On the Budyko-Sellers energy balance climate model with ice line coupling

James Walsh,Christopher Rackauckas

doi:10.3934/dcdsb.2015.20.2187

Abstract

Over 40 years ago, M. Budyko and W. Sellers independently introduced low-order climate models that continue to play an important role in the mathematical modeling of climate. Each model has one spatial variable, and each was introduced to investigate the role ice-albedo feedback plays in influencing surface temperature. This paper serves in part as a tutorial on the Budyko-Sellers model, with particular focus placed on the coupling of this model with an ice sheet that is allowed to respond to changes in temperature, as introduced in recent work by E. Widiasih. We review known results regarding the dynamics of this coupled model, with both continuous (``Sellers-type) and discontinuous (``Budyko-type) equations. We also introduce two new Budyko-type models that are highly effective in modeling the extreme glacial events of the Neoproterozoic Era. We prove in each case the existence of a stable equilibrium solution for which the ice sheet edge rests in tropical latitudes. Mathematical tools used in the analysis include geometric singular perturbation theory and Filippov's theory of differential inclusions.

Highlights

The hierarchy of climate models begins on one end with conceptual energy balance and low-order models, progresses through intermediate complexity models, and culminates with ever-evolving Earth system models
We prove the existence of a stable equilibrium solution for finite approximations of the infinite-dimensional coupled model having an ice edge in tropical latitudes
We present a new Budyko-type Energy balance models (EBM) that admits a stable, large ice cap equilibrium solution

Summary

Introduction

The hierarchy of climate models begins on one end with conceptual energy balance and low-order models, progresses through intermediate complexity models, and culminates with ever-evolving Earth system models. We prove the existence of a stable equilibrium solution for finite approximations of the infinite-dimensional coupled model having an ice edge in tropical latitudes This so-called Jormungand climate state was proposed in [1] as an alternative to the Snowball Earth hypothesis, in which the Earth is posited to have been completely covered with ice as a consequence of ice-albedo feedback, as discussed below. We introduce two new Budyko-type albedo functions, proving in each case the existence of a stable equilibrium solution with ice line near the equator. On the other hand, assumed a continuous transition in the albedo function across the ice line, and EBM with this property are of “Sellers-type.” We use this terminology in all that follows. We turn first to results concerning the Sellers-type version of (11) presented in [40]

The temperature–ice line model

The Jormungand model

Findings

Conclusion