The middle Pleistocene transition as a generic bifurcation on a slow manifold

Abstract

The Quaternary period has been characterised by a cyclical series of glaciations, which are attributed to the change in the insolation (incoming solar radiation) from changes in the Earth’s orbit around the Sun. The spectral power in the climate record is very different from that of the orbital forcing: prior to 1000 kyr before present most of the spectral power is in the 41 kyr band while since then the power has been in the 100 kyr band. The change defines the middle Pleistocene transition (MPT). The MPT does not indicate any noticeable difference in the orbital forcing. The climate response to the insolation is thus far from linear, and appears to be structurally different before and after the MPT. This paper presents a low order conceptual model for the oscillatory dynamics of the ice sheets in terms of a relaxation oscillator with multiple levels subject to the Milankovitch forcing. The model exhibits smooth transitions between three different climate states; an interglacial (i), a mild glacial (g) and a deep glacial (G) as proposed by Paillard (Nature 391:378–381, 1998). The model suggests a dynamical explanation in terms of the structure of a slow manifold for the observed allowed and “forbidden” transitions between the three climate states. With the model, the pacing of the climate oscillations by the astronomical forcing is through the mechanism of phase-resetting of relaxation oscillations in which the internal phase of the oscillation is affected by the forcing. In spite of its simplicity as a forced ODE, the model is able to reproduce not only general features but also many of the details of oscillations observed in the climate record. A particular novelty is that it includes a slow drift in the form of the slow manifold that reproduces the observed dynamical change at the MPT. We explain this change in terms of a transcritical bifurcation in the fast dynamics on varying the slow variable; this bifurcation can induce a sudden change in periodicity and amplitude of the cycle and we suggest that this is associated with a branch of “canard oscillations” that appear for a small range of parameters. The model is remarkably robust at simulating the climate record before, during and after the MPT. Even though the conceptual model does not point to specific mechanisms, the physical implication is that the major reorganisation of the climate response to the orbital forcing does not necessarily imply that there was a big change in the environmental conditions.

Appendix: The functional form of the slow manifold

We choose the following form for $H(y,v,\lambda )$:

$$\begin{aligned} H(y,v,\lambda )&= h_0 \tanh ^{-1}(y)+h_1y+h_2v+h_3\nonumber \\&\quad + h_4\frac{(y+h_6)e^{-h_5v}}{1+h_7(y+h_8)^2}+\lambda \end{aligned}$$

(11)

where $h_i$ are all non-negative constants that will be chosen. Setting $h_4=0$ and choosing $h_{0,1,2}$ appropriately gives hysteresis between stable sheets of the slow manifold close to $y\approx \pm$1 for fixed v and varying $\lambda$ there will be a range of $\lambda$ with two stable sheets ($i$ and $G$) while for $\lambda \rightarrow \pm \infty$ there will only be one stable sheet near $y\approx \pm$1. This can be seen by approximating $\tanh ^{-1}(y)=y+y^3/3+{{\mathcal {O}}}(y^5)$, thus for $y$ small $H(y,v,\lambda )=0$ becomes $(h_0+h_1)y+h_0 y^3/3+h_3+\lambda =h_2v$. Setting $h_4>0$ introduces an additional “cusp” to the slow manifold that gives an extra possible stable value of −1 < y < 1 ($g$) for fixed v (namely three states) and allows us to see transitions between the equilibrium states follow the selection-rules proposed by Paillard. The constants $h_i$ are chosen for (11) as follows:

$$\begin{aligned} \begin{array}{l} h_0=4,\quad h_1=-6.9,\quad h_2=-7,\quad h_3=2.80847,\quad h_4=50,\\ h_5=5,\quad h_6=0.1,\quad h_7=80,\quad h_8=0.2. \end{array} \end{aligned}$$

(12)

This choice gives a topology for the slow manifold that is robust (small changes in parameters do not change the sheets and the transitions between sheet of the slow manifold). The value of $h_3$ is chosen so that we have a change in the selection rules as we decrease $\lambda$ through 0 in (5); more precisely, $h_3$ is chosen so that there is a critical point $(v,y,\lambda )=(\tilde{v},\tilde{y},0)$ where $H_{y}=H_{v}=0$.