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

Automated summary

We study a two-layer energy balance model that considers the vertical exchanges between the surface layer and the atmosphere. The coupling between the surface temperature and the atmospheric temperature is influenced by the emission of infrared radiation and non-radiative vertical energy exchanges. The absorptivity of the atmosphere, denoted as epsilon a, plays a crucial role and is affected by greenhouse gases. The research proves the existence of solutions when epsilon a is within the range of (0, 2) and indicates finite time blow up if epsilon a > 2. The long term dynamics are explained for epsilon a ∈ (0, 2), showing convergence to equilibrium points. Additionally, the dependence of equilibrium points on the involved parameters is studied, highlighting the monotonic increase of surface temperature with respect to epsilon a, which is the mathematical manifestation of the greenhouse effect.