glaciological parameterisations, under two climate state-types (cold and dry with strong ocean, warm and wet with weak ocean), with the only difference applied being the mantle viscosity defined in the elastic lithosphere, relaxing asthenosphere solid Earth model. Asymmetry arises in our original ensemble under the cold climate state with a viscosity of 1.0 × 1021Pas−1, so we explore values spanning four orders of magnitude from very weak (1.0 × 1019Pas−1) to very strong (1.0 × 1023Pas−1), to bracket plausible scenarios for both Greenland and Antarctica (Fig. 4). Where the mantle viscosity is low, such as is typical for West Antarctica today, our 100 kyr forcing using a cold climate state with strong ocean results in ice volume oscillations that are near-identical to one another, regardless of the mantle viscosity imposed (Fig. 4a), reaching maxima during glacial climates and minima during interglacials. Although the fluctuations are not perfectly sinusoidal, they are nonetheless symmetrical. Increasing mantle viscosity from 5.0 × 1020Pas−1 (Fig. 4a) to 1.0 × 1021Pas−1 (Fig. 4b), however, results is a markedly different response, with much greater loss of ice volume during deglacial warming and prolonged recovery of the ice sheet during the interglacial, leading to an asymmetric waveform. Further increasing the mantle viscosity by the same increment, however, produces a response in which ice volume fluctuations once again become symmetrical with respect to applied climate forcing, but which lead to complete loss of the ice sheet during the warmest periods. At viscosity values of 5.0 × 1022Pas−1 and 1.0 × 1023Pas−1 (Fig. 4c), ice regrowth during successive glacials is increasingly prevented because the topographically-depressed bed is very slow to rebound. As a consequence, by the time the land surface is high enough to encourage ice advance, the ocean is too warm, and regrowth is delayed until the subsequent cooling phase. This is especially the case under cold, dry atmospheres with strong ocean forcing because precipitation rates are too low to offset oceanic melt. In warmer, wetter, scenarios this asymmetry does not arise because precipitation rates are much higher, and more effectively drive ice sheet regrowth. Ultimately, this combined solid Earth–climate control on ice volume cycling means that its phasing with climate forcing becomes increasingly less clear, and eventually the ice sheet is entirely prevented from regrowing for several glacial cycles.

Fig. 4: Mantle-sensitivity of ice volume response.
Fig. 4: Mantle-sensitivity of ice volume response.
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a Ice volume variability under a cold, dry, climate state with strong ocean forcing, and with a relatively weak mantle (1e19--5e20 Pa s-1). b Identical simulations as (a) but with a stiffer mantle (1e21--1e22 Pa s-1), and c very stiff mantle (5e22--1e23 Pa s-1). d–f Identical simulations to those shown in (a--c) but under a warmer, wetter, climate with weak ocean forcing. The default mantle viscosity used in the main experiments is 1e21 Pa s-1. Note that the asymmetric cycles only occur in (b).

Repeating this ensemble under a warmer, wetter, climate with a weak ocean forcing illustrates that the majority of these mantle-dependent behaviours are consistent, irrespective of climate (Fig. 4d–f). The only notable difference between the two ensemble subsets is that the asymmetric cycles evident in the cold atmosphere, strong ocean scenario are not evident in the equivalent simulation using a warm atmosphere and weak ocean. From these simulations we conclude that modelled ice sheet evolution is particularly sensitive to mantle rheology, but only within a certain range. A very weak mantle prevents asymmetry arising, whereas a very strong mantle initially prevents retreat, then prevents regrowth. Ultimately, ice volume becomes decoupled from climate and ‘skips’ one or more glacial–interglacial cycles. Although our mantle experiments only explore the 100 kyr forcing, a future research avenue might be to investigate whether ‘skipped’ 23 kyr cycles could be an alternative mechanism for the  ≈ 50 kyr periodicity seen in the proxy records described above. In the 100 kyr experiments, however, the period-doubling ice volume response seen under intermediate or transitional climate states does not appear in either of the scenarios investigated here, suggesting that—at least at this timescale—the harmonic response is climatically-driven, and is independent of solid Earth feedbacks.

