Analyzing a dynamical system describing the global climate variations
requires, in principle, exploring a large space spanned by the numerous
parameters involved in this model. Dimensional analysis is traditionally
employed to deal with equations governing physical phenomena to reduce the
number of parameters to be explored, but it does not work well with
dynamical ice-age models, because, as a rule, the number of parameters in
such systems is much larger than the number of independent dimensions.
Physical reasoning may, however, allow us to reduce the number of effective
parameters and apply dimensional analysis in a way that is insightful. We
show this with a specific ice-age model (Verbitsky et al., 2018), which is a
low-order dynamical system based on ice-flow physics coupled with a linear
climate feedback. In this model, the ratio of positive-to-negative feedback
is effectively captured by a dimensionless number called the “

Mathematical modeling of Pleistocene ice ages using astronomically forced
spatially resolving models of continental ice sheets, the ocean, and the
atmosphere has always been and remains a computational challenge.
Therefore, though higher-resolution models (e.g., Abe-Ouchi et al., 2013) and
models of intermediate complexity (e.g., Verbitsky and Chalikov, 1986;
Chalikov and Verbitsky, 1990; Gallée et al., 1991; Ganopolski et al.,
2010) are gaining popularity, it has been argued for a long time that
significantly less computationally demanding dynamical models may provide
just as much insight as the models with more degrees of freedom (Saltzman,
1990). However, even though the computational load for solving dynamical
equations is minimal, the work and number of experiments needed for spanning
the full parameter space is easily overwhelming. Analyzing a dynamical
system of ice ages is thus, in principle, a difficult task. In mathematical
physics, the method of dimensional analysis (e.g., Barenblatt, 2003) has
been traditionally employed to take advantage of symmetry or invariance
principles and, as a result, to reduce the number of effective parameters.
It has not been applied to low-order models of the Pleistocene climate,
because in such models the number of governing parameters is much larger
than the number of independent dimensions. Indeed, the number of independent
dimensions in a dynamical system does not exceed the number of variables (it
may be smaller if some variables have the same or dependent dimensions), to
which one adds time, which is always present in a dynamical system. For
example, the dynamical system of Saltzman and Verbitsky (1993) described the
evolution of four variables: ice volume (in cubic meters),

In Verbitsky et al. (2018), we derived a dynamical model of the Pleistocene
climate from the scaled conservation equations of viscous non-Newtonian ice
and combined them with an equation describing the evolution of the climate temperature. The work was motivated by the prospect of delivering a
low-order, parsimonious approach to the problem of understanding
glacial–interglacial cycles. The state of the ice–climate system is
summarized by a 3-dimensional vector: glaciation area

In this paper we will apply this approach systematically to all model variables. This will allow us to demonstrate that, in the model, glacial area and climate temperature are scale invariant in the orbital frequencies domain (in the case of the climate temperature – even beyond this domain) and observe that this property does not depend on the specific physical nature of the climate system feedbacks. This observation is important. The empirical analysis of paleoclimate series shows that there is a rich spectral content and points to the existence of “spectral slopes” (e.g., Huybers and Curry, 2006; Lovejoy and Schertzer, 2013). Lovejoy and Schertzer (2013) evoke some generic process, such as the principle of “cascades” which is tightly linked to the concept of scale invariance of the equations. For example, the scale invariance of fluid-dynamics equations is exploited to provide inferences about spectral slopes of turbulent flows. However, to our knowledge, there is no available theory supporting scale invariance in regimes associated with glacial–interglacial dynamics. Yet, paleoclimate simulations with more sophisticated models, including the seminal paper by Abe-Ouchi et al. (2013) and the simulations with CLIMBER provided by Ganopolski et al. (2010), tend to focus on the response of the ice-sheet climate system to orbital forcing and discuss the respective amplitudes of the 100, 41, and 21–23 kyr periods, but none discuss the slope of the power spectrum down to the millennium scale. Therefore, we believe that our research will provide at least some important elements that should help us to bridge both approaches

Accordingly, our paper is structured as follows. First, we will briefly
recapture equations, parameters, and dimensions of the Verbitsky et al. (2018) model. Then we will remind readers of the essence of the

The nonlinear dynamical model of the global climate system (Verbitsky et al., 2018) is derived from the scaled equations of ice sheet thermodynamics,
combined with a linear feedback equation involving an effective
“temperature”, which describes the climate state outside the ice region.

Model parameters along with their dimensions are as follows:

Physical reasoning and numerical experiments (Verbitsky et al., 2018) led us
to the suggestion that the system response is essentially determined by the

If we assume that the

We previously noticed (Verbitsky et al., 2018) that with weak climate
positive feedback (

Figure 1 presents a sketch of what the function

A typical illustrative

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

We begin again with the most general hypothesis. We suggest that the
amplitude of glacial area variations

The system response to a single-sinusoid external forcing of constant
amplitude and different periods: (1)

The amplitude spectrum of the

As

Since Eq. (3) for

So far, we have based our implications of scaling relationships on the
significance of a dimensionless number (in our case, the

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

This, incidentally, shows how difficult it is to disambiguate the physical
mechanisms responsible for a given behavior. Different assemblages yielding
the same

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:

Thus far we have assumed a single-sinusoid external forcing with an amplitude

We now repeat the same reasoning for the corresponding amplitudes of the
system response:

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:

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).

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

Twenty-seven years ago, Saltzman and Verbitsky (1993) discussed their model
of 18 parameters, nine of which were physically unconstrained (i.e., free
parameters) and formulated a challenge “to account for as much of the
variance with fewer free parameters. A challenge is thus posed to ourselves
and other theoretical paleoclimatologists to construct a more parsimonious
model in this regard that can supersede our present effort.” We may now
conclude that this challenge has been met. All parameters in our model
(Verbitsky et al., 2018) are physically constrained. Moreover, dimensional
analysis reveals that there are

The analysis indicates that the amplitudes of glacial area variations and of
climate temperature are

Retrospectively, we could have inferred scale invariance from the mere
assumption that the behavior of the continental glacial area (measured in
square meters) depends on the mass influx to its surface (in meters per second) and the periodicity
of the mass influx variation (in seconds), but perhaps these assumptions are too
simple to be convincing. In our study, we have chosen a bit more
sophisticated but more credible approach. We derived a dynamical model from
the scaled conservation equations of viscous non-Newtonian ice combined with
an equation describing the evolution of the climate temperature. We observed
that most of the dynamical system behavior can be explained by a balance
between positive and negative feedbacks. This observation, finally,
illuminated the crucial role of the mass influx and its periodicity, making
application of the

Certainly, we cannot claim to have a full picture of the mechanisms of ice
ages, but if ice age physics are well captured by the mathematical structure
that we have obtained, then this scale invariance linking response
amplitudes and periods applies.
We further suggest that a model that would indeed be a bit different than the Verbitsky et al. (2018) model (because it includes some other important, may be nonlinear, mechanisms) might still
retain an important property that we have discovered: there is a connection between the sensitivity of the fixed point (since the

The MATLAB R2015b code and data to calculate
model response to periodical forcing as presented in Fig. 2 are available at

MYV conceived the research and developed the formalism. MYV and MC contributed equally to writing the paper.

The authors declare that they have no conflict of interest.

We are grateful to Eyal Heifetz and two reviewers for their helpful comments and to Dmitry Volobuev for his help in producing Fig. 2.

This paper was edited by Gabriele Messori and reviewed by Eyal Heifetz and two anonymous referees.