We start with the mass conservation equation integrated over the entire ice-sheet surface: d(HS)dt=AS. Here, A is total snow accumulation minus ablation and calving. According to Eq. (5), H=ζS1/4,ζ=μa(ρg)n12n+2; then, assuming ζ = const, Eq. (18) can be written as dSdt=45ζ-1S3/4A.

It is interesting that the fundamental mass balance Eq. (19) provides some support for the hypothetical model of Huybers (2009). If we consider the right side of Eq. (19) to be the rate of deglaciation, then explicit finite differencing of Eq. (19) would give us St=St-1+45ζ-1St-13/4AΔt, where the rate of deglaciation depends on the dimension of the previous glaciation, St-1. It does not imply that ice can “remember” its previous state; Eq. (19) simply tells us that deglaciation of bigger ice sheets occurs on a bigger area.

We adopt the following parameterization for the total mass balance A appearing in Eq. (19): A=A1-A2-A3-A4.

Here, the following statements are true. a.

A1=a is the snow accumulation rate.

Scales (Eqs. 14–16) were obtained for the equilibrium state of the ice sheet. We now account for the fact that variations described by Eqs. (14)–(16) follow climate change with a time lag equal to a characteristic vertical advection rate H/a: dθdt=Δθa+ΔθT+ΔθS-θH/a. We substitute scales (Eqs. 5 and 14–16) into Eq. (21). We combine Δθa and ΔθT together, assuming that Δθa is proportional to ΔθT and ΔθT is proportional to ω: Δθa+ΔθT∼αω. We replace ΔθS∼ΔS/S in Eq. (16) with ΔθS∼β(S-S0), where S0 is a reference glaciation area. Here, α and β are sensitivity coefficients. We also modify a like in Eq. (20), but without basal sliding because basal sliding is associated with horizontal movements and does not contribute to vertical advection: A1-A2-A3=a-εFS-κω. Thus Eq. (21) takes the following form: dθdt=ζ-1S-1/4a-εFS-κωαω+βS-S0-θ.

Since the ocean is the main component of the climate system on ice-age timescales, for climate temperature ω we adopt the ocean temperature equation used by Saltzman and Verbitsky (1993a, b) as a cumulative proxy for the outside-of-glacier climate: dωdt=γ1-γ2S-S0-γ3ω. Here, ω is a characteristic climate temperature, γ1 and γ3 define its steady-state value, and 1/γ3 is a response-time constant. It is reasonable to assume here that ω is controlled by polar ice sheets and associated ice shelves (-γ2[S-S0], γ2 being a measure of the strength of such control) but the term -γ2[S-S0] may also contain effects of albedo change or other atmospheric feedbacks. The timescale 1/γ3 is of the order of magnitude of a few thousand years to capture the involvement of the deep ocean.

Our three-variable dynamical global climate system then takes its final shape. dSdt=45ζ-1S3/4a-εFS-κω-cθdθdt=ζ-1S-1/4a-εFS-κωαω+βS-S0-θdωdt=γ1-γ2S-S0-γ3ω

First, we will inspect feedback loops in Eqs. (24)–(26) as they are portrayed in Fig. 1.

There are three major feedback loops in our system, one negative feedback and two positive feedbacks. Basal temperature θ provides negative feedback to the glaciation area S: when ice sheet grows it increases basal temperature (βS) and creates more favorable conditions for basal sliding, which tends to reduce glaciation dimensions (-cθ). When ice retreats, the bottom temperature is reduced and tends to preserve glaciation area. There are two positive feedback loops associated with climate temperature ω: when ice grows it reduces climate (e.g., ocean) temperature (-γ2S), which in turn increases mass balance on the ice-sheet surface (-κω). It also reduces the temperature on the ice-sheet surface (αω), and this temperature change will propagate to the bottom. Both of these changes will help to build even bigger ice sheet. When ice sheet retreats, climate (e.g., ocean) temperature rises, and through increased ablation and increased bottom temperature it makes this retreat even faster.

