Climate Stability for a Sellers-Type Model

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  • 1 Courant Institute of Mathematical Sciences, New York University, New York 10012
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Abstract

We study a diffusive energy-balance climate model, governed by a nonlinear parabolic partial differential equation. Three positive steady-state solutions of this equation are found; they correspond to three possible climates of our planet: an interglacial (nearly identical to the present climate), a glacial, and a completely ice-covered earth. We consider also models similar to the main one studied, and determine the number of their steady states. All the models have albedo continuously varying with latitude and temperature, and entirely diffusive horizontal heat transfer. The diffusion is taken to be nonlinear as well as linear.

We investigate the stability under small perturbations of the main model's climates. A stability criterion is derived, and its application shows that the “present climate” and the “deep freeze” are stable, whereas the model's glacial is unstable. A variational principle is introduced to confirm the results of this stability analysis.

We examine the dependence of the number of steady states and of their stability on the average solar radiation. The main result is that for a sufficient decrease in solar radiation (∼2%) the glacial and interglacial solutions disappear, leaving the ice-covered earth as the only possible climate.

Abstract

We study a diffusive energy-balance climate model, governed by a nonlinear parabolic partial differential equation. Three positive steady-state solutions of this equation are found; they correspond to three possible climates of our planet: an interglacial (nearly identical to the present climate), a glacial, and a completely ice-covered earth. We consider also models similar to the main one studied, and determine the number of their steady states. All the models have albedo continuously varying with latitude and temperature, and entirely diffusive horizontal heat transfer. The diffusion is taken to be nonlinear as well as linear.

We investigate the stability under small perturbations of the main model's climates. A stability criterion is derived, and its application shows that the “present climate” and the “deep freeze” are stable, whereas the model's glacial is unstable. A variational principle is introduced to confirm the results of this stability analysis.

We examine the dependence of the number of steady states and of their stability on the average solar radiation. The main result is that for a sufficient decrease in solar radiation (∼2%) the glacial and interglacial solutions disappear, leaving the ice-covered earth as the only possible climate.

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