A radiative-convective model computing precipitations with the maximum entropy production hypothesis

What do we need to compute the pertinent variables for climate? Some highly detailed models exist, called Earth System Models, where all the relevant components of climate are present: the atmosphere, the ocean, the vegetation and the ice sheets. As many as possible phenomena are represented, and for accuracy, there are two ways of doing it. The first is to solve dynamics equations with a grid size as small as possible. This method induces high economic and computational costs. The second method is to compute the sub-grid processes with smart parameterizations adapted to the grid size. This method induces a massive amount of parameterizations. Some simpler models exist, e.g. 1D radiative-convective model, but like the other models, they use parameterizations. For example, to compute the material energy fluxes provoked by temperature gradients, one may use a Fourier law, saying that energy fluxes are locally proportional to temperature gradients. While this law has a well-defined parameter value at the microscopic scale, the parameter needs to be better defined for the climate scale. More than that, the button for this parameter can be turned to make the model closer to observations. This process is called tuning and exists in all accurate climate models. This article uses a new method to compute temperatures and energy fluxes, where tuning is impossible. We hope this method is more physical and universal as we have less range to tell the model to give the desired result beforehand. Therefore, it could be used for climates where few are known, such as paleoclimate or climates of other planets. The method used is based on a thermodynamic hypothesis, the maximum entropy production. For simplicity, we restrict the model to be 1D vertical for a tropical atmosphere. With conservation laws, the problem is an optimization problem under constraints. It is solved with an algorithm making a gradient descent from an initial condition. The result is the maximum of the objective function, the entropy production, where the constraints are satisfied. As constraints, energy conservation and mass conservation already give a suitable temperature profile. This article adds a new constraint on the water cycle. The water vapour is allowed to disappear, leading to precipitations, but it is not allowed to be created. The result for this new set of equations (or constraints) shows precipitations nil everywhere and positive at the top of the troposphere. It is like "cumulonimbus" precipitations. It seems coherent to what happens in the tropics, where the Intertropical Convergence Zone leads to deep convection. Moreover, the computed order of magnitude is correct. Fundamentally, although the water cycle is often described as a complex and multidisciplinary problem, the correct order of magnitude of precipitations can be computed with almost only the knowledge of radiative transfer.