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The equations of the multi-phase humid atmosphere expressed as a quasi variational inequality

Abstract:
In this talk we introduce a new formulation of the multi-phase humid atmosphere equations, generalizing the models studied in earlier works by M. Coti-Zelati, J. Tribbia, R. Temam and others, see e.g., Coti Zelati et al, Physica D. (2013)  and Temam and Wu, J. Funct. Anal. (2015). More precisely, we consider the situation where the humid quantities in the clouds comprise the water vapor, the cloud-condensates, and rain water with relative mass ratios q_v, q_c and q_r, respectively. Meanwhile, the saturation vapor concentration q_{vs} is a diagnostic variable depending itself on the state (i.e., the temperature T and pressure p). The bulk microphysics model for q_v, q_c,  q_r  and for the temperature T  are in the spirit of Kessler (1969) and Klemp and Wilhelmson(1978).
When considering the condensation of water vapor to cloud water and the inverse evaporation process, it is often assumed, in particular for warm clouds, that the vapor-to-cloud water conversion is instantaneous. Accordingly, the water vapor mass ratio q_v can not exceed the saturation ratio q_{vs} in general when cloud condensate is formed instantaneously. From the mathematical viewpoint, we are faced with the challenges brought by the nonlinear constraint q_v ≤ q_{vs} which depends itself on the solution, and the discontinuity in the source terms corresponding to the phase change. We are led to introduce and handle a system of equations and quasi-variational inequalities, for which we prove the global existence of solutions satisfying the constraint using regularization and penalization techniques.