**Research article**| 10 May 2022

# Inarticulate past: similarity properties of the ice–climate system and their implications for paleo-record attribution

Mikhail Y. Verbitsky

^{1,2}

**Mikhail Y. Verbitsky**Mikhail Y. Verbitsky

^{1,2}

^{1}Gen5 Group, LLC, Newton, MA, USA^{2}Earth and Life Institute, UCLouvain, Louvain-la-Neuve, Belgium

^{1}Gen5 Group, LLC, Newton, MA, USA^{2}Earth and Life Institute, UCLouvain, Louvain-la-Neuve, Belgium

**Correspondence**: Mikhail Y. Verbitsky (verbitskys@gmail.com)

**Correspondence**: Mikhail Y. Verbitsky (verbitskys@gmail.com)

Received: 02 Jul 2021 – Discussion started: 30 Aug 2021 – Revised: 11 Mar 2022 – Accepted: 22 Apr 2022 – Published: 10 May 2022

Reconstruction and explanation of past climate evolution using proxy records is the essence of paleoclimatology. In this study, we use dimensional analysis of a dynamical model on orbital timescales to recognize theoretical limits of such forensic inquiries. Specifically, we demonstrate that major past events could have been produced by physically unsimilar processes making the task of paleo-record attribution to a particular phenomenon fundamentally difficult if not impossible. It also means that any future scenario may not have a unique cause and, in this sense, the orbital timescale future may be to some extent less sensitive to specific terrestrial circumstances.

The interpretation of most prominent events of climate history such as the Middle Pleistocene transition (Ruddiman et al., 1986, Lisiecki and Raymo, 2005, Clark et al., 2021) has been an inspiration for several generations of climate modelers (see for a review Saltzman, 2002; Tziperman et al., 2006; Crucifix, 2013; Mitsui and Aihara, 2014; Paillard, 2015; Ashwin and Ditlevsen, 2015; Verbitsky et al., 2018; Willeit et al., 2019; Riechers et al., 2022). While specific physical mechanisms invoked to explain changing glacial rhythmicity vary, they all include slow changes in ocean–atmosphere governing parameters (e.g., Saltzman and Verbitsky, 1993; Raymo, 1997; Paillard and Parrenin, 2004) or glaciation parameters (Clark and Pollard, 1998). On a more general level, all these theories in fact assume slow changes in the intensities of positive (such as long-term variations in carbon dioxide concentration, e.g., Saltzman and Verbitsky, 1993) or negative (for example, regolith erosion, Clark and Pollard, 1998, or the diminished role of the geothermal heat flux relative to the vertical temperature advection in growing ice sheets, Verbitsky and Crucifix, 2021) system feedbacks. Though all physical phenomena invoked are, indeed, real and may be plausible, the following question still remains unanswered: is it possible to disambiguate the past and elevate a single “correct” theory? Answering this question is the goal of our study.

Indeed, this is the classical attribution challenge that has been successfully addressed in the context of another well-known problem of geophysics: the causality of the observed global warming. For this purpose, the most comprehensive space-resolving models have been employed to reproduce observed time series under different conditions and to prove (or discredit) a candidate physical phenomenon (e.g., Stocker, 2014). Certainly, these models cannot be employed on extremely long orbital timescales (10–100 kyr) due to computational constraints. In search of an alternative, we turn here to dimensional analysis. Historically, dimensional analysis and concepts of similarity have been used for studying physical phenomena, complementing even the most sophisticated computational tools and providing physical insight in situations where physical interpretation of the higher-complexity modeling results may be difficult. Here, on orbital timescales, when we retreat from physics-abundant space-resolving models to more conceptual dynamical models, dimensional analysis may be promoted from a supporting to a more prominent role.