Without astronomical forcing, the system (Eqs. 24–26) evolves to equilibrium (Fig. 2). The results of Fig. 2 were obtained with the following values of system parameters (Table 2).

For a steady-state solution in Fig. 2 (panel a) ε=0. To test the stability of the equilibrium point, we repeated calculations with a weak insolation forcing, ε=0.01. The results, presented in Fig. 2 (panel b), show that the equilibrium point is stable.

Initial conditions are S(0)=10106km2θ(0)=0∘Cω(0)=2∘C.

This steady-state solution can be obtained from the system (Eqs. 24–26) if we set all derivatives equal to zero. 0=45ζ-1S3/4a-εFS-κω-cθ0=ζ-1S-1/4a-εFS-κωαω+βS-S0-θ0=γ1-γ2S-S0-γ3ω Then, for ε=0 〈S〉=S0+ac-α+κcγ1γ3β-α+κcγ2γ3,〈θ〉=a-κωc,〈ω〉=γ1-γ2〈S〉-S0γ3. Here, 〈S〉, 〈θ〉, and 〈ω〉 are the steady-state solutions of corresponding variables.

It can be seen that without climate feedbacks involved (γ1=γ2=0, or α=κ=0), the steady-state extent of glaciation is defined by ice-sheet properties only, 〈S〉-S0=a/(βc). This glaciation extent may be reduced by a climate damping factor (α+κ/c)γ1/γ3 or enhanced by climate positive feedback (α+κ/c)γ2/γ3.

We will measure the total climate impact Ω as a difference between the strength of climate positive feedback and a damping factor Ω=α+κcγ2γ3-1S0α+κcγ1γ3 and introduce a dimensionless number, V (variability number), as a ratio of the intensity of climate impact Ω on the intensity of ice-sheet own negative feedback: V=Ωβ=α+κcγ2γ3-1S0γ1γ3β. This number is small if the positive feedback is weak and it is high otherwise. We will now demonstrate that the variability number, V, defines different modes of glacial rhythmicity.

We will now present the results of dynamical system evolution for the same set of parameters as in Sect. 4.1 but ε=0.11 km kyr-1. Very similar results have been produced for other sets of parameters as long as V=0.75±0.02; for example, α=2.1 and κ=0.006 km kyr-1 ∘C-1 (slightly increased positive feedback), β=1.8 ∘C 10-6 km-2) (slightly decreased negative feedback), γ2 and γ3 are proportionally increased by about 10 %, or κ and c are proportionally increased by about 10 %, and so on. In Fig. 3 we show the results of calculations of the glaciation extent for the last 1 000 000 years together with the LR04 benthic foraminifera δ18O stack (Lisiecki and Raymo, 2005).

The major events of the last million years are reproduced reasonably well, except for the interglacial of 400 kyr ago (marine isotopic stage 11), which has usually been described as a long and deep interglacial and obviously the last interglacial starting 30 000 years ago instead of 10 000 years ago, driving the present day into a glaciation. The timing of all other interglacials coincides with Past Interglacial Working Group of PAGES (2016) data. The deglaciation associated with termination V leading to stage 11 creates a difficult problem indeed (see, e.g., Berger and Wefer, 2003). It is remarkable that the model (Eqs. 24–26) actually simulates a deglaciation around 430 ka BP, while the celebrated Imbrie and Imbrie (1980) model, for example, missed it and that termination remains a challenge for a model of intermediate complexity with an interactive carbon cycle such as CLIMBER (Victor Brovkin, personal communication, 2018). Furthermore, the Fig. 3 results illustrate the fact that the termination V was a slower process than other deglaciations, which is reasonably consistent with sea-level reconstructions (Rohling et al., 2010, 2014). However, the simulated sea level during stage 11 is not as high as the observations suggest, and it is possible that the model (Eqs. 24–26), by using mid-June insolation as the forcing metric, is not giving enough weight to obliquity. Some tuning efforts at this point could be exercised to get an even better fit, but we believe that exact replication should not be expected from a simple model like ours taking into account all uncertainties involved in parameters values. It should also be taken into consideration that the V number (and each of the eight parameters involved in its definition) does not have to be a constant as has been implied in this section, but may experience a slow trend. For example, when the V number evolves from its low early Pleistocene value to its higher late Pleistocene value to simulate the mid-Pleistocene transition (see Sect. 4.5 below), the glacial variability, consistent with instrumental records, is at maximum amplitude around 400 kyr ago. The timing of the last glacial cycle for an evolving V number is also close to the LR04 record. It should also be recalled that the benthic curve is not strictly a measure of only ice volume.

