We previously noticed (Verbitsky et al., 2018) that with weak climate positive feedback (V∼0), the system, exhibits fluctuations in response to the astronomical forcing with a dominating period of about 40 kyr, which may arise as either a direct response to obliquity or a doubled-period response to the forcing associated with climatic precession (2×20kyr). When the climate positive feedback intensifies such that V∼0.75 and external forcing is strong, the system evolves with a doubled obliquity period. We can therefore assume that the period of the system response to the external forcing, P, is a function of the V number, the amplitude of the external forcing, ε, and the period of the external forcing, T. We thus begin with the most general hypothesis: 5P=ψV,a,ε,T. It is at this stage that the π theorem intervenes. Specifically, it stipulates that a physical relationship should not depend on a system of units, and therefore, in the dimensionless form, the number of dimensionless arguments is equal to the total number of the governing parameters minus the number of governing parameters with independent dimensions (Buckingham, 1914). If we select dimensions of ε and T as independent dimensions, then application of the π theorem to the Eq. (5) gives us the following: 6P/T=ΨV,ε/a7P=TΨΠ1,Π2;Π1=V,Π2=ε/a.

Figure 1 presents a sketch of what the function ΨV,ε/a may look like, qualitatively. The underlying idea is that the Pleistocene history of the climate system may be understood as a trajectory in the V,ε/a space (Crucifix and Verbitsky, 2019). The shape and location of the period-doubling domain Ψ=2 is expected to depend on the forcing period.

It is interesting that Fig. 1 is consistent with a similar map produced by a conceptual model built on a completely different principle, i.e., the simple oscillator type model of Daruka and Ditlevsen (2016). In both cases, the obliquity period doubling requires relatively intense external forcing in combination with the relatively high V number (or reduced damping in the case of Daruka and Ditlevsen, 2016). This similarity implies that the importance of the V number for a climate system dynamics may extend well beyond the Verbitsky et al. (2018) model.

We begin again with the most general hypothesis. We suggest that the amplitude of glacial area variations Ś is a function of the V number, the characteristic rate of snow precipitation, a, the amplitude of the external forcing ε, and the period of the system response P as it is described by Eq. (7). The relationship between the period of the response and that of the forcing may therefore be nontrivial. It means that the system response may exhibit original forcing periods or multiples of them. 8Ś=φV,a,ε,P If the hypothesis Eq. (8) is true, then, taking dimensions of ε and P as independent dimensions and using the π theorem, we obtain: Ś/ε2P2=ΦV,ε/a, and finally: 9Ś=ε2P2ΦΠ1,Π2. Neither Π1 nor Π2 contains P. Equation (9) therefore implies that, at constant amplitude of the external forcing ε, the amplitude of glacial area variations is scale invariant with a frequency slope equal to 2. Figure 2 (Ś with reference parameter values) presents a numerical test of the hypothesis Eq. (8) and of its implication Eq. (9). Here, we measure the system response to single-sinusoid forcings of constant amplitude and periods T varying from 5 to 50 kyr. The system responds to this forcing with periods P ranging from 5 to 100 kyr, because forcing periods T of 40 and 50 kyr produce response periods P of 80 and 100 kyr, correspondingly. It can be seen that the Ś-amplitude frequency slope, βa, is close to 2 (i.e., βa=1.8) for periods between 30 and 100 kyr. It means that the amplitude of glacial area variations is scale invariant in the orbital domain.

The amplitude spectrum of the θ variable cannot be derived unambiguously from the same simple considerations as we have employed for P and Ś because of the following reasons: (a) we cannot constrain ourselves with only parameters V, a, ε, and P, since the basal temperature θ is measured in degrees Celsius, but none of ε, a, or P contains degrees Celsius and (b) as soon as we disassemble the V number, i.e., use all individual model parameters instead of V, the advantage of using the π theorem is lost. Nevertheless, if we disassemble the V number wisely, we can minimize the number of dimensional parameters, and, as a result, we may be rewarded by discovering the identities of critical groups that define the scaling properties of θ. Accordingly, we will disassemble the V number using not the individual parameters involved but instead using dimensionless groups that are present in the V number: α,κc,γ1βγ3,S0, andγ2βγ3. If we consider that the group γ1βγ3S0 is a dimensionless representation of the parameter γ1 and the group γ2βγ3 is a dimensionless representation of the parameter β, then the remaining parameters γ2,γ3, and S0 need to be represented individually in the dimensional form. Taking this together, this yields the following hypothesis: 10θ´=χα,κc,γ1βγ3S0,γ2βγ3,γ2,γ3,S0,a,ε,P. Taking γ2,S0, and P as independent dimensions, the π theorem implies: 11θ´=γ2S0PXα,κc,γ1βγ3S0,γ2βγ3,γ3P,aPS0-1/2,εPS0-1/2, or, combining groups α,κc,γ1βγ3S0, andγ2βγ3 back into V number as follows: 12θ´=γ2S0PXV,γ3P,aPS0-1/2,εPS0-1/2. Since Π2=ε/a, 13θ´=γ2S0PXΠ1,Π2,Π3,Π4, where Π3=γ3P, and Π4=εPS0-1/2.

