A Nonhydrostatic Finite-Element Model for Three-Dimensional Stratified Oceanic Flows. Part I: Model Formulation

Abstract

The vast majority of advanced numerical ocean models in use today, while performing extremely well, especially for certain classes of problem, do not necessarily take full advantage of current trends in numerical analysis and scientific computing. Here, a three-dimensional finite-element model is presented for use in oceanic simulations. The main aim is to fully exploit the use of unstructured meshes in both the horizontal and vertical directions, in order to conform well to topography and coastlines, and also to enable the straightforward use of dynamic variable mesh resolution reflecting fluid flow. In addition, the model should be accurate and efficient under typical oceanographic conditions, and not make the hydrostatic approximation. For simplicity here however the model does assume the presence of a rigid lid. To cope with inherent instabilities present in finite-element simulations of incompressible flow, caused by the Lagrange multiplier role that pressure plays in satisfying incompressibility, a mixed formulation for representing velocity and pressure is employed. Additionally, instabilities occurring due to the advection-dominated nature of the flow are dealt with using linear Petrov?Galerkin methods. In the course of this work a different type of instability has also been observed, which has some similarities with the sigma coordinate pressure gradient problem. The instability results from the mixed nature of the finite-element formulation and consequent poor satisfaction of hydrostatic balance, which in turn manifests itself in errors and spurious velocities on distorted meshes, such as those typical over topography. A satisfactory solution to this problem is presented that involves a splitting of pressure and allows efficient computations, whether the flow be close to or far from a state of hydrostatic balance.