Energy balance models (EBM) are simple climate models that try to predict the average surface temperature of the Earth from solar radiation, emission of radiation to outer space, and Earth’s energy absorption and greenhouse effects.
In a steady state, the Earth has an effective temperature corresponding to the amount of radiation it emits, according to the Stefan-Boltzmann law?. This effective temperature is not the same as the average surface temperature however, because the surface temperature depends on radiation and absorption effects of the atmosphere and the oceans. Energy balance models try to account for these effects and try to predict the average surface temperature of the Earth accordingly: The temperature is per definitionem the only output of an EBM.
For some outline data, see solar radiation.
There is ongoing research about EBM with stochastic influences, using stochastic differential equations, for an example see:
This line of research is about getting an idea of how the probability distributions of important climate variables look like, when the plethora of influences is modeled by white noise. The authors of the preceding paper find evidence that a simple model could result in distributions with “fat tails”, such that usual cost functions don’t have a finite expectation value. In such a situation, these cost functions cannot be used to deduce mitigation strategies.
If these results can be reproduced with full fledged GCMs remains an open problem, however.
Kendal McGuffie, Ann Henderson-Sellers: A Climate Modelling Primer (see recommended reading)
John Marshall and R.Alan Plumb: Atmosphere, Ocean, and Climate Dynamics: an Introductory Text
Luc Tartar: An Introduction to Navier Stokes Equation and Oceanography (Springer 2006, ZMATH)