An Analysis of the Klemp and Durran Radiation Boundary Condition as Applied to Dissipative Internal Waves

Abstract

Numerical simulations of the oceanic (atmospheric) boundary layer are complicated by the need to specify appropriate ?outflow? or ?radiation? boundary conditions at the artificial lower (upper) boundary of the computational domain. If the boundary layer is stratified, particular care is necessary to insure that internal-gravity-wave disturbances generated within the domain are not artificially reflected by the computational boundary. A major advance was made almost 20 years ago by Klemp and Durran; their radiation condition relates the Fourier transformed pressure fluctuation to the Fourier transformed vertical-velocity perturbation along the artificial boundary. Because it is local in time, the Klemp and Durran (KD) condition is easily incorporated into a wide variety of numerical models for only a minor computational expense. Indeed, it has been widely used in the atmospheric and oceanic sciences communities. For simulations of dissipative systems, however, perturbation-flux conditions must also be specified at the artificial boundary?these are in addition to the KD condition (or some other constraint) on the normal velocity component at that boundary. This article considers the performance of the KD condition in conjunction with zero perturbation stress and zero perturbation buoyancy-flux conditions (?KDZ? conditions, collectively), because the latter are generally assumed to be appropriate for simulations of boundary layer phenomena. Analysis of the response of a weakly dissipative, uniformly stratified fluid to forcing concentrated at a given depth reveals two potentially serious drawbacks of the KDZ conditions. First, nonhydrostatic dynamics are not adequately treated by the KD condition, itself. Moreover, the imposition of zero perturbation-flux conditions causes artificial boundary layers to form along the outflow boundary. Although these boundary layers are passive, they are unlikely to be resolved in numerical simulations; thus, discretization of the KDZ conditions may cause further errors in the simulated internal-wave dynamics. A consistent set of boundary conditions for simulations of dissipative, stratified fluids is proposed.