Analysis of a two-layer energy balance model: long time behaviour and greenhouse effect

Cannarsa, P., Lucarini, V. ORCID: https://orcid.org/0000-0001-9392-1471, Martinez, P., Urbani, C. and Vancostenoble, J. (2023) Analysis of a two-layer energy balance model: long time behaviour and greenhouse effect. Chaos: An Interdisciplinary Journal of Nonlinear Science, 33 (11). 113111. ISSN 1089-7682 doi: 10.1063/5.0136673

Abstract/Summary

We study a two-layer energy balance model that allows for vertical exchanges between a surface layer and the atmosphere. The evolution equations of the surface temperature and the atmospheric temperature are coupled by the emission of infrared radiation by one level, that emission being partly captured by the other layer, and the effect of all non-radiative vertical exchanges of energy. Therefore, an essential parameter is the absorptivity of the atmosphere, denoted εa⁠. The value of εa depends critically on greenhouse gases: increasing concentrations of CO2 and CH4 lead to a more opaque atmosphere with higher values of ϵa⁠. First, we prove that global existence of solutions of the system holds if and only if εa∈(0,2) and blow up in finite time occurs if εa>2⁠. (Note that the physical range of values for εa is (0,1]⁠.) Next, we explain the long time dynamics for εa∈(0,2)⁠, and we prove that all solutions converge to some equilibrium point. Finally, motivated by the physical context, we study the dependence of the equilibrium points with respect to the involved parameters, and we prove, in particular, that the surface temperature increases monotonically with respect to εa⁠. This is the key mathematical manifestation of the greenhouse effect.

Item Type	Article
URI	https://reading-clone.eprints-hosting.org/id/eprint/113559
Identification Number/DOI	10.1063/5.0136673
Refereed	Yes
Divisions	Interdisciplinary Research Centres (IDRCs) > Centre for the Mathematics of Planet Earth (CMPE) Science > School of Mathematical, Physical and Computational Sciences > Department of Mathematics and Statistics
Publisher	American Institute of Physics
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