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

Piermarco Cannarsa; Valerio Lucarini; Patrick Martinez; Cristina Urbani; Judith Vancostenoble

doi:10.1063/5.0136673

paper

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

arXiv:2211.15430 · doi:10.1063/5.0136673

Abstract

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 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 $ε_{a}$ . The value of $ε_{a}$ depends critically on greenhouse gases: increasing concentrations of $C O_{2}$ and $C H_{4}$ lead to a more opaque atmosphere with higher values of $ε_{a}$ . First we prove that global existence of solutions of the system holds if and only if $ε_{a} \in (0, 2)$ , and blow up in finite time occurs if $ε_{a} > 2$ . (Note that the physical range of values for $ε_{a}$ is $(0, 1]$ .) Next, we explain the long time dynamics for $ε_{a} \in (0, 2)$ , 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 with respect to $ε_{a}$ . This is the key mathematical manifestation of the greenhouse effect.

arXiv comment56 pages, 17 figues

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

Abstract

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