Analysis of a two-layer energy balance model: Long time behavior and greenhouse effect

Author:

Cannarsa P.1ORCID,Lucarini V.23ORCID,Martinez P.4ORCID,Urbani C.5ORCID,Vancostenoble J.4ORCID

Affiliation:

1. Dipartimento degli Studi di Matematica, Università di Roma “Tor Vergata” 1 , Via della Ricerca Scientifica, 00133 Roma, Italy

2. Department of Mathematics and Statistics, University of Reading 2 , Reading RG6 6AX, United Kingdom and , Reading RG6 6AX, United Kingdom

3. Centre for the Mathematics of Planet Earth, University of Reading 2 , Reading RG6 6AX, United Kingdom and , Reading RG6 6AX, United Kingdom

4. Institut de Mathématiques de Toulouse, UMR 5219, Université de Toulouse; CNRS UPS IMT 3 , F-31062 Toulouse Cedex 9, France

5. Dipartimento di Scienze Tecnologiche e dell’Innovazione, Università Mercatorum 4 , Piazza Mattei 10, 00186 Roma, Italy

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 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.

Funder

Istituto Nazionale di Alta Matematica "Francesco Severi"

Ministero dell'Istruzione, dell'Università e della Ricerca

Accademia Nazionale dei Lincei

Horizon 2020 Framework Programme

Marie Curie ITN CriticalEarth

Engineering and Physical Sciences Research Council

Agence Nationale de la Recherche

Publisher

AIP Publishing

Subject

Applied Mathematics,General Physics and Astronomy,Mathematical Physics,Statistical and Nonlinear Physics

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