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

TL;DR

This paper establishes the global well-posedness of the coupled ocean primitive equations with the Gent-McWilliams-Redi eddy parametrization, a key model in large-scale ocean circulation, including complex nonlinear diffusion effects.

Contribution

It provides the first rigorous proof of global well-posedness for the primitive equations with this advanced eddy parametrization, accounting for nonlinear diffusion and tracer interactions.

Findings

Existence of global weak solutions with regularized density

Proof of global well-posedness for the full coupled system

Applicability to the small-slope approximation in ocean models

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 modelling 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-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.