On the equations of the large-scale ocean

TLDR: In this article, the authors study the mathematical formulations and attractors of three systems of equations of the ocean, i.e., primitive equations (the PEs), the PEV2s, and the Boussinesq equations.

Abstract: As a preliminary step towards understanding the dynamics of the ocean and the impact of the ocean on the global climate system and weather prediction, the authors study the mathematical formulations and attractors of three systems of equations of the ocean, i.e. the primitive equations (the PEs), the primitive equations with vertical viscosity (the PEV2s), and the Boussinesq equations (the BEs), of the ocean. These equations are fundamental equations of the ocean. The BEs are obtained from the general equations of a compressible fluid under the Boussinesq approximation, i.e. the density differences are neglected in the system except in the buoyancy term and in the equation of state. The PEs are derived from the BEs under the hydrostatic approximation for the vertical momentum equation. The PEV2s are the PEs with the viscosity for the vertical velocity retained. This retention is partially based on the important role played by the viscosity in studying the long time behaviour of the ocean, and the Earth's climate.