The Duality between the Boussinesq and Non-Boussinesq Hydrostatic Equations of Motion

Abstract

The hydrostatic equations of motion for ocean circulation, written in terms of pressure as the vertical coordinate, and without making the Boussinesq approximation in the continuity equation, correspond very closely with the hydrostatic Boussinesq equations written in terms of depth as the vertical coordinate. Two mathematical equivalences between these non-Boussinesq and Boussinesq equation sets are demonstrated: first, for motions over a level bottom; second, for general motions with a rigid lid. A third non-Boussinesq equation set, for general motions with a free surface, is derived and is shown to possess a similar duality with the Boussinesq set after making due allowance for exchange of the roles of bottom pressure and sea surface height in the boundary conditions, a reversal of the direction of integration of the hydrostatic equation, and substitution of specific volume for density in the hydrostatic equation. The crucial simplification in these equations of motion comes from the hydrostatic approximation, not the Boussinesq approximation. A practical consequence is that numerical ocean circulation models that are based on the Boussinesq equations can, with very minimal rearrangement and reinterpretation, be made free of the strictures of the Boussinesq approximation, especially the ones that follow from its neglect of density dilatation in the conservation of mass.