As Π3 and Π4 include P, then, generally speaking, the amplitude of basal temperature variations is not expected to be scale invariant. Under some circumstances though, the function XΠ1,Π2,Π3,Π4 may become P-independent and the amplitude of the basal temperature variations may develop the property of scale invariance. For example, we observed experimentally that when Π4→0 (e.g., the amplitude of the external forcing, ε, is reduced), the Eq. (13) becomes scale invariant with a frequency slope equal to 1. In this case θ´=γ2S0PXΠ1,Π2,Π3/Π4.

Since Eq. (3) for ω is linear, it may provide us with a hint about the response scaling characteristics of this variable. In the orbital domain Π3=γ3P≫1 so that Eq. (3) may be approximated as γ3ω≈γ1-γ2S-S0. Hence, ω´=γ2γ3Ś. We may hypothesize therefore that in the orbital domain and possibly even beyond: 14ω´=νV,γ2γ3,a,ε,P. Taking the dimensions of γ2γ3,ε, and P as independent and applying again π-theorem reasoning, we should expect that 15ω´=γ2γ3ε2P2NΠ1,Π2. At constant amplitude of the external forcing ε, Eq. (15) implies that the amplitude of climate temperature variations ω´ grows with the square of the response period. The results presented in Fig. 2 (ω´ with reference parameter values) support the hypothesis (14) and its implication (15): the ω variable amplitude frequency slope is close to 2 (i.e., βa=1.8) for periods between 5 and 100 kyr. It means that in the orbital and millennial domains, the amplitude of the climate temperature is scale invariant.

So far, we have based our implications of scaling relationships on the significance of a dimensionless number (in our case, the V number) quantifying a mean ratio between positive and negative feedbacks. That is, the scaling relationships found should be robust across changes in the composition of V, provided that the value of V is unchanged. To illustrate this implication, we conducted four numerical experiments. In the first experiment, we increase coefficients α and κ 2-fold and reduce γ2 by a half relative to their reference values. This does not change the reference value of the V number (see the Eq. 4, and note that the reference value of γ1=0) that is V=0.75 but transforms system Eqs. (1)–(3) into a system where the positive feedback is dominated by the climate temperature affecting ice-sheet mass balance and its temperature regime. We then measure the system response to the single-sinusoid forcing of the same amplitude and periods T=5–50 kyr. (Note that periods T=40 and 50 kyr produce system response of periods P=80 and 100 kyr, correspondingly). In the second experiment, we decrease coefficients α and κ by 50 % and increase γ2 2-fold relative to their reference values. Again, this does not change the reference value of the V number, V=0.75, but transforms system Eqs. (1)–(3) to a system where the positive feedback is dominated by the albedo feedback. In the third experiment, we increase coefficients α and κ by 50 % as well as the coefficient β, thus creating the system with intensive climate–temperature positive feedback and intensive ice-sheet basal temperature negative feedback, the V number still being equal to 0.75. And finally, we decrease coefficients α, κ, and β by 50 %, making a system with weak climate–temperature positive and ice-sheet basal temperature negative feedbacks. The response of all four systems to the external forcing is shown in Fig. 2. Despite different underlying physics, all four systems demonstrate the same outcomes: in the orbital domain, their amplitudes of glacial area variations are scale invariant with 1.8 frequency slope, and the amplitudes of the climate temperature are scale invariant in the orbital and millennial domains with the same slope.

This robustness is comforting. As we know, the physical interpretation of a low-order dynamical model can be partly ambiguous. For example, the mechanisms responsible for the changes in the effective climate temperature and how it impacts the ice mass balance are not fully described in this model. It is therefore reassuring to have been able to identify what seems to be the key ingredient for the scaling relationship, in this case, that a single quantity (the V number) grossly determines the dynamics of the system response. In other words, it relies on the fact that the number of effective parameters is smaller than is apparent from a more detailed description of the system.

