Robert L. Higdon

Absorbing boundary conditions for acoustic and elastic waves in stratified media

Abstract Absorbing boundary conditions are needed for computing numerical models of wave motions in unbounded spatial domains. Prior progress on this problem for acoustic and elastic waves has generally been concerned with waves propagating through uniform media. The present paper is concerned with waves in stratified media, which are of interest, for example, in geophysical problems. Suppose that the medium consists of homogeneous layers separated by parallel horizontal interfaces, and suppose that absorbing boundary conditions are needed along a vertical computational boundary. The boundary conditions that are described in this paper are based on a quantity known as the ”ray parameter.“ According to Snells law, this parameter remains the same when a plane wave propagates through a stratified medium and undergoes reflection, refraction, and, in the case of elastic waves, conversion. Once can therefore use the same absorbing boundary conditions in all layers. For acoustic waves, the absorption properties are the same in all layers. For elastic waves, the absorption properties vary somewhat from one layer to another; however, one still obtains good absorption in all layers, even in the presence of strong contrasts between layers. The boundary conditions are also effective in absorbing Rayleigh waves, which propagate along free surfaces of elastic media. The boundary formulas developed here can be applied without modification to problems in both two and three dimensions.

Numerical modelling of ocean circulation

Computational simulations of ocean circulation rely on the numerical solution of partial differential equations of fluid dynamics, as applied to a relatively thin layer of stratified fluid on a rotating globe. This paper describes some of the physical and mathematical properties of the solutions being sought, some of the issues that are encountered when the governing equations are solved numerically, and some of the numerical methods that are being used in this area.

Absorbing Boundary Conditions for Dispersive Waves

The goal of an absorbing boundary condition is to simulate the outward radiation of energy at an artificial computational boundary. To that end, the boundary condition should make a valid statement about outgoing waves, at least in an approximate sense, but this statement should not require apriori knowledge of the pointwise values of the wave field. Typically, such a boundary condition involves a wave velocity. However, in the case of dispersive waves there are two kinds of velocity, phase velocity and group velocity, and each of the velocities varies with wavenumber and frequency. This variation of velocity is a central issue in the construction of absorbing boundary conditions for dispersive waves. The present abstract summarizes a technique developed by [Higdon, 1994].

Physical and Computational Issues in the Numerical Modeling of Ocean Circulation

The large-scale circulation of the world’s oceans can be modeled by systems of partial differential equations of fluid dynamics, as scaled and parameterized for oceanic flows. This paper outlines some physical, mathematical, and computational aspects of such modeling. The topics include multiple length, time, and mixing scales; the choice of vertical coordinate; properties of the shallow water equations for a single-layer fluid, including effects of the rotating reference frame; a statement of the governing equations for a three-dimensional stratified fluid with an arbitrary vertical coordinate; time-stepping and multiple time scales; and various options for spatial discretizations.

Multiple time scales and pressure forcing in discontinuous Galerkin approximations to layered ocean models

This paper addresses some issues involving the application of discontinuous Galerkin (DG) methods to ocean circulation models having a generalized vertical coordinate. These issues include the following. (1) Determine the pressure forcing at cell edges, where the dependent variables can be discontinuous. In principle, this could be accomplished by solving a Riemann problem for the full system, but some ideas related to barotropic-baroclinic time splitting can be used to reduce the Riemann problem to a much simpler system of lower dimension. Such splittings were originally developed in order to address the multiple time scales that are present in the system. (2) Adapt the general idea of barotropic-baroclinic splitting to a DG implementation. A significant step is enforcing consistency between the numerical solution of the layer equations and the numerical solution of the vertically-integrated barotropic equations. The method used here has the effect of introducing a type of time filtering into the forcing for the layer equations, which are solved with a long time step. (3) Test these ideas in a model problem involving geostrophic adjustment in a multi-layer fluid. In certain situations, the DG formulation can give significantly better results than those obtained with a standard finite difference formulation.

Pressure forcing and dispersion analysis for discontinuous Galerkin approximations to oceanic fluid flows

This paper is part of an effort to examine the application of discontinuous Galerkin (DG) methods to the numerical modeling of the general circulation of the ocean. One step performed here is to develop an integral weak formulation of the lateral pressure forcing that is suitable for usage with a DG method and with a generalized vertical coordinate that includes level, terrain-fitted, isopycnic, and hybrid coordinates as examples. This formulation is then tested, in special cases, with analyses of dispersion relations and numerical stability and with some computational experiments. These results suggest that the advantages of DG methods may significantly outweigh their disadvantages, in the settings tested here. This paper also outlines some other issues that need to be addressed in future work.