Over the Quaternary, ice volume variations are “paced” by astronomy. However, the precise way in which the astronomical parameters influence the glacial–interglacial cycles is not clear. The origin of the 100 kyr cycles over the last 1 million years and of the switch from 40 to 100 kyr cycles over the Mid-Pleistocene Transition (MPT) remain largely unexplained.
By representing the climate system as oscillating between two states, glaciation and deglaciation, switching once glaciation and deglaciation thresholds are crossed, the main features of the ice volume record can be reproduced

Palaeoclimate records over the Quaternary (last 2.6 Myr), such as ice cores

The nature and physics of this link has been a central question since the discovery of previous warm and cold periods, and long before the obtention of continuous

The obtention of 100 kyr cycles is not possible with a linear theory like that of Milankovitch

Several conceptual models have been developed to try to solve these questions.

One of the critical questions for conceptual models is to decide which insolation to use as input.
Milankovitch's work utilized “caloric seasons” at 65

In our work, we will consider several summer insolation forcings at 65

The model used in our study is an adapted and simplified version of the conceptual model of

In our model, the evolution of the ice volume in these two states is described by

A critical point is to define the criteria for the switch between the glaciation and deglaciation states. To enter the deglaciation state, both ice volume and insolation seem to play a role

The idea that the deglaciation threshold is linked to both insolation and ice volume is not new

The conceptual model defined previously uses summer insolation as input. It is therefore important to consider which insolation should be used. Insolation is usually taken at 65

Here, for the first time, we want to examine the effect of this choice on the dynamics of a conceptual model. We therefore use four different summer insolation forcings, and compare the model results obtained with each of them. We use the summer solstice insolation, the caloric season and ISI above two different thresholds (300 and 400 W m

These four insolation forcings differ, and have different contributions from precession and obliquity, as can be seen in Fig.

To compute ISI

The conceptual model relies on a small number of model parameters:

To estimate which model parameters lead to model results closer to the data, the definition of an objective criteria is needed. The choice of such a criteria is not straightforward, and the use of different criteria could have led to slightly different results. Our model is simple and does not aim at precisely reproducing the ice volume evolution, but rather at reproducing the main qualitative features, such as the shape and frequency of the oscillations. Therefore, we used a criteria based on the state of the system: glaciation or deglaciation. Similar results can be obtained using a simple correlation coefficient (see Sect. S2). The definition of the deglaciation state in the data is explained in Sect.

We defined a criteria for each of these two conditions, and assembled them in a global criteria. To determine if deglaciation is well reproduced by our model, we look at the state of the model (glaciation or deglaciation) at the time halfway between the start of the deglaciation and the end of the deglaciation. If the model state at that time is deglaciation, the deglaciation is considered as correctly reproduced. Otherwise, it is considered as a “missed” deglaciation. We simply defined the criteria

In order to study the evolution of the optimal deglaciation threshold

For each insolation forcing, the best fit over the Quaternary is defined as the simulation over the whole Quaternary (0 to 2500 ka) with

To calculate our accuracy criteria

For each insolation, we computed the deglaciation threshold

Optimal deglaciation threshold

In some cases, several values of the deglaciation threshold

Over the same time period, different insolation values lead to slightly different optimal

For each insolation, the accuracy corresponding to the optimal

Accuracy over the five 500 kyr periods for the four different summer insolation forcings at 65

It is first noticeable that the accuracy is higher over the last 1 million-year period, regardless the input summer insolation used. In the last 1 million years, the summer solstice insolation as input produces the best results. However, this is no longer the case for older time periods: the solstice insolation gives the worst results at the start of the Quaternary. The accuracy obtained for the whole Quaternary period (fixed

In the earlier part of the Quaternary (periods earlier than 1.5 Ma), the results are less robust. This is due to increased uncertainties in the LR04 record, and the associated definition of interglacial periods, which affects our accuracy criteria. In their classification,

Moreover, the

The increase in the optimal deglaciation threshold

The overall lowest accuracy in the older part of the record suggests that our non-linear threshold model is less adapted for this time period. Indeed, the ice volume might respond more linearly to the insolation forcing before the MPT, as some studies suggest

Best model fit over the whole Quaternary and corresponding spectral analysis.

Our conceptual model is able to reproduce qualitatively well the data (LR04 normalized curve) over the whole Quaternary. The model’s best fit over the Quaternary for each insolation forcing, as defined in Sect.

For the last 1 million years, it is possible to reproduce all terminations with the right timing, apart from the last deglaciation, for all insolation forcings, by using a single value of the

Despite the accurate timing of terminations, the spectral analysis of the model results over the last 1 million years differs from the spectral analysis of the data. For all forcings except the summer solstice insolation, obliquity continues to dominate after the MPT. The spectral analysis shows secondary and third peaks of lower frequency, but does not show a sharp 100 kyr cyclicity as in the LR04 record. Compared to the data, all the model outputs over the post-MPT period have a more pronounced obliquity and precession component and a less pronounced 100 kyr component. This feature is most probably due to the model formulation, and more specifically the direct dependence of ice volume evolution to insolation via the

In the first part of the Quaternary (2.6 to 1 Ma), the spectral analysis of the data is dominated by a 41 kyr (obliquity) peak. It is also the case for the model results, for each type of insolation. However, the model outputs also show a precession component (19 to 23 kyr), especially for the summer solstice and the ISI above 400 W m

On the oldest part of the Quaternary, the caloric season forcing and the ISI above 300 W m

Over the last 1 million years, the highest accuracy is obtained with the summer solstice insolation as input forcing (

Normalized model results over the last 1 million years, with the different summer insolation forcings: insolation at the summer solstice, caloric season, ISI above 300 W m

To model future natural evolutions of the climate system, possible evolutions of the

However, this exercise is purely academic, as we are not taking into account the role of anthropogenic

We have used a conceptual model with very few tunable parameters that represents the climatic system with multiple equilibria and relaxation oscillation. Only one parameter was varied, the deglaciation threshold parameter

The model code, insolation input files, spectral analysis and code needed to reproduce the figures are available for download:

The supplement related to this article is available online at:

GL and DP designed the study. GL performed the simulations, and also wrote the manuscript under the supervision of DP.

The contact author has declared that neither they nor their co-author has any competing interests.

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

This article is part of the special issue “A century of Milankovic’s theory of climate changes: achievements and challenges (NPG/CP inter-journal SI)”. It is not associated with a conference.

We acknowledge the use of the LSCE storage and computing facilities. We also thank the reviewers for their helpful comments and suggestions.

This research has been supported by ANDRA (contract no. 20080970).

This paper was edited by Marie-France Loutre and reviewed by Mikhail Verbitsky, Andrey Ganopolski, and one anonymous referee.