Calving dynamics

Having established that the climatically-driven phasing of ice volume (and by inference, global sea level) is strongly tied to a solid Earth-modulated balance between precipitation-dominated surface mass balance and ocean-driven ice-shelf melt, both under steady state as well as transient cooling scenarios, our final mechanistic exploration investigates how such behaviours might be expressed in other proxy records, such as those that infer ice dynamics from records of iceberg-rafted debris (IBRD). To do this we examine how ice discharge, specifically in terms of iceberg calving, is related to climate forcing under different climate states. As with ice volume (and area), our matrix of correlation coefficients illustrates a switching between cold atmosphere / strong ocean and warm atmosphere / weak ocean states when melt rate amplitude as well as magnitude is varied (Fig. 1; Supplementary Fig. S4)). Under such scenarios, the transition from anti-phase to in-phase behaviour maps out a diagonal across the suite of considered environmental combinations. When melt rate amplitude is held constant, however, this switching is absent in the 100 and 23 kyr-forced subsets, and only very weakly expressed in the 41 kyr subset (Supplementary Fig. S4). These relationships, and their dependencies, are similar to those described above in relation to ice volume. Yet closer inspection of individual calving timeseries reveals a more complex behaviour than is captured by time-integrated correlation coefficients. Under many scenarios, calving fluxes rise and fall periodically in a similarly sinusoidal way to ice volume oscillations (Supplementary Figs. S5–S7). Yet in some scenarios, calving flux variability is distinctly leptokurtic, dominated by short, sharp, spikes in discharge, with these spikes occurring asymmetrically with respect to the imposed forcing cycle (Supplementary Figs. S8–S10).

Taking one example of such a response, Fig. 5 illustrates calving fluxes under constant atmosphere and strong ocean forcing, at 41 kyr periodicity, where peakiness and asymmetry are both well-expressed. In this example, the sinusoidal melt rate forcing (Fig. 5a, blue line) exhibits maximum rates of warming (purple lines) midway between minima and subsequent maxima (red lines). The timings of peak calving fluxes (Fig. 5b, green line) do not align with the timings of maximum melt rate or maximum rate of melt rate change, but rather with times when melt rates are changing most quickly (Fig. 5b, brown bars). Peak calving discharge in this scenario is characterised by an abrupt (decadal-scale) onset and rapid increase over centuries to a millennium, that then persists for around 6000 years before abruptly decreasing to a much lower rate for the remainder of the 41,000-year cycle. The rapid onset of modelled iceberg pulses is consistent with geological records of calving events from ice sheets44, and reflects an abrupt switching mechanism that responds sensitively not to the magnitude or rate-of-change of sub-ice shelf melt rates, but to peak melt-rate acceleration (Fig. 5b, brown bars). Calving pulses are typically shorter and more abrupt under higher rather than lower frequency forcings, typically amounting to 15–25% of the wavelength of the climate oscillation driving them (Supplementary Figs. S5–S7).

Fig. 5: Iceberg discharge dynamics under a cold climate scenario.
Fig. 5: Iceberg discharge dynamics under a cold climate scenario.
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a Blue curve indicates sinusoidal sub-ice shelf melt rate variability at 41 kyr periodicity. Vertical lines denote times of maximum melt (red) and fastest increase in melt rate (purple). b Timeseries of iceberg calving flux showing multi-millennial-scale pulses coincident with, or shortly following, periods when melt rate acceleration (blue curve) is greatest (brown bars). Timeseries resolution is 10 years. Onset of calving pulses initiates within decades, consistent with geological records44. Alternative plots showing calving patterns at 100 and 23 kyr periodicities are shown in Supplementary Fig. S11.

Conclusions

Our ensemble of 198 simplified geometry ice sheet simulations reveal mechanistic insights into the way that climate forcing drives ice sheet and global sea level variability over centennial to multi-millennial timescales. Comparison of these experiments to previously-published proxy and model-based studies supports the general conclusions that whereas surface mass balance is responsible for controlling smoothly-varying ice volume changes that are typically in phase with temperature, it is primarily ice shelf basal melt changes that drive more rapid mass loss or gain that is anti-phased with temperature. The sign of glacial–interglacial phasing, that is, the ice sheet’s resonance with imposed climate, depends heavily on the prevailing climate state, meaning that the direction of the response can switch from one sign to another if the background climate warms or cools sufficiently. Simulations bridging intermediate climate states reveal a resonance mechanism that, particularly under our 100 kyr forcing, gives rise to half-wavelength harmonic responses as a consequence of the delicate balance between surface accumulation and basal melt that occurs during phase (sign) switching. We also find that our modelled asymmetric ice volume cyclicity, characteristic of the Late Pleistocene, primarily occurs in our 100 kyr (rather than 41 or 23 kyr) forcing experiments, and in our model setup stems from a particular combination of oceanic forcing and solid earth feedbacks. Collectively, our suite of experiments implies that gradual cooling and/or warming of the atmosphere, combined with strong ocean-driven basal melt, could have plausibly led to fundamental changes in the periodicity, phasing, and asymmetry of past ice sheet growth and decay, and may similarly do so under warming levels likely to be encountered in coming centuries.

