Chapter 6 The foundations of oceanic dynamics and climate modeling

The main goal of this article is to derive the mathematical foundations for oceanic dynamics, with applications to the study of the climate of the Earth. This article builds upon the work of Pliss and Sell, [55], in which the physical basis for the related mathematical model is presented. A key feature described herein is an in{Thorn}nitesimal Interface Model for the ocean. The role of this model is to aggregate the net effects of all the relevant atmospheric activity at the surface of the ocean. The Interface Model contains an important aspect of the boundary forces acting on the ocean. The basic model is a system of partial differential equations, consisting of the 3D Navier{Eth} Stokes equations and two transport equations for the temperature and the salinity. Since the boundary forces acting on the ocean vary with time, the oceanic equations areÑof necessityÑ nonautonomous equations. We assume here the Quasi Periodic Ansatz (QPA), which states that all the time-dependent forces acting on the ocean are quasi periodic functions of time. The associate frequency vector is determined by the natural frequencies of the planetary motion in the solar system. The resulting inhomogeneous boundary conditions describe forces which may add energy to the ocean. This includes the transfer of radiant heat and other atmospheric effects. We calculate an upper bound on the latent energy absorbed by the ocean from the boundary forces, see (3.37). If the boundary forces are too strong, this can destabilize the dynamics of the ocean. The existence and uniqueness theory for the strong solutions is fully developed in this article. Our approach is based on the method of Leray and Hopf for the Navier{Eth}Stokes equations. The QPA enables one to obtain a skew product semißow π(t) on T^k+1 × V¹, see Theorem 4.1. An important dynamical property of this semißow is the QP-Herculean Theorem 4.2, see Section 4. Among other things, the latter theorem states that any bounded, invariant set in T^k+1 × V¹ is contained in T^k+1 × V², where V² is the domain D(A) of the related Stokes operator for the oceanic equations. Finally we show that related research in the area of thin domain dynamics offers a hopeful sign for obtaining good information concerning the global attractor for the oceanic dynamics, and thereby for the climate of the Earth.