How much is not enough? One-, two- and three-atmospheric-layer models for earth’s energy balance

Simple energy balance models of planetary systems are of fundamental importance to understanding equilibrium temperatures. Most textbooks that discuss energy balance take a further step and include the effects of an atmosphere on the surface temperature. It is noticeable, however, that in such discussions of planetary surface energy balance some fundamental physical characteristics of energy balance are incorrectly portrayed, but rarely commented upon. Here it is shown that zero-dimensional energy balance models with either one or two atmospheric layers will necessarily lead to results that lack correct physical content, although they may or may not give temperature results that are numerically approximately correct. Ignoring the role of convective heat transfer and having even rough magnitudes of both upward- and downward-welling long-wavelength radiation wrong means that fundamental physical processes are missing. Adding a third atmospheric layer to a simple model that also includes a stylised representation of convection can restore both reasonable numerical accuracy and the correct physical properties, while at the same time being simple enough for easy physical interpretation.