A Spectral Limited-Area Model Formulation with Time-dependent Boundary Conditions Applied to the Shallow-Water Equations

Abstract

The spectral technique is frequently used for the horizontal discretization in global atmospheric models. This paper presents a method where double Fourier series are used in a limited-area model (LAM). The method uses fast Fourier transforms (FFT) in both horizontal directions and takes into account time-dependent boundary conditions. The basic idea is to extend the time-dependent boundary fields into a zone outside the integration area in such a way that periodic fields are obtained. These fields in the extension zone and the forecasted fields inside the integration area are connected by use of a narrow relaxation zone along the boundaries of the limited area. The extension technique is applied to the shallow-water equations. A simple explicit (leapfrog) integration is shown to give results that are almost identical to the hemispherical forecast used as boundary fields. A nonlinear normal-mode initialization scheme developed in the framework of the spectral formulation is shown to work satisfactorily. The initialization scheme is furthermore used in a normal-mode time extrapolation scheme. Combined with the leapfrog scheme this method is stable for time steps similar to those used in the semi-implicit scheme and has the advantage that it is able to reduce the noise introduced in the forecast from unbalanced boundary fields. Experiments are made where the semi-Lagrangian treatment of advection is combined with either a semi-implicit or a normal-mode adjustment scheme. Both combinations yield comparably good results for moderately long time steps, though the semi-Lagrangian semi-implicit scheme is more accurate and more stable for long time steps. An efficient semi-Lagrangian scheme without any interpolations is introduced and shown to be unconditionally stable and nondamping for advection by a constant wind field. This scheme is tested and compared with the usual semi-Lagrangian schemes where interpolations are involved. The overall efficiency and accuracy of the proposed spectral formulation applied to the shallow-water model encouraged the development of a baroclinic spectral LAM, now in progress.