Computing precipitations with a vertical radiative-convective model with no adjustable parameters, using the maximum entropy production hypothesis.
Authors: Pikeroen Quentin , Paillard Didier , Dubrulle Bérengère , Watrin Karine 
Published: February 2023

The state-of-the-art General Circulation Models or Earth System Models are based on conservation equations like the conservation of mass, momentum (Navier-Stokes), energy, and water... These equations are written in the form of partial derivative equations and are resolved on a grid whose spatial increment is a few tens or hundreds of kilometers and whose time increment is a few minutes. This means that phenomena acting below the numerical resolution are not computed. But because of the nonlinearity of equations, large scales are not independent of small scales, therefore the cutoff in resolution induces errors. For example, the linear relation between energy fluxes and temperature gradients (Fourier law) is not true for a grid of this size. To overcome this issue, it is usual to add new equations in order to close the conservation equations. In these new equations, new parameters are added and are generally tuned to fit observations. Though they are all based on the same physics, every climate model has a different set of "closure equations" and tuned parameters, leading to different results. For instance, while model comparisons are satisfying when looking at temperatures, results may differ significantly between two models when looking at precipitations.Now, I am going to present an alternative way of resolving the climate system using zero tunable parameters. To achieve this, a paradigm change is needed. Partial derivative equations are no longer used, and variables are resolved with an optimization problem: maximizing a function under constraints (of conservations). The maximized function is the entropy production due to energy transfers and depends on temperatures only. Because solving the optimization problem isn't straightforward, the climate system is for now reduced to a vertical atmosphere, with only vertical energy fluxes. Such kind of model is sometimes called "radiative-convective" model and can be compared to tropical atmospheric observations because horizontal fluxes are less important there. The constraints imposed are the conservation of energy, the conservation of mass, and the conservation of water. Surprisingly, adding this last constraint to the model enables us to predict precipitations of about 1.2 m/year, in the good order of magnitude of average tropical precipitations. Theoretically, this means that precipitations depend mostly on the radiative transfer in the atmosphere.The maximization of entropy production is probably not a generic "law of Nature" and might not apply to any out-of-equilibrium system. Here, we choose not to enter the debate whether it should be true for the climate or not, but only to show that this procedure can be a useful and efficient tool to close equations without introducing any tunable parameters, even when applied to precipitations. Though the optimization problem may rapidly become intractable, we can still envision building a more complete model of the atmospheric water cycle on these premises.