# Relaxation-oscillations in a conceptual climate model

Conceptual climate models are relatively simple mathematical constructs that can capture essential features of the nonlinear nature of the climate. They usually focus only on several ingredients of the physical description and model their interplay. As can be easily noticed, due to their simplicity, they cannot be used as computation tools that provide detailed simulations of reality (as opposed to General Climate Models (GCMs). However, they still are valuable enough to help us to understand basic laws of climate dynamics. Their most important feature is a relatively low dimension what makes them feasible to study analytically and simulate cheaply.

One of the remarkable features of the paleoclimatic record is the occurence of relaxation-oscillations of the climate in the Pleistocene epoch . These are asymmetrical variations of the ice sheet extent that may suggest that the climate is an astronomically driven dynamical system that encompasses various feedbacks. The characteristic feature of these relaxation-oscillations is a slow growth of the ice sheet followed by a rapid deglaciation . During the last several million years the ice sheets came and went quasi-periodically leading to glacial periods and interglacials. Conceptual models that try to explain this phenomena are reviewed in .

One member our of the whole family of conceptual climate models is so-called KCG (Källén, Crafoord, Ghil) (see  for an original account and [2,3] for our results). It constitutes a dynamical system of two nonlinear equations describing planetary energy and mass (ice) balances. Schematically, the energy equation is the following $c \frac{dT}{dt} = \frac{1}{4}\left(1-\alpha\right)Q - A + BT,$

where $T$ is the globally averaged temperature, $Q$ is the solar constant, $\alpha$ is albedo (the ratio of incoming to reflected radiation), and $A$, $B$ are empirical constants. The mass balance, on the other hand, is derived based on a simple model of ice sheet presented in Fig. 1 (see ). The flow of the bulk of ice is modelled a plastic flow in the southward direction from the Arctic Ocean. The mass of ice is nourished by the snowfall that can occur only when the temperature is low enough. This is demarcated by the snow-line, i.e. $0^\circ$C isotherm (dashed line in Fig. 1). The ice sheet looses its volume by melting (ablation) in the southern part. This balance can be encapsulated in the following equation $\frac{dV}{dt} = a l_0 - m \left(l-l_0\right),$

where $a$, and $m$ are accumulation, and ablation ratios. The number $l_0$ is the margin between these two zones, while $l$ is the southward extent of the ice-sheet.

The model is closed with providing two feedbacks:

• ice-albedo feedback – more ice leads to higher reflectivity of the surface,
• temperature-precipitation feedback – higher temperature implies more evaporation which increases precipitation.

Mathematically speaking, the above requires that $\alpha = \alpha(T,l)$ and $a = a(T)$. Therefore, the KCG model is a dynamical system composed of two coupled nonlinear equations.

In [2,3] we have investigated a generalized KCG model that with general parametrisations of aforementioned feedbacks. We have classified all the critical points of this system and showed that under some realistic assumptions it exhibits relaxation-oscillations which are also manifestly present in the real palaeoclimatological data (see Fig. 2). Further, under some suitable and justified simplification we show that the governing system can concisely be written in the form of a singularly perturbed system $\begin{cases} \epsilon\frac{dx}{dt} = f(x)-y,\\ \frac{dy}{dt} = \sqrt{y}\left(g(x)-y\right),\end{cases}$

where $\epsilon$ is a small parameter. When supplied with realistic values of the parameters equations reproduce the ice age oscillations with decent accuracy for a model of such simplicity. This may show that an asymmetric quasi-periodic evolution of the climate is an internal feature which does not require any external forcing.

Apart from a complete analysis of the phase plane of the most general system our main result concerns an exact leading order formula for the period of climate’s oscillation-relaxations. Moreover, we give some useful and simple estimates that bound the period into the correct time scale of palaeoclimatological data. To sum up we have found that:

• After a Hopf bifurcation climate oscillates around a critical point that represents its present state.
• Relaxation-oscillations naturally emerge from nonlinear interactions of energy and mass balances.
• No external (astronomical) forcing is needed to produce a realistic period of ice age oscillations.

This model is a starting point for further study of externally forced climate and understanding its consequences on the onset and bifurcations in the ice age oscillations.

## References

 Källén E., Crafoord C., Ghil M. Free oscillations in a climate model with ice-sheet dynamics. Journal of the Atmospheric Sciences, 36(12): 2292-2303, 1979.
 Płociniczak Ł. Hopf bifurcation in a conceptual climate model with ice-albedo and precipitation temperature feedbacks. Nonlinear Analysis: Real World Applications, 51, 102967, 2020.
 Płociniczak Ł. Asymptotic analysis of internal relaxation-oscillations in a conceptual climate model. IMA Journal of Applied Mathematics 85(3): 467-494, 2020.
 Lisiecki LE., Raymo ME. A pliocene-pleistocene stack of 57 globally distributed benthic $\delta$18o records. Paleoceanography, 20(1), 2005.
 Crucifix M. Oscillators and relaxation phenomena in pleistocene climate theory. Phil. Trans. R. Soc. A, 370(1962):1140-1165, 2012.
 Weertman J. Milankovitch solar radiation variations and ice age ice sheet sizes. Nature, 261:17-20, 1976.

By Łukasz Płociniczak, Faculty of Pure and Applied Mathematics, Wrocław University of Science and Technology