MEP solution for a minimal climate model: success and limitation of a variational problem

Abstract

Abstract. Maximum Entropy Production conjecture (MEP) is applied to a minimal four-box model of climate which accounts for both horizontal and vertical material heat fluxes. It is shown that, under condition of fixed insolation, a MEP solution is found with reasonably realistic temperature and heat fluxes, thus generalising results from independent two-box horizontal or vertical models. It is also shown that the meridional and the vertical entropy production terms are independently involved in the maximisation and thus MEP can be applied to each subsystem with fixed boundary conditions. We then extend the four-box model by increasing its number of degrees of freedom, and test its realism by comparing it with a GCM output. An order-of-magnitude evaluation of contributions to the material entropy production (≈50 mW m−2 K−1) due to horizontal and vertical processes within the climate system is carried out by using ad hoc temperature fields. It turns out that approximately 40 mW m−2 K−1 is the entropy production due to vertical heat transport and 5–7 mW m−2 K−1 to horizontal heat transport. A MEP solution is found which is fairly realistic as far as the horizontal large scale organisation of the surface climate is concerned whereas the vertical structure looks to be unrealistic and presents seriously unstable features. Finally a more general problem is investigated in which the longwave transmissivity is varied simultaneously with the temperature. This leads to a MEP solution characterised by a much warmer climate, with very vigorous vertical heat fluxes, in which the atmosphere is opaque to longwave radiation. A critical discussion about how to interpret MEP and how to apply it in a physically correct way concludes the paper.