A Perfectly Matched Layer Approach to the Linearized Shallow Water Equations Models

Abstract

A limited-area model of linearized shallow water equations (SWE) on an f plane for a rectangular domain is considered. The rectangular domain is extended to include the so-called perfectly matched layer (PML) as an absorbing boundary condition. Following the proponent of the original method, the equations are obtained in this layer by splitting the shallow water equations in the coordinate directions and introducing the absorption coefficients. The performance of the PML as an absorbing boundary treatment is demonstrated using a commonly employed bell-shaped Gaussian initially introduced at the center of the rectangular physical domain. Three typical cases are studied: A stationary Gaussian where adjustment waves radiate out of the area. A geostrophically balanced disturbance being advected through the boundary parallel to the PML. This advective case has an analytical solution allowing one to compare forecasts. The same bell being advected at an angle of 45° so that it leaves the domain through a corner. For the purpose of comparison, a reference solution is obtained on a fine grid on the extended domain with the characteristic boundary conditions. Also computed are the rms difference between the 48-h forecast and the analytical solution as well as the 48-h evolution of the mean absolute divergence, which is related to geostrophic balance. The authors found that the PML equations for the linearized shallow water equations on an f plane support unstable solutions when the mean flow is not unidirectional. Use of a damping term consisting of a 9-point smoother added to the discretized PML equations stabilizes the PML equations. The reflection/transmission is analyzed along with the case of instability for glancing propagation of the bell disturbance. A numerical illustration is provided showing that the stabilized PML for glancing bell propagation performs well with the addition of the damping term.