  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.
