Global Well-Posedness of the Primitive Equations of Large-Scale Ocean Dynamics with the Gent–McWilliams–Redi Eddy Parametrization Model

Abstract

We prove global well-posedness of the ocean primitive equations coupled to advection-diffusion equations of the oceanic tracers temperature and salinity that are supplemented by the eddy parametrization model due to Gent, McWilliams, and Redi. This parametrization forms a milestone in global ocean modeling and constitutes a central part of any general ocean circulation model computation. The eddy parametrization adds a secondary transport velocity to the tracer equation and renders the original Laplacian operators in the advection-diffusion equations nonlinear, with a diffusion matrix that depends via the equation of state in a nonlinear fashion on both tracers simultaneously. The eddy parametrization of Gent, McWilliams, and Redi augments the complexity of the mathematical analysis of the whole system which we present here. We show first that weak solutions exist globally in time, provided the parametrization uses a regularized density. Then we prove by a detailed analysis of the eddy operators the global well-posedness. Our results apply also to the “small-slope approximation” that is commonly used in global ocean simulations.