Several key terms need to be introduced before we outline the structure of
our paper. We will be using the definitions of similarities as they have
been articulated by Barenblatt (2003). Suppose we have a physical
phenomenon that is governed by *n* physical parameters, *k* parameters of which
are parameters with independent dimensions. Then, according to the *π* theorem (Buckingham, 1914), the phenomenon can be described by *n*−*k*
adimensional similarity parameters *π*_{1}, *π*_{2},
… *π*_{i} …, *π*_{n−k}. We will consider two
phenomena as being *physically similar* if they are described by identical similarity
parameters *π*_{1}, *π*_{2}, …*π*_{i} …, *π*_{n−k}. The dimensionless time series of physically similar
processes are also identical. If a similarity parameter *π*_{i} can
be excluded from the description of a physical process (a phenomenon becomes
independent of it in the limit that *π*_{i} tends to zero or
infinity), we can talk about *complete similarity* of this physical process in this parameter:
regardless of the parameter's specific value, the process does not depend on it.
And, finally, we may observe *incomplete similarity* when none of similarity parameters *π*_{1}, *π*_{2}, … *π*_{i} …, *π*_{n−k} can
be neglected even if they are too small (or too big), but the number of
effective parameters may still be reduced because a phenomenon depends not
on actual values of similarity parameters but on their products in some
power degree (i.e., conglomerate similarity groups):
${\mathrm{\Pi}}_{j}=\left({\mathit{\pi}}_{\mathrm{1}}^{{\mathit{\alpha}}_{j}}\right)\left({\mathit{\pi}}_{\mathrm{2}}^{{\mathit{\beta}}_{j}}\right)\mathrm{\dots}\left({\mathit{\pi}}_{i}^{{\mathit{\lambda}}_{j}}\right)\mathrm{\dots}({\mathit{\pi}}_{n-k}^{{\mathit{\chi}}_{j}}$) ($j=\mathrm{1},\mathrm{2},\mathrm{\dots},l;l<n-k$). Here
*α*_{j}, *β*_{j}, …, *λ*_{j}, …,
*χ*_{j} are power degrees of *π*_{1}, *π*_{2}, … *π*_{i} …, *π*_{n−k} involved into Π_{j} formulation.

We are now ready to proceed with the structure of our paper. (a) First, we
will introduce our dynamical model and describe the major physical processes
involved. (b) Using dimensional analysis, we will define eight similarity
parameters *π*_{1}−*π*_{8} that completely define the model's behavior.
(c) Since our system does not have a property of complete similarity in any
of the individual parameters *π*_{1}−*π*_{8}, we will attempt to
discover incomplete similarity and find conglomerate similarity groups.
Unfortunately, there are no specific algorithms that can help us to
determine governing conglomerate similarity groups Π_{j} if they do indeed exist. Therefore, we will articulate such conglomerate similarity
groups based on observed system behavior. (d) We will then discuss
implications of our findings for the attribution challenge and illustrate
our reasoning with a numerical experiment. (e) We will conclude our study
with some thoughts relating our results to the real-world climate system.

For our experiments we employ the Verbitsky et al. (2018), VCV18 hereafter, dynamical model of the ice–climate system. It has been derived from the scaled mass- and heat-balance equations of the non-Newtonian ice flow, i.e., Eqs. (1) and (2), respectively, and has been combined with an energy-balance equation of the global climate temperature (Eq. 3):

