A SPECTRAL LIMITED-AREA MODEL FORMULATION WITH TIME-DEPENDENT BOUNDARY-CONDITIONS APPLIED TO THE SHALLOW-WATER EQUATIONS

The spectral technique is frequently used for the horizontal discretiz ation in global atmospheric models. This paper presents a method where double Fourier series are used in a limited-area model (LAM). The met hod uses fast Fourier transforms (FFT) in both horizontal directions a nd takes into account time-dependent boundary conditions. The basic id ea 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 t he integration area are connected by use of a narrow relaxation zone a long the boundaries of the limited area. The extension technique is ap plied to the shallow-water equations. A simple explicit (leapfrog) int egration is shown to give results that are almost identical to the hem ispherical forecast used as boundary fields. A nonlinear normal-mode i nitialization scheme developed in the framework of the spectral formul ation is shown to work satisfactorily. The initialization scheme is fu rthermore used in a normal-mode time extrapolation scheme. Combined wi th 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 treat ment of advection is combined with either a semi-implicit or a normal- mode adjustment scheme. Both combinations yield comparably good result s for moderately long time steps, though the semi-Lagrangian semi-impl icit scheme is more accurate and more stable for long time steps. An e fficient semi-Lagrangian scheme without any interpolