Palaeoclimate records display a continuous background of variability connecting centennial to 100 kyr periods. Hence, the dynamics at the centennial, millennial, and astronomical timescales should not be treated separately. Here, we show that the nonlinear character of ice sheet dynamics, which was derived naturally from the ice-flow conservation laws, provides the scaling constraints to explain the structure of the observed spectrum of variability.

Most theories of Quaternary climates consider that glacial–interglacial cycles emerge from components of the climate system interacting with each other and responding to the forcing generated by the variations in summer insolation caused by climatic precession and the changes in obliquity and in eccentricity. A common approach is to represent these interactions and responses by ordinary differential equations. In a low-order dynamical system, the state vector only includes a handful of variables, which vary on roughly the same timescales as the forcing. Barry Saltzman has long promoted this approach, and his models' state variables represented the volume of continental ice sheets, deep ocean temperature, carbon dioxide concentration, and in some models the lithospheric depression (e.g., Saltzman and Verbitsky, 1993). Similar models featuring other mechanisms were published more recently (e.g., Omta et al., 2016). The purpose of these models is to explain the temporal structure of ice age cycles, but the spectrum of variability at centennial and millennial timescales is generally ignored. This approach is commonly justified by a hypothesis of the separation of timescales, as formulated by Saltzman (1990). However, this hypothesis is questionable. Indeed, the observational records display a continuous background of variability connecting centennial to 100 kyr periods (Huybers and Curry, 2006). For this reason, the dynamics at the centennial, millennial, and astronomical timescales should not be considered separately. Here, we address this concern and show that the ice dynamics are an effective vehicle for propagating high-frequency forcing upscale.

To make this case, we use the dynamical model previously
presented in Verbitsky et al. (2018). This nonlinear dynamical system was
derived from scaled conservation equations of ice flow, combined with an
equation describing the evolution of a variable synthesizing the state of
rest of the climate, called “climate temperature”. The three variables are
thus the area of glaciation, ice sheet basal temperature, and climate
temperature. Without astronomical forcing, the system evolves to an
equilibrium. When the astronomical forcing is present, the system exhibits
different modes of nonlinearity leading to different periods of ice age
rhythmicity. Specifically, when the ratio of positive climate feedback to
negative glaciation feedback (quantified by the

In the reference experiment presented in Verbitsky et al. (2018), the system is
driven, following standard practice, by mid-June insolation at 65

When our system is forced by a pure 5 kyr sinusoid of small amplitude,
the system remains in the vicinity of its equilibrium point, with a glaciation
area of

This shift of the time mean glaciation area and temperature has a dramatic
effect on ice age dynamics. When insolation forcing is combined with strong
millennial forcing, the latter moves the system into the domain where
obliquity period doubling no longer occurs because ice no longer grows to
the level needed to enable the strong positive deglaciation feedback.
Consequently, the 100 kyr variability almost vanishes – Fig. 1b. We term this suppression of ice age variability by millennial
variability “hijacking”. This result by itself invalidates the classical timescale
separation hypothesis: we see here that increased

Millennial forcings can be aggregated: Several sinusoids of smaller amplitudes and of different millennial periods create the same hijacking effect as a single 5 kyr high-amplitude sinusoid, moving the system into the phase-plane domain of higher temperatures and lower ice volume (Fig. 1c).

Acting alone, low-amplitude millennial sinusoids preserve their original
frequencies. However, several components may generate low-frequency
beatings, which are then demodulated by the system. Through this mechanism,
millennial forcing may induce responses at periods close to the orbital
periods, e.g., periods of precession and obliquity (Fig. 1d). For example,
the 41 kyr mode in Fig. 1d, which one might be tempted to attribute to obliquity,
is in fact the demodulation of the envelope generated by the interplay of
the 6 and 7 kyr forcing sinusoids:

It is possible to anticipate the disruptive effect of forcing at other
periods. Indeed, let us measure this disruption potential as the distance

Similar scaling arguments can be applied to the amplitude of the

For multi-period forcing in a frequency domain

Accordingly, since amplitudes of high-frequency variability,

To our knowledge, only Le Treut and Ghil (1983) have previously adopted a deterministic framework to model a background spectrum connecting millennial to astronomical timescales. Unfortunately, their model did not generate credible ice age time series. The more common route for simulating the centennial and millennial spectrum is to introduce a stochastic forcing (e.g., Wunsch, 2003; Ditlevsen and Crucifix, 2017). Such stochastic forcing may in principle be justified by the existence of chaotic or turbulent motion in the atmosphere–ocean continuum. However, whether such forcing is large enough to integrate all the way up to timescales of several tens of thousands of years is speculative. The deterministic theory proposed here presents the advantage of using the nonlinear character of ice sheet dynamics, which was derived naturally from the conservation laws and therefore provides a clear physical interpretation of the nonlinear origin of the cascade. Our approach is thus remarkably parsimonious because it requires no more physics than the minimum needed to explain ice ages plus the existence of centennial or millennium modes of motions. The latter may very plausibly arise as modes of ocean motion (Dijkstra and Ghil, 2005; Peltier and Vettoretti, 2014). Of course, stochastic forcing may still be added, and its cumulated effects would then be estimated by Eq. (2).

In summary, using a deterministic nonlinear dynamical model of the global
climate, we demonstrated that astronomical timescale variability cannot be
considered separately from millennial phenomena and that the ice dynamics are an effective vehicle for propagating high-frequency forcing into the orbital
timescale. This may imply that the knowledge of millennial and centennial
variability is needed to fully understand and replicate ice age history. As
we have seen,

The MatLab R2015b code and data to calculate the model
response to astronomical and millennial forcing (Verbitsky et al., 2019) are
available at

MYV conceived the research and developed the model. MYV and MC wrote the paper. DMV developed the numerical scheme and MatLab code.

The authors declare that they have no conflict of interest.

Michel Crucifix is funded by the Belgian National Fund of Scientific Research. Dmitry M. Volobuev is funded in part by the Russian Foundation for Basic Research, grant 19-02-00088-a. We are grateful to our reviewers Gerrit Lohmann and Niklas Boers for their helpful comments.

This paper was edited by Anders Levermann and reviewed by Gerrit Lohmann and Niklas Boers.