Here, *S* (m^{2}) is the area of glaciation, *θ* (^{∘}C) is the basal
ice-sheet temperature, and *ω* (^{∘}C) is the global temperature of
the ocean–atmosphere (rest of the climate) system. In deriving Eqs. (1)
and (2) we consider ice sheets in the thin-boundary-layer approximation such
that their inertial forces are negligible relative to stress gradients, and
motion equations with very high accuracy can be written in a quasi-static
form. For such an approximation, a characteristic ice thickness *H* is connected
to ice area *S* as $H=\mathit{\zeta}{S}^{\mathrm{1}/\mathrm{4}}$, where *ζ* (m${}^{\mathrm{1}/\mathrm{2}}$) is a profile factor
assumed to be constant (Verbitsky and Chalikov, 1986, VCV18). Further,
Eq. (1) represents global ice balance $\frac{\mathrm{d}\left(HS\right)}{\mathrm{d}t}=AS$,
where, again, $H=\mathit{\zeta}{S}^{\mathrm{1}/\mathrm{4}}$ and $A=a-\mathit{\epsilon}{F}_{\mathrm{S}}-\mathit{\kappa}\mathit{\omega}-c\mathit{\theta}$ is the surface mass influx. Equation (2) describes vertical ice
temperature advection with a time scale$\phantom{\rule{0.125em}{0ex}}H/(a-\mathit{\epsilon}{F}_{\mathrm{S}}-\mathit{\kappa}\mathit{\omega}$), and Eq. (3) is the global energy-balance equation. The
parameter *a* (m s^{−1}) is the snow precipitation rate; *F*_{S} is normalized
external forcing, specifically, mid-July insolation at 65^{∘} N
(Berger and Loutre, 1991), of the amplitude *ε*(m s^{−1}) such
that *ε**F*_{S} describes the ice ablation rate due to
astronomical forcing; *κ**ω* is the ice ablation rate representing
the cumulative effect of the global climate on ice-sheet mass balance;
*c**θ* represents ice discharge due to ice-sheet basal sliding; *α**ω* is the basal temperature response to global climate temperature change, and *β*[*S*−*S*_{0}] is the basal temperature reaction to the changes in ice geometry; $-\mathit{\gamma}\left[S-{S}_{\mathrm{0}}\right]$ describes the global
temperature response to ice geometry changes (e.g., albedo); *κ* (m s^{−1} ^{∘}C^{−1}), *c* (m s^{−1} ^{∘}C^{−1}), *α* (adimensional),
*β* (^{∘}C m^{−2}), and *γ* (^{∘}C m^{−2} s^{−1}) are
sensitivity coefficients; *S*_{0} (m^{2}) is a reference glaciation area;
and *τ* (s) is the timescale for *ω*. When orbitally forced,
the model reproduced events of the last million years reasonably well,
except for the interglacial of 400 kyr ago (Marine Isotopic Stage 11). The
timing of all other interglacials coincides with Past Interglacial Working
Group of PAGES (2016) data (VCV18).

We will now focus on the most remarkable feature of the historical records –
a period *P* of climate response to the astronomical forcing. Indeed, it is the
change in the climate variability from the predominant period *P*= 40 kyr to
the main periods of *P*= 80–120 kyr that makes the Middle Pleistocene
transition so extraordinary. Though the amplitude increase was considered,
until recently, to be a necessary attribute of this transition, its presence
in the paleo-records is now questioned (Clark et al., 2021). We begin with
the dimensional analysis of the VCV18 Eqs. (1)–(3). Indeed, it has 11
governing parameters (including the amplitude *ε* and the period
*T* of the external forcing). If we choose *ε*, *T* and *γ* to
be parameters with independent dimensions, then in accordance with the *π* theorem a period of the system response can be fully described by eight
dimensionless similarity parameters *π*_{1}−*π*_{8}:
${\mathit{\pi}}_{\mathrm{1}}=\frac{\mathit{\epsilon}}{a},{\mathit{\pi}}_{\mathrm{2}}=\mathit{\alpha},{\mathit{\pi}}_{\mathrm{3}}=\mathit{\kappa}\mathit{\gamma}\mathit{\epsilon}{T}^{\mathrm{3}},{\mathit{\pi}}_{\mathrm{4}}=c\mathit{\gamma}\mathit{\epsilon}{T}^{\mathrm{3}},{\mathit{\pi}}_{\mathrm{5}}=\frac{T}{\mathit{\tau}},{\mathit{\pi}}_{\mathrm{6}}=\frac{\mathit{\gamma}T}{\mathit{\beta}},\phantom{\rule{0.125em}{0ex}}{\mathit{\pi}}_{\mathrm{7}}=\frac{{S}_{\mathrm{0}}}{{\mathit{\epsilon}}^{\mathrm{2}}{T}^{\mathrm{2}}},{\mathit{\pi}}_{\mathrm{8}}=\frac{\mathit{\varsigma}}{{\mathit{\epsilon}}^{\mathrm{1}/\mathrm{2}}{T}^{\mathrm{1}/\mathrm{2}}},$ and

