Dispersion Relation Analysis of the $P^NC_1 - P^_1$ Finite-Element Pair in Shallow-Water Models

Abstract

Shallow-water models typically employ gridpoint, finite, and spectral-element techniques. For most of these methods the coupling between the momentum and continuity equations is a delicate problem and usually leads to spurious solutions in the representation of inertia-gravity waves. The spurious modes have a wide range of characteristics and may take the form of pressure (surface-elevation), velocity, and/or Coriolis modes. The modes usually cause aliasing and an accumulation of energy in the smallest-resolvable scale, leading to noisy solutions. The triangular finite-element pair $P^{NC}_{1}-P^{}_{1}$ is shown to "properly" model the dispersion of the inertia-gravity waves, and this is achieved at a reasonable computational cost compared to traditional finite-difference schemes. Two tests are proposed. The first test examines the propagation and dispersion of fast surface gravity waves in a circular basin and their reflection at the lateral boundary. In the second test, results of the Stommel problem are presented using the proposed finite-element pair and are compared with those obtained with the C- and CD-finite-difference grids. They illustrate the promise of the proposed approach.