Figure 4 compares the modeled climate temperature ω with the reconstruction of tropical ocean temperatures provided by Herbert et al. (2010). It can be noted that calculated temperature changes and ocean temperature data generally evolve in phase, though, like in the case with glaciation extent (Fig. 3), the last interglacial slightly leads the data.

In Fig. 5 we compare spectral diagrams for calculated glaciation volume (ζS5/4) against LR04 benthic δ18O data and for calculated ω values against Herbert et al. (2010) tropical ocean temperature data. In both cases, the results are reasonably consistent: in all cases periods of nearly 100 kyr clearly dominate. Precession and obliquity periods can be easily distinguished, though in both empirical diagrams 40 kyr periods are more visible than in the model.

The sequence of events that produce 100 kyr periods can be visualized by zooming into a typical cycle. For this purpose, in Fig. 6 we show the evolution of the derivatives of the model variables, together with insolation changes.

It can be seen that when ice grows, its basal temperature increases (with a time lag) due to increased internal friction and due to a more prominent role of the geothermal heat flux. As we discussed earlier, it provides a negative feedback to ice-sheet growth. At the same time, climate temperature is getting lower as well, and it creates a positive feedback for ice growth. Four times during this cycle the astronomical forcing (precession) “challenges” ice growth by trying to switch it to the disintegration mode. These attempts fail three times: ice volume is not high enough and positive feedback from the climate acting through its mass balance (κω) and through basal temperature (αω) is not strong enough to counteract ice-sheet own negative feedback (βS). The fourth attempt succeeds. Ice is high enough, positive feedback is strong, and disintegration proceeds until ice almost disappears. This scenario is consistent with the idea that glacial cycles skip insolation cycles until ice has grown to a point that precipitates its full disintegration, as conceptualized among others by Paillard (1998) and Tzedakis et al. (2017). It is to some extent reminiscent of the phase-locking mechanism mathematically analyzed by Tziperman et al. (2006) and Crucifix (2013). In our model, though, the role of insolation is much bigger than in phase-locking scenarios: insolation variations define not only the phase of ice fluctuations but, being nonlinearly amplified, they actually define the period of glacial rhythmicity as well. The mechanism involved in the glacial frequency setting is similar to a “period-two” (period doubling) response to astronomical forcing exhibited by a conceptual model of Daruka and Ditlevsen (2016). To illustrate this “period-two” behavior, we solve our system (Eqs. 24–26) with astronomical forcing replaced by a single obliquity period sinusoid FS=εsin⁡(2πt/41 kyr). The results of the calculations, presented in Fig. 7, show that for a relatively high amplitude of forcing (ε>0.07) our model produces fluctuations of a doubled obliquity period. It is interesting that the signatures of such period doubling have been found in LR04 variability (Vakulenko et al., 2011).

Until full dynamical system analysis is performed, which is clearly outside of the scope of this paper, we will (loosely) reference this mechanism as “nonlinear amplification” and we will demonstrate that this mechanism manifests itself differently depending on the V number, which is a measure of the positive-to-negative feedback ratio.

We will now present the results of dynamical system evolution for the same set of parameters as in Sect. 4.2 but α=0 and κ=0 (or γ2=γ1=0); i.e., positive feedback is weak, V=0. The results of the calculations are shown in Fig. 8.