The numerical experiments with Eqs. (1)–(3) demonstrate that
individual similarity parameters *π*_{1}−*π*_{8} cannot be
discarded using simply “too big” or “too small” arguments. It means that
Eqs. (1)–(3) does not have a property of complete similarity in any
of the individual parameters *π*_{1}−*π*_{8}. At the same time, we
observed earlier (Verbitsky and Crucifix, 2020) that the period of Eqs. (1)–(3) response to the obliquity forcing of period *T* is mostly
governed by two dimensionless parameters: by the ratio of the
astronomical forcing amplitude to terrestrial ice-sheet snow precipitation
rate, $\mathit{\epsilon}/a$, and by the adimensional *V* number. The physical
meaning of the *V* number in the orbital domain becomes most evident if we take
a closer look at the structure of positive and negative feedbacks as they
appear in Eqs. (1)–(3). The time-dependent negative feedback is
proportional to the ice-sheet area size as *β*(*S*−*S*_{0}). The
coefficient *β* is defined by thermodynamical properties of an ice
sheet, most importantly by the Péclet number, $\mathit{Pe}=\widehat{A}H/k$. $\widehat{A}$ is a characteristic mass influx, i.e., accumulation minus ablation, and *k* is
ice temperature diffusivity (VCV18, Verbitsky and Crucifix, 2021). This
negative feedback acts on ice-sheet mass balance with a vertical-advection
time delay and is amplified by a sensitivity coefficient *c* that reflects the
intensity of basal sliding. The time-dependent positive feedback is global
temperature *ω*. In the orbital domain, *τ*≪*T* (*π*_{5}≫1), *ω* is approximately proportional to $-\mathit{\gamma}\mathit{\tau}(S-{S}_{\mathrm{0}}$). The global temperature acts on the ice-sheet mass balance
“instantly” as *κ**ω* and with the vertical-advection
time delay as a component of basal temperature conditions, *α**ω**c*. Thus, the *V* number is emerging in the orbital domain as a ratio
of amplitudes of time-dependent positive and negative feedbacks.

Specifically, when *V*∼0.75 and $\mathit{\epsilon}/a\sim \mathrm{1}$, the system exhibits the obliquity-period doubling. When the positive
feedback and the obliquity forcing are less articulated, the system responds
with the 40 kyr period. Thus, slow changes in the *V* number (for example, from
*V*= 0.5 at *t*= 3000 kyr ago to *V*= 0.75 at *t*= 0) and of the
$\mathit{\epsilon}/a$ ratio (for example, from $\mathit{\epsilon}/a=$ 0.3 to
$\mathit{\epsilon}/a=$ 1.7 over the same time span) produce a change in the
ice–climate behavior similar to the Middle Pleistocene transition.

We now notice that the *V* number can be presented in terms of similarity
parameters *π*_{1}−*π*_{8}, specifically

We also experimentally established that the period doubling sustains (Ψ=2) if, under fixed $\mathit{\epsilon}/a$ and *V*, the period of the external
forcing changes from let us say *T*= 35 kyr to *T*= 50 kyr. It can only happen if
in this domain similarity parameters *π*_{7} and *π*_{8} make another
conglomerate similarity group that does not depend on *T*, specifically$\phantom{\rule{0.125em}{0ex}}\frac{{\mathit{\pi}}_{\mathrm{8}}^{\mathrm{4}}}{{\mathit{\pi}}_{\mathrm{7}}}$.

