Modeling the Dynamics of Glacial Cycles

Abstract

This article is concerned with the dynamics of glacial cycles observed in the geological record of the Pleistocene Epoch. It focuses on a conceptual model proposed by Maasch and Saltzman [J. Geophys. Res., 95, D2 (1990), pp. 1955-1963, which is based on physical arguments and emphasizes the role of atmospheric CO 2 in the generation and persistence of periodic orbits (limit cycles). The model consists of three ordinary differential equations with four parameters for the anomalies of the total global ice mass, the atmospheric CO 2 concentration, and the volume of the North Atlantic Deep Water. In this article, it is shown that a simplified two-dimensional symmetric version displays many of the essential features of the full model, including equilibrium states, limit cycles, their basic bifurcations, and a Bogdanov-Takens point that serves as an organizing center for the local and global dynamics. Also, symmetry breaking splits the Bogdanov-Takens point into two, with different local dynamics in their neighborhoods.

Effects of Periodic Forcing on a Paleoclimate Delay Model

“…Summary of modification of stable manifold algorithm by England, Osinga and Krauskopf [12]. The supplementary material describes how one can modify the search circle (SC) algorithm for stable manifolds in [12] for maps M given implicitly through (13) M : dom L x → y ∈ dom L, where y is the solution of RM +1 Lx = RM Ly.…”

“…In particular, the algorithm [12] does not rely on root-finding using Newton iterations, but rather on a bisection to find the intersection between image of the search circle and previous manifold. In principle, this algorithm could be applied directly, if one solves the defining system (13), RM +1 Lx = RM Ly, for y every time the original algorithm applies its map (in our case M ) to a point x ∈ R 2 . However, a modification of the SC algorithm avoids the need to solve the nonlinear equation (13).…”

“…In principle, this algorithm could be applied directly, if one solves the defining system (13), RM +1 Lx = RM Ly, for y every time the original algorithm applies its map (in our case M ) to a point x ∈ R 2 . However, a modification of the SC algorithm avoids the need to solve the nonlinear equation (13). Instead of a single sequence (and interpolating curve) S k one maintains two curves, (note the power of M in the mapping) to S k R .…”

“…Although DDEs are infinitedimensional, the phase portrait in Figure 2 gives initial evidence that the dynamics are confined to a two-dimensional slow manifold. Engler et al [13] derived a two-dimensional slow manifold for the original model [34] through considering the deep ocean timescale as instantaneous (τ = 1). Here we consider the case where the deep ocean timescale is not instantaneous (τ > 1).…”

“…The dominant periodicity of the oscillations also changed from approximately 41 kyr in the beginning of the Pleistocene to roughly 100 kyr towards the end of the Pleistocene, together with an increase in amplitude and degree of asymmetry in the oscillations [16,24]. This shift in dynamics is known as the Mid-Pleistocene Transition (MPT), and the exact timing of it is believed to be sometime between 1200 and 700 kyr BP [24,31,8,10,13]. The oscillations and the MPT can be observed through proxy records as shown in Figure 1.…”