The local mid-July insolation at 65∘ N, FS (Berger and Loutre, 1991), is dominated by a precession period of about 23 kyr (Fig. 8b), while the early Pleistocene paleoclimatic data indicate a stronger response in the 40 kyr period band. The integrated summer insolation, which is dominated by the 41 kyr obliquity cycle (Huybers, 2006), is often invoked to explain this discrepancy. It is interesting that our model produces fluctuations with a dominating obliquity period (Fig. 8c) without needing integrated summer forcing. It happens due to period-doubling dynamics, in this case precession periods. To illustrate this phenomenon, we solve our system (Eqs. 24–26) with astronomical forcing replaced by a single precession period sinusoid FS=εsin⁡(2πt/23 kyr). The results of the calculations, presented in Fig. 9, show that when V=0 our model produces fluctuations of a doubled precession period.

If we also set S0=2 (106 km2) we will get fluctuations with a lower mean level of ice-area extension, which may be relevant to early Pleistocene fluctuations (Fig. 10). Astronomical periods are still very visible in the ice spectral diagram of ice fluctuations.

In this section we show that the system would produce fluctuations of about 400 kyr if the positive feedback increased further (V→1). Specifically, we present the results of dynamical system evolution (Fig. 11) for the same set of parameters as in Sect. 4.2 but β=1.57 ∘C 10-6 km-2), i.e., V=0.94. Very similar results have been produced for different sets of parameters as long as V∼0.95; for example, γ2=0.26 ∘C 10-6 km-2 kyr-1.

The results may seem to be counterintuitive at first because one would expect faster ice growth as the positive feedback increases. However, it makes sense by observing that with stronger positive feedback every astronomical “challenge” is more successful in retreating ice. Consequently, it takes longer to grow ice to the level at which its retreat can be really strong. The ratio between positive and negative feedbacks determines how long it takes for ice to grow until a level at which it is vulnerable to the astronomical “challenge” (in our model it corresponds roughly to S∼20×106 km2). This example also shows that if the ratio between positive and negative feedbacks were wrong (i.e., the wrong V number), then the astronomical forcing would in no case reproduce the correct sequence of events. In other words, incorrect physics cannot be rescued by tuning the strength of the astronomical forcing.

If we attribute the V=0 mode to the early Pleistocene and the V=0.75 mode to the late Pleistocene, then the V number may provide some guidance about possible scenarios to explain the mid-Pleistocene transition. Since the V number is a combination of multiple parameters, it implies that multiple scenarios are possible. For memory, V=(1/β)(α+κ/c)(γ2/γ3-γ1/γ3/S0), where the parameters β, c, and S0 are glaciation parameters, α and κ are parameters associated with atmospheric circulations, and γ1, γ2, and γ3 are ocean parameters. Slow variations in atmospheric, ocean, or even glaciation parameters (Clark and Pollard, 1998) are quite possible. It is outside of the scope of this paper to provide a full list of possible scenarios, but at least one plausible suggestion can be made. A commonly invoked scenario involves tectonic forcing, including volcanism and weathering processes, which could produce long-term variations in carbon dioxide such that it dropped from above 300 ppm during the early Pleistocene to its current values of about 250 ppm. The scenario remains commonly invoked (Saltzman and Maasch, 1991; Saltzman and Verbitsky, 1993a, b; Raymo, 1997; Paillard and Parrenin, 2004) even though it is fair to admit that the observations remain uncertain (Zhang et al., 2013) and that this scenario has been challenged (Honisch et al., 2009). Whether or not this specific mechanism is responsible for the regime change, it still seems reasonable to assume that the slow trend in climatic condition occurred and that it can cause a drift in the climate positive feedback (e.g., γ2), specifically an increase from lower to higher values. As an illustration, in Figs. 12–14 we present an example of such a transition. In this instance, at t=-3000 kyr, γ2, S0, and ε have been reduced by 60 % from their Mode I values and then increased linearly so that they regained 100 % of their Mode I values at present (t=0). A forced change in the glaciation reference line S0 defines a gradual increase in the global ice volume; changes in the sensitivity parameter ε cause an increase in fluctuation amplitudes. The most dramatic change, i.e., the transition from about 40 kyr period fluctuations to predominantly 100 kyr period fluctuations, is due to the intensified climate positive feedback , γ2, which caused a gradual increase in the V number from V=0.3 at t=-3000 kyr to V=0.75 at present (t=0). The example shows reasonable consistency between model results and the data: in model calculations and in the records of instrumental measurements (Fig. 13), the early Pleistocene is dominated by mostly 40 kyr fluctuations. At about 1.5–1.3 Myr ago, 100 kyr rhythmicity becomes predominant. Its amplitude is at a maximum about 400 kyr ago in both instrumental and model time series. The evolution of the cross-wavelet spectrum (Grinsted et al., 2004) between benthic foraminifera δ18O data and the system-produced (Eqs. 24–26) evolution of glaciation area S (Fig. 14) shows a mostly in-phase relationship between the model and measurement records on 40 and 100 kyr periods.