Methods

Model setup & experiments

We use the Parallel Ice Sheet Model, a thermodynamic coupled ice-sheet—ice-shelf model, which employs a hybrid stress balance approach that solves equations for ice flow by vertical shear and by horizontal stretching across the entire domain45,46. We parameterise our model using values previously found to work well for the Antarctic Ice Sheet47,48 and which yield a modelled ice dome that is of an approximately similar aspect ratio to present-day East Antarctic catchments (Supplementary Fig. S2). Alternative parameter combinations could no doubt be found that might also give rise to a modelled ice sheet of similar geometry and dynamics, but given the conceptual nature of the experiments being conducted we do not explore other possibilities. For computational tractability of the extremely long periods of simulation we employ a simplified geometry model based on a radially symmetric domain measuring 3200 x 3200 km (x, y). Although not explicitly intended to mimic any single extant ice sheet, we nonetheless define the climate for our present-day baseline climate state according to conditions typical for the East Antarctic Ice Sheet at the South Pole. Within our climate field we include an annual cycle for air temperature with an additional white noise signal that allows for unforced (stochastic) daily variability. Surface melting is calculated using a positive degree-day approach. Climatic feedbacks are captured in three principal ways. In the spatial domain we define a vertical atmospheric temperature lapse rate of -8 °C / km, such that increasing surface elevation results in colder air temperatures. We also define a precipitation lapse rate of -100 mm yr−1/ km based on observations that air masses at higher elevations are drier due to orographically-forced loss of moisture over the ice sheet49,50. In the temporal domain we also allow our prescribed present-day precipitation field to scale with air temperature, such that colder climates are drier than warmer climates5,51. Air temperatures themselves vary through time according to a simple sinusoid oscillating at one of the three primary Milankovitch periodicities (23, 41, and 100 kyr). These single-frequency forcings serve to isolate how different responses of the ice sheet might arise, but do not fully reflect real-world insolation variability arising from the combination of all three periodicities. As such, we are not able to specifically explore non-linearities in ice response that might occur due to, for example, eccentricity-forced amplitude modulation of the precession signal52.

To capture ice sheet behaviours under a range of background climate states, we impose a uniform adjustment relative to our initial conditions, such that the modelled mean climate anomalies range from -2.5 °C to +5.5 °C in steps of 2 °C (Supplementary Fig. S1a). The amplitude of glacial-interglacial variability is held constant at 5 °C, informed by palaeoclimate reconstructions of the Plio-Pleistocene53,54. Our model is fully isostatic and makes use of a non-local Earth deformation model based on a Fast Fourier Transform55. We keep sea level fixed for all simulations, to enable clearer comparisons between the results. Where grounded ice enters the ocean and achieves flotation, our hybrid shallow ice approximation (SIA)/shallow shelf approximation (SSA) model allows ice shelves to form. The horizontal extent of these shelves is controlled through a calving scheme that calculates mass loss as a function of ice thickness or horizontal strain rate, depending on which condition dominates in each cell56. The thickness of ice shelves is modified by sub-ice shelf melting, which we define as a spatially-uniform basal melt rate that follows a sinusoidal timeseries in phase with air temperature fluctuations. In one set of experiments (‘primary ensemble’) we increase the amplitude of glacial–interglacial basal melt rate variability in a way that maintains the same melt rate during each successive glacial period, but allows the interglacials to warm by increasing amounts (Supplementary Fig. S1b, solid lines). In a duplicate set of experiments (‘secondary ensemble’) we make uniform baseline adjustments to the basal melt forcing in a comparable way to air temperature perturbations, but keep the amplitude the same (Supplementary Fig. S1b, dashed lines). In real terms this means that experiments using the ocean forcings ‘constant’, ‘cold’, and ‘weak’ are identical between the two ensembles. Using this dual approach allows us to compare the relative impacts of melt-rate magnitude and melt-rate variability (amplitude), the latter of which can be considered a surrogate for the temperature-dependent non-linearity of sub-ice shelf melt sensitivity57. The junction between grounded and floating ice is dynamically calculated at every time step using a two-dimensional sub-grid-scale scheme that improves grounding line behaviour58. We do not, however, use sub-grid melt interpolation.

