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

DOI10.1063/5.0136673arXiv2211.15430OpenAlexW4388411184MaRDI QIDQ6418732

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

Publication date: 28 November 2022

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

e p s i l o n_{a}

. The value of

e p s i l o n_{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

e p s i l o n_{a}

. First we prove that global existence of solutions of the system holds if and only if

e p s i l o n_{a} i n (0, 2)

, and blow up in finite time occurs if

e p s i l o n_{a} > 2

. (Note that the physical range of values for

e p s i l o n_{a}

is

(0, 1]

.) Next, we explain the long time dynamics for

e p s i l o n_{a} i n (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 grows as

e p s i l o n_{a}

grows. This is the key mathematical manifestation of the greenhouse effect.

Full work available at URL: https://doi.org/10.1063/5.0136673

Mathematics Subject Classification ID

Dynamical systems in fluid mechanics, oceanography and meteorology (37N10) Climate science and climate modeling (86A08)