Thus, Eq. (4) can be written as

which is the pure case of incomplete similarity as we defined it above: none of the similarity parameters can be neglected, but instead of eight governing parameters we have been able to migrate to four governing conglomerate similarity groups. Finally we may notice that $\frac{{\mathit{\pi}}_{\mathrm{8}}^{\mathrm{4}}}{{\mathit{\pi}}_{\mathrm{7}}}=\frac{{H}^{\mathrm{4}}}{{S}_{\mathrm{0}}^{\mathrm{2}}}\ll \mathrm{1}$ for all large ice sheets. If, we set it to be constant, we can re-write Eq. (7) in a more simple form as

Recognition of governing conglomerate similarity groups is important because
it provides us with a powerful insight: different combinations of similarity
parameters *π*_{i} may produce the same *V* number, i.e., *physically unsimilar processes *(formed by non-identical *π*_{i}) *may cause the same outcome.* This observation is critical for our attribution
challenge. Certainly, precise disambiguation of historical records is always
a difficult task because even two physically similar processes having
identical adimensional similarity parameters and demonstrating the same
behavior may have been produced by different values of the physical parameters
involved, unless these parameters are physical constants or well defined.
The situation becomes especially challenging when we deal with conglomerate
similarity groups because, as we just stated, the same results may be
produced by non-identical similarity parameters (physically unsimilar
processes). This is the theoretical limit that we aspire to expose.

We will now apply our findings to the Middle Pleistocene transition. Since
the physical interpretation of the governing parameters incorporated in the
conglomerate *V* number is very straightforward, we may observe a similar (in
terms of the period-*P* bifurcation) system response to changes of a completely
different physical nature. For example, parameter *β*, as we have
discussed above, defines the intensity of the negative feedback and is formed as
a result of interplay between vertical ice advection, internal friction, and
geothermal heat flux (VCV18). An increased Péclet number of a growing ice sheet
diminishes the role of the geothermal heat flux and may reduce parameter
*β* thus increasing the *V* number. The same period-*P* bifurcation can also
be caused, for example, by slow changes in the parameter *γ* that
defines the intensity of the positive feedback and incorporates the effects of
the albedo change or other atmospheric feedbacks. We solve Eqs. (1)–(3) for these two cases. In both cases we invoke a global cooling trend. In
our first experiment (Fig. 1a), this trend is translated into a reduction of
*β*, i.e., a weakening of the ice-sheet negative feedback, and a corresponding increase in the *V* number from *V*= 0.5 to *V*= 0.75. The
increased continentality of the climate (reduced intensity of the snowfall
during colder climate) is accounted by the $\mathit{\epsilon}/a$ ratio increase
from $\mathit{\epsilon}/a=$ 0.3 to $\mathit{\epsilon}/a=$ 1.7. In the second
experiment (Fig. 1b), the *V* number also evolves from *V*= 0.5 to *V*= 0.75,
but this time it is achieved by the increased intensity of the positive feedback
(*γ*). In both experiments, we used mid-July insolation at
65^{∘} N (Berger and Loutre, 1991) for the last 3 Myr as an
astronomical forcing. The millennial forcing is added to *ε**F*_{S}
as a single sinusoid of 5 kyr period and doubled (2*ε*) amplitude.
It is important to note that in the first experiment (changing *a* and *β*) only similarity parameters *π*_{1} and *π*_{6} are being changed,
but in the second experiment (changing *a* and *γ*) the same changes in
the *V* number are caused by changing *π*_{1}, *π*_{3}, *π*_{4},
*π*_{6}. It means that the processes involved in these two experiments are
not physically similar. Though the time series produced in these two cases
are obviously non-identical (see Fig. 1 inserts), we can observe that
different physical phenomena may produce the same changes in the
conglomerate *V* number and the same large-scale effect, i.e., the
period-doubling bifurcation at about 1 Myr ago.

We do not attempt here to fully reproduce paleo-records such as Lisiecki
and Raymo (2005) and a discussion of whether a period doubling should be
accompanied by the amplitude increase is outside of the current paper's
scope. We will just remark that the amplitude of the system response is the
function of not just the period *P* but also of the $\mathit{\epsilon}/a$ ratio
(Verbitsky and Crucifix, 2020) and, for example, less articulated
continentality of colder climates may explain diminished amplitude contrasts
as has been recently advocated by Clark et al. (2021).

