Periodic Orbits for a Discontinuous Vector Field Arising from a Conceptual Model of Glacial Cycles

Conceptual climate models provide an approach to understanding climate processes through a mathematical analysis of an approximation to reality. Recently, these models have also provided interesting examples of nonsmooth dynamical systems. Here we discuss a conceptual model of glacial cycles consisting of a system of three ordinary differential equations defining a discontinuous vector field. We show that this system has a large periodic orbit crossing the discontinuity boundary. This orbit can be interpreted as an intrinsic cycling of the Earth's climate giving rise to alternating glaciations and deglaciations

Mixed mode oscillations in a conceptual climate model

Much work has been done on relaxation oscillations and other simple oscillators in conceptual climate models. However, the oscillatory patterns in climate data are often more complicated than what can be described by such mechanisms. This paper examines complex oscillatory behavior in climate data through the lens of mixed-mode oscillations. As a case study, a conceptual climate model with governing equations for global mean temperature, atmospheric carbon, and oceanic carbon is analyzed. The nondimensionalized model is a fast/slow system with one fast variable (corresponding to ice volume) and two slow variables (corresponding to the two carbon stores). Geometric singular perturbation theory is used to demonstrate the existence of a folded node singularity. A parameter regime is found in which (singular) trajectories that pass through the folded node are returned to the singular funnel in the limiting case where $\epsilon = 0$. In this parameter regime, the model has a stable periodic orbit of type $1^s$ for some $s>0$. To our knowledge, it is the first conceptual climate model demonstrated to have the capability to produce an MMO pattern.Comment: 28 pages, 11 figure

Esther Richiyani Widiasih's Quick Files

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A Discontinuous ODE Model of the Glacial Cycles with Diffusive Heat Transport

We present a new discontinuous ordinary differential equation (ODE) model of the glacial cycles. Model trajectories flip from a glacial to an interglacial state, and vice versa, via a switching mechanism motivated by ice sheet mass balance principles. Filippov’s theory of differential inclusions is used to analyze the system, which can be viewed as a nonsmooth geometric singular perturbation problem. We prove the existence of a unique limit cycle, corresponding to the Earth’s glacial cycles. The diffusive heat transport component of the model is ideally suited for investigating the competing temperature gradient and transport efficiency feedbacks, each associated with ice-albedo feedback. It is the interplay of these feedbacks that determines the maximal extent of the ice sheet. In the nonautonomous setting, model glacial cycles persist when subjected to external forcing brought on by changes in Earth’s orbital parameters over geologic time. The system also exhibits various bifurcation scenarios as key parameters vary

A dynamics approach to a low order climate model

Energy Balance Models (EBM) are conceptual models which have proved useful in the study of planetary climate. The focus of EBM is placed on large scale climate components such as incoming solar radiation, albedo, outgoing longwave radiation and heat transport, and their interactions. Until recently, their study has centered on equilibrium solutions of an associated model equation, with no consideration of the dynamical nature of these solutions. In this paper we continue and expand upon recent efforts aimed at placing EBM in a more mathematical, dynamical systems context. In particular, the dynamical behavior of several variants of the Budyko-Sellers model, all but one of which involve the movement of glaciers, is shown to reduce to the study of the system on an attracting one-dimensional invariant manifold in an appropriately defined state space