# American Institute of Mathematical Sciences

September  2015, 20(7): 2187-2216. doi: 10.3934/dcdsb.2015.20.2187

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

 1 Department of Mathematics, Oberlin College, 10 N. Professor St, Oberlin, OH 44074 2 Department of Mathematics, University of California{Irvine, Irvine, CA 92697, United States

Received  September 2014 Revised  January 2015 Published  July 2015

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.
Citation: James Walsh, Christopher Rackauckas. On the Budyko-Sellers energy balance climate model with ice line coupling. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 2187-2216. doi: 10.3934/dcdsb.2015.20.2187
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