Indeed, as we have already indicated, we used mid-July insolation at
65^{∘} N for the last 3 Myr as an astronomical forcing.
Apart from that, these examples may also serve as an illustration of some
future scenarios of the climate system behavior under post-industrial
atmospheric carbon dioxide concentration reduction as implied by Ridgwell
and Hargreaves (2007). Again, regardless of the physical nature of the
underlying dynamical system, it exhibits a 40 kyr rhythmicity in the first 1.5 Myr of its evolution and consequent obliquity-period doubling.
This probable renaissance of ice ages is different from the one envisioned
by Talento and Ganapolski (2021), which is based on the model tuned to the
Late Pleistocene (last 800 kyr) ice-volume data and thus postulates only
100 kyr period variability for the future.

The idea of the current presentation is simple, but its implication may be important: if the ice–climate system is defined by conglomerate similarity groups, then we may be limited in our ability to disambiguate historical records and different physical processes may produce the same future scenarios. The latter is intriguing because since Saltzman (1962) and Lorenz (1963) discovered a hydrodynamic system's sensitivity to initial conditions, the concept of deterministic chaos has become a dominant concept of weather and climate theory. Our findings suggest that if we consider orbital time scales and, instead of time series, focus on their more generalized attributes such as the period of the system response to the astronomical forcing, we may observe that the behavior of these attributes may be, to some extent, less sensitive to the physical nature of the terrestrial governing processes.

But do conglomerate similarity groups indeed govern the dynamics of the real
orbital-scale climate system? So far, these groups have been found only in
our VCV18 low-order dynamical model, and although this model has been
explicitly derived from the conservation laws, our concept will remain
hypothetical until it is supported by empirical data. We speculate, though,
that existing historical records may perhaps provide some support to our
theory. To evaluate the feasibility of a diagnostic approach, we entertain
here a simple scaling exercise. Suppose that an empirical time series, such
as a *δ*^{18}O record, is created by a parent system (other than the
VCV18) which is controlled by *n* physical parameters (*k* of them having
independent dimensions). If we choose the period of the astronomical forcing
*T* to be among parameters with independent dimensions, then in accordance with
the *π* theorem we have

The wavelet spectrum of the Late Pleistocene *δ*^{18}O variability
in response to the precession (∼ 20 kyr period) and obliquity
(∼ 40 kyr period) forcing shows the dominance of 40 kyr and
80 kyr periods (Fig. 1c). If we are willing to accept it as a hint of Ψ = 2 for *T*= 20 kyr and for *T*= 40 kyr, then, since some of the similarity
parameters $\phantom{\rule{0.125em}{0ex}}{\mathit{\pi}}_{\mathrm{1}},\phantom{\rule{0.125em}{0ex}}{\mathit{\pi}}_{\mathrm{2}},\phantom{\rule{0.125em}{0ex}}\mathrm{\dots},\phantom{\rule{0.125em}{0ex}}{\mathit{\pi}}_{n-k}$
depend on *T*, the period-*T* independence of Ψ may only happen when *π*_{1}, *π*_{2}, …, *π*_{n−k} make conglomerate
*T*-independent groups. In other words, *period independence of the* Ψ *function may be a fingerprint of conglomerate similarity groups*. Indeed, the diagnostics of the Ψ
function may require much more sophisticated instruments than our ad hoc
reasoning, and the records will likely not explicitly reveal what the
conglomerate similarity groups look like; nevertheless, their mere existence
would corroborate the idea of this paper.

The MATLAB R2015b code and data to reproduce the results of the numerical experiment as they are presented in Fig. 1 are available at https://doi.org/10.5281/zenodo.6525434 (Verbitsky, 2022).

This paper refers exclusively to published research articles and their data. We refer the reader to the cited literature for access to data.

The author has declared that there are no competing interests.

Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The author is grateful to Michel Crucifix for multiple discussions related to this topic and to Jeremy Bassis and anonymous reviewer for their helpful comments.

This paper was edited by Anders Levermann and reviewed by Jeremy Bassis and one anonymous referee.

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