This, incidentally, shows how difficult it is to disambiguate the physical mechanisms responsible for a given behavior. Different assemblages yielding the same V number will, indeed, produce slightly different solutions but less different than one could have perhaps expected. The dimensionless functions like, for example, function ΦV,ε/a in the Eq. (9), Ś=ε2P2ΦV,ε/a, and function Φ′V,ε/a, corresponding to the same value of the V number but formed by the different physics (different set of parameters), Ś=ε2P2Φ′V,ε/a, though not identical, yield the same scaling behavior. If the amplitude of the external forcing ε is constant, the period P shows up only as a power-law monomial ∼Pn, and its power n makes the same scale-invariant amplitude-spectrum slope regardless of the specific physics defining the V number. In other words, though the functions ψV,a,ε,T, φV,a,ε,P, χV,a,ε,P, and ν(V,γ2γ3,a,ε,P) may change depending on the specific physics forming the V number, their governing parameters always remain the same because they are determined by the structure of the system Eqs. (1)–(3). Accordingly, the functions ΨΠ1,Π2, ΦΠ1,Π2, XΠ1,Π2,Π3,Π4, and NΠ1,Π2 may also change, but their dimensionless arguments (Π groups) remain unaffected. As long as their groups, like, for example, Π1 and Π2, do not contain P, we have a possibility of scale-invariance. This observation makes the scale invariance a very general and expected property of the climate system.

The physical interpretation of the dynamical model we employ in this study (Verbitsky et al., 2018) is very straightforward as far as Eqs. (1) and (2) are concerned; these are scaled equations of mass and energy conservation of viscous ice flow. We must admit, though, that Eq. (3) of the climate temperature is, indeed, ambiguous. In other words, we are uncertain about some key mechanisms that we have chosen to describe using the rest-of-the-climate linear equation. Among others, these may be nonlinear effects related to the carbon cycle, nonlinear effects of sea-level destabilization of ice sheets and related synchronization, nonlinear effects associated with atmospheric circulation, or nonlinear effects related to biogenic calcifiers and their action on alkalinity, etc. A challenger might thus claim that these effects are so important that they should be considered more explicitly. Indeed, we have the hope that even after accounting for these processes, we might end up with a model that still has grossly the same mathematical structure as the Verbitsky et al. (2018) model, even though the meaning of some of the variables will have changed. Specifically, since Eq. (3) is linear, it can be split into several equations: ω=ω1+ω2+…+ωndω1dt=γ11-γ21S-S0-γ3ω1dω2dt=γ12-γ22S-S0-γ3ω2…dωndt=γ1n-γ2nS-S0-γ3ωn. Each of the above equations may represent different feedback mechanisms. Therefore our experiments with increased (or reduced) γ2 may be also understood as experiments with additional feedbacks of different nature (γ2=γ21+γ22+…+γ2n), though of the same timescale 1/γ3.

Thus far we have assumed a single-sinusoid external forcing with an amplitude ε and a period T. When we force our system with normalized mid-July insolation at 65∘ N (Berger and Loutre, 1991), this assumption is not valid any longer because both the amplitudes and the periods of precession and obliquity are different. Therefore, the hypothesis Eq. (5) must be rewritten as: 16P=ψV,a,ε1,T1,ε2,T2. Here P is a period of the system response to a specific forcing component (a peak of the response spectrum); index “1” corresponds to obliquity, and index “2” corresponds to precession. Taking dimensions of ε1 and T1 as independent dimensions and using the π theorem, we obtain: 17P1=T1Ψ1V,ε1/a,ε1/ε2,T1/T2. Here P1 is a period of the system response to the obliquity forcing. Similarly, taking dimensions of ε2 and T2 as independent dimensions, and using the π theorem, we have 18P2=T2Ψ2V,ε2/a,ε1/ε2,T1/T2. Here P2 is a period of the system response to the precession forcing. Since in the case of the orbital forcing ε1/ε2 and T1/T2 are invariant, we can apply generalized π theorem (Sonin, 2004) and to rewrite Eqs. (17) and (18) as 19P1=T1Ψ1V,ε1/a20P2=T2Ψ2V,ε2/a. It can be seen that Eqs. (19) and (20) are identical to Eq. (7), and the response periods to obliquity and to precession do not depend on each other. This result is not by any means intuitive.

