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

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.