# American Institute of Mathematical Sciences

September  2017, 22(7): 2687-2715. doi: 10.3934/dcdsb.2017131

## Diffusive heat transport in Budyko's energy balance climate model with a dynamic ice line

 Department of Mathematics, Oberlin College, 10 N. Professor St, Oberlin, OH 44074, USA

Received  July 2016 Revised  September 2016 Published  April 2017

Fund Project: The author recognizes and appreciates the support of the Mathematics and Climate Research Network (http://www.mathclimate.org).

M. Budyko and W. Sellers independently introduced seminal energy balance climate models in 1969, each with a goal of investigating the role played by positive ice albedo feedback in climate dynamics. In this paper we replace the relaxation to the mean horizontal heat transport mechanism used in the models of Budyko and Sellers with diffusive heat transport. We couple the resulting surface temperature equation with an equation for movement of the edge of the ice sheet (called the ice line), recently introduced by E. Widiasih. We apply the spectral method to the temperature-ice line system and consider finite approximations. We prove there exists a stable equilibrium solution with a small ice cap, and an unstable equilibrium solution with a large ice cap, for a range of parameter values. If the diffusive transport is too efficient, however, the small ice cap disappears and an ice free Earth becomes a limiting state. In addition, we analyze a variant of the coupled diffusion equations appropriate as a model for extensive glacial episodes in the Neoproterozoic Era. Although the model equations are no longer smooth due to the existence of a switching boundary, we prove there exists a unique stable equilibrium solution with the ice line in tropical latitudes, a climate event known as a Jormungand or Waterbelt state. As the systems introduced here contain variables with differing time scales, the main tool used in the analysis is geometric singular perturbation theory.

Citation: James Walsh. Diffusive heat transport in Budyko's energy balance climate model with a dynamic ice line. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2687-2715. doi: 10.3934/dcdsb.2017131
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##### References:
Solid: Plot of (3) with $\beta=24.5^\circ$. Dashed: Quadratic approximation (6)
Plots of function (25). (a) $N=1$ in (15), $D=0.45$. (b) $N=1, D=0.35$. (c) D=0.35. Solid: $N=2$. Dashed: $N=5$
Plots of function (25) with $D=0.394$. (a) and (b) $N=1$ in (15). (c) Solid: $N=2$ in (15). Dashed: $N=5$ in (15).
Plots of the Jormungand diffusion model functions $h^-(\eta) \,$ ($\eta<\rho) \,$ and $h^+(\eta) \,$ $\, (\eta\geq \rho$) for $D=0.25$. (a) $N=1$ in (15). (b) Solid: $N=2$ in (15). Dashed: $N=5$ in (15)
Bifurcation plot for the Jormungand model with diffusive heat transport, with $N=5$ in (15)
Plots of function $h(\eta)$ given by (52). (a) $N=1$ in (15). (b) Including higher-order terms in (4). Solid: $N=2$ ($s_4=-0.044$). Dashed: $N=5$ ($s_6=0.006, s_8=0.016, s_{10}=0.006$). Parameters as in (26), $C=3.09$
Plots of functions $h^-(\eta)$ (64) (for $\eta<\rho$) and $h^+(\eta)$ (for $\eta> \rho$) (a) $N=1$ in (15). (b) $N=5$ ($s_4, ... s_{10}$ as in Figure 6). Parameters for (a) and (b) as in Figure 13 in [37]. (c) $N=1$, parameters as in (40)
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