Primitive Equation Alternatives to the Wave Equation Formulation

In the late 1970's, the wave equation formulation of the shallow water equations appeared to be one of the few options for finite-element model development. Spurious pressure modes occurred in most primitive equation formulations because of the coupling between the gravity wave terms in the continuity and momentum equations. However, several new and rediscovered elements make a primitive equation approach viable and offer advantages over the wave equation methods. Some of the strengths of the wave equation formulation are the amplitude and phase accuracy for the explicit version, good efficiency, and absence of spurious pressure modes. Some major shortcomings are poor accuracy for the implicit version and poor stability when advection becomes important. A finite element that provides a close replacement for the wave equation formulation is the P₁^NC-P₁ element which has linear non-conforming bases for velocity and linear conforming bases for sea level. The latter are the same as the linear bases used with the wave equation approach; hence, there is a close correspondence in data structure between the two approaches. Used in conjunction with the primitive equations, the approach with this element provides all the same strengths as the wave equation formulation but not the weaknesses. This approach has better amplitude and phase accuracy for both explicit and implicit methods, has good efficiency, has no spurious modes, and can be used with a wide variety of advection operators including ELM and semi-Lagrangian methods. In addition, investments in software infrastructure can be retained because of the similar data structure in the two approaches.