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arXiv · 2211.15430

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

Piermarco Cannarsa·Valerio Lucarini·Patrick Martinez·Cristina Urbani·Judith Vancostenoble
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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 $CO_2$ and $CH_4$ 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.

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Piermarco Cannarsa, Valerio Lucarini, Patrick Martinez, Cristina Urbani, Judith Vancostenoble. 2023-09-29. Analysis of a two-layer energy balance model: long time behaviour and greenhouse effect. https://doi.org/10.1063/5.0136673

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