Well-posedness of a non local ocean-atmosphere coupling model: study of a 1D Ekman boundary layer problem with non-local KPP-type turbulent viscosity profile

This paper addresses the mathematical analysis of the ocean-atmosphere coupling problem, including Coriolis force, non-local turbulent closure and realistic nonlinear interface conditions. We introduce a 1D vertical model corresponding to a coupled Ekman boundary layer problem with non-local turbulent viscosities. The interest of this model lies in its proximity to realistic ones by considering the numerical strategies employed to take into account the turbulent scale. Well-posedness is first studied in stationary and non-stationary states considering generalized parameterized turbulent viscosities. We establish sufficient criteria on the viscosity profiles for the uniqueness of solution and find that they are not met for parameters in the order of magnitude used in ocean and atmosphere models. To identify precisely the conditions of well-posedness, we therefor establish a necessary and sufficient criterion for the stationary state. We show that there is non-uniqueness of the solution when considering typical viscosity profiles from ocean and atmosphere models. Eventually, we illustrate that non-uniqueness is produced by an inconsistency between the viscosity profile and the boundary layer parametrisation.