All simulations start from an initially ice-free state, with ice sheet inception taking place on a horizontal bed 100 m above sea level in the central part of the domain. Ice accumulation leads to expansion of the ice sheet into the ocean, as well as to isostatic loading that depresses the bed below sea level. Once this expansion has taken place, the ice sheet is influenced by changes in both oceanic as well as atmospheric forcing, to a degree modulated by the extent of its fringing ice shelves. Within the first approximately 5000 years of the simulations the ice sheet has reached a thickness and lateral extent comparable to current East Antarctic drainage basins (Supplementary Fig. S2).

Limitations

As with any conceptual modelling approach, the simplifications necessary to make the investigation viable mean that any results must be considered in the context of the methodological approximations. The physics of our model setup are as comprehensive as our previous implementations in which realistic geometries and forcings have been applied47,48, so the limitations of the current study lie more in the simplified representations of ice geometry, bedrock topography, and temporal pattern of our climatic forcing. In terms of ice geometry, our radially-symmetric ice dome captures the approximate aspect ratio of extant East Antarctic drainage basins, such as the Wilkes Subglacial Basin. Considerable variability exists, however, both within the modern ice sheets of Antarctica and Greenland (Supplementary Fig. S2), and in former northern hemisphere ice sheets, such as the Laurentide, Scandinavian, and Barents-Kara systems. Large sectors of those former ice sheets were marine-based, for example in the Hudson Bay area of the Laurentide Ice Sheet59 and in the Baltic/Gulf of Bothnia sector60,61 and northwest margin62 of the Scandinavian Ice Sheet. The Barents-Kara system is thought to have been almost entirely marine-based63. So whilst our domain configuration is configured for, and comparable to, currently marine-based sectors of the EAIS, and similar in terms of ice thickness and bedrock elevation to major glaciers of West Antarctica and Greenland (Supplementary Fig. S2c, d) it should also be reflective of marine-based sectors of palaeo-ice sheets. It may, however, be less representative of palaeo and present ice sheet catchments that have or had land or lake-terminating margins40,41.

From a topographic perspective, our simplified experiments lack the roughness and complexity that characterises the real Antarctic or Greenland bed (Supplementary Fig. S2). Consequently, ice flow is less impeded than it might be, and because there is no topographic variability perpendicular to ice flow, spatial patterns of the velocity field evolve entirely due to internal feedbacks rather than from external control. An additional question is the extent to which our model represents the palaeo topography of, for example, East and West Antarctica. Whilst constraints are sparse, comprehensive topographic reconstructions show that most of the present West Antarctic Ice Sheet bed was close to or below present-day sea level from at least around 23 Ma onwards, and possibly even earlier64. Our ice-free initial bed elevation of +100 m is therefore a reasonable approximation of what is known for the former geometry of West, as well as East, Antarctica.

Climatologically we capture many of the relevant atmospheric relationships between, for example, temperature, precipitation, and elevation. Furthermore, our simulations span a very wide breadth of climatic environments (both in terms of air temperature and ocean forcing) that are representative of climates past, present, and future for extant and former ice sheets. As such, we believe that our modelled ice sheet responses should be generalisable to a wide range of present and former ice sheets. We do not, however, have the capacity to include albedo–temperature feedbacks that might arise from changing snow and ice cover. This process could lead to a non-linear climatic response to external (insolation) changes, which in turn could influence ice sheet fluctuations43. Furthermore, the time-dependent anomalies we apply are simple sinusoids and do not reflect ‘real-world’ palaeoclimate scenarios in which the symmetry, amplitude, and internal variability of former glacial cycles deviated substantially from a time-invariant sinusoidal shape. As a result, our modelled ice sheet responses will not truly reflect former ice sheet behaviours arising from more complex forcing scenarios. Similarly, it is possible that our modelled relationships could be different if the prescribed amplitude of atmospheric and oceanic fluctuations was greater or smaller. Our adopted value of glacial–interglacial air temperature fluctuation was chosen based on an approximation of Late Pleistocene climate53, reduced slightly to account for less cold glacial maxima of older periods, such as the Pliocene54. The oceanic forcing range was chosen based on values that yield realistic oscillations of the ice margin under a present-day to glacial maximum climate.

These limitations notwithstanding, the value of the simplified approach is that it facilitates a clearer analysis of theoretical ice sheet responses to simple patterns of climate change, without the complications and obfuscations introduced by noisy and non-stationary environmental change imposed upon an ice sheet environment that is prevented from responding cleanly by its spatial complexity. Furthermore, because marine-based sectors of the former AIS, Laurentide Ice Sheet, Scandinavian and Barents-Kara ice sheets were so extensive, the ice sheet responses we simulate should be detectable in empirical reconstructions of former ice masses. In short, our conceptual approach affords the greatest opportunity to separate signal from noise in orbital-scale ice sheet–climate interactions.