  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 $\epsilon_a$. 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 $\epsilon_a
\in (0, 2)$, 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 $\epsilon_a \in (0, 2)$, 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 $\epsilon_a$. This is the key
mathematical manifestation of the greenhouse effect.
