Smooth solutions and attractor dimension bounds for planetary geostrophic ocean models

Abstract

Recent results on the existence and uniqueness of global strong solutions of the planetary geostrophic model of large-scale ocean circulation, with traditional parametrizations of horizontal and vertical diffusion of heat and horizontal momentum, are summarized. The new results are stronger than existing results for the hydrostatic primitive equations. A rigorous upper bound on attractor dimension for the planetary geostrophic model is presented, which scales as the cube of the product of the Reynolds number Re and the Peclet number Pe, or as Re 3/2 .Pe 9/4 if some additional assumptions are made. The dimension bound demonstrates that the model dynamics are controlled asymptotically by a finite number of effective degrees of freedom, and raises new physical questions regarding the associated estimate of this number. Available numerical evidence appears to indicate that, in contrast to analogous results for homogeneous turbulence described by the Navier-Stokes equations, substantial improvements to these bounds may be possible.