The system (Eqs. 24–26) can be expanded to include Southern Hemisphere (Antarctic) ice dynamics: dSdt=45ζ-1S3/4a-εFS-κω-cθdθdt=ζ-1S-1/4a-εFS-κωαω+βS-So-θdωdt=γ1-γ2S-So+SA-SAmax-γ3ωdSAdt=45ζ-1SA3/4aA-εFSA-κω-cAθA;SA≤SAmaxdθAdt=ζ-1SA-1/4aA-εFSA-κωαω+βSA-So-θA. Here, Eqs. (34) and (35) are identical to Eqs. (24) and (25). Equations (37) and (38) describe the evolution of the glaciation area of the Antarctic ice-sheet SA and its basal temperature θA; FSA is mid-January insolation at 65∘ S (Berger and Loutre, 1991); SAmax is the area of the Antarctic continent; aA is the rate of snow precipitation and cA is the intensity of basal sliding for the Antarctic ice sheet. All other parameters in Eqs. (37) and (38) are intentionally left the same as in Eqs. (34) and (35), following the assumption that the Antarctic ice sheet and Northern Hemisphere ice sheets are governed by the same physics. Equation (36) is the same as Eq. (26) except that it includes an additional term to reflect a potential control by the Antarctic ice sheet and associated ice shelves (-γ2[SA-SAmax]). The system (Eqs. 34–38) is represented graphically in Fig. 15.

The system (Eqs. 34–38) has been solved for the same set of parameters as in Sect. 4.1 but aA=0.5 km ky-1 and cA=0.01 km ky-1 ∘C-1 (implying more positive mass balance of the Antarctic ice sheet than Northern Hemisphere ice sheets). For this set of parameters SA=SAmax = const and the system (Eqs. 34–38) takes the following shape: dSdt=45ζ-1S3/4a-εFS-κω-cθdθdt=ζ-1S-1/4a-εFS-κωαω+βS-So-θdωdt=γ1-γ2S-So-γ3ωSA=SAmaxdθAdt=ζ-1SA-1/4aA-εFSA-κωαω+βSAmax-So-θA. Now, Eqs. (39)–(41) are exactly the same as in our original system (Eqs. 24–26). It means that all previous calculations were conducted with the assumption that the area of Antarctic glaciation remains constant. Interestingly, the system (Eqs. 39–43) has a “diode-like” structure (Fig. 16): Northern Hemisphere insolation has an important impact on Northern Hemisphere glaciation, the extent of glaciation effects on climate temperature, and climate temperature effects on Antarctic mass balance, surface temperature, and eventually ice-sheet basal temperature. FS→S→ω→αωandκω→θA But since SA=SAmax = const, there is no “opportunity” for Southern Hemisphere insolation to be amplified by climate positive feedback and to propagate its signal north. This produces a simple explanation for the synchronous response of the Northern and Southern Hemisphere to Northern Hemisphere insolation variations.