We now repeat the same reasoning for the corresponding amplitudes of the system response: 21Ś1=φ1V,a,ε1,P1,ε2,P222Ś2=φ2V,a,ε1,P1,ε2,P223Ś1=ε12P12Φ1V,ε1/a,ε1/ε2,P1/P224Ś2=ε22P22Φ2V,ε2/a,ε1/ε2,P1/P2. Though in the case of the orbital forcing ε1/ε2 and T1/T2 are invariant, P1/P2 is not an invariant (see Fig. 1); therefore 25Ś1=ε12P12Φ1V,ε1/a,P1/P226Ś2=ε22P22Φ2V,ε2/a,P1/P2. We can see that although periods of the system response to the precession and obliquity forcings are independent, the amplitudes of the corresponding variations are interdependent and thus may deviate from a pure square-period law. This observation may have an important implication for our understanding of the paleodata. As we demonstrated before (Verbitsky et al., 2018), P1/P2 evolves over time, specifically P1/P2=1 for the early Pleistocene due to precession period doubling and P1/P2=4 for the late Pleistocene due to obliquity period doubling. It means that the slope of the spectrum of the system response may also evolve.

Introduction of more sinusoids (for example, accounting for the millennial forcing) makes the situation even more complex. In such a case, a period of the system response to a specific forcing component depends on the amplitudes and the periods of all sinusoids: 27P=ψV,a,ε1,T1,ε2,T2,…εi,Ti…. Then, for example, P1, the period of the system response to obliquity forcing, can be presented as 28P1=T1Ψ1V,ε1a,…,ε1εi,T1Ti,…, and corresponding amplitude of the glaciation area response 29Ś1=φ1V,a,ε1,P1,ε2,P2,…εi,Pi…30Ś1=ε12P12Φ1V,ε1a,…,ε1εi,P1Pi,…. Equations (28) and (30) show that, generally speaking, every peak P and corresponding amplitude Ś of the system response depend on each forcing sinusoid. Such a dependence may break the scale invariance we discussed earlier. For example, we have demonstrated in our previous study (Verbitsky et al., 2019a) that introduction of the millennial variability of significant amplitude (i.e., ε1/εi→0) may disrupt the response of the system to the orbital forcing and essentially reduce the slope βa. The empirical energy density spectrum of Huybers and Curry (2006) has a slope of B≈2 in the orbital domain. Since the energy density slope B relates to the fluctuation amplitude slope βa as B=2βa+1, B≈2 corresponds to βa=0.5<2. We may therefore speculate that the observed spectrum of the climate variability could be significantly influenced by the millennial forcing propagated into the orbital domain.

It is apparent that not every dynamical model has the property of scale invariance, which is encoded in its dynamical equations. As an illustration, let us consider the van der Pol oscillator. It was previously suggested as a minimal model capturing ice-age dynamics (Crucifix, 2012). 31dxdt=-y+β+γFτ32dydt=-αy33-y-xτ Here all variables and parameters, except τ, are dimensionless; τ is measured in units of time. Variable x is thought to represent the global ice volume, and variable y makes the “rest-of-the climate” response. Using the same π-theorem technique, let us determine the period P and the amplitude x′ of the system response to the external forcing F of the period T. 33P=ψα,β,γ,τ,T34P=TΨα,β,γ,τ/T Since α, β, and γare constants, 35P=TΨτ/T. Similarly, 36x′=φα,β,γ,τ,P37x′=Φα,β,γ,τ/P=Φτ/P. It means that the amplitudes of forced fluctuations in the van der Pol model are not necessarily scale invariant. We have tested this conclusion experimentally for τ=36.2 kyr and a forcing period T ranging from 5 to 100 kyr. The response shows slope breaks near approximately 90 and 50 kyr, which are clearly related to the auto-oscillation of the 100 kyr dominant period and its 50 kyr overtone.

Therefore, in a search for the most adequate ice-age physics, it would indeed be useful to see whether more sophisticated ice sheet–ocean–atmosphere models have the property of scale invariance. We suspect that potential universality of this property may stem from the universality of the Eq. (1). Equation (1) represents the global ice volume balance and simply says that changes in the ice volume are equal to the mass influx to the ice-sheet surface. This statement is valid for each and every climate model of any complexity. Therefore, if a model can be diagnosed with a single dimensionless number similar to the V number that would effectively capture most of the climate dynamics, then the scale invariance of the glaciation area variations (in square meters) can be reduced from the simple observation that it depends on the mass influx to its surface (in meters per second) and the periodicity of the mass influx variation (in seconds). This might not be too difficult to verify with an adequate set of experiments, but we must obviously leave this task to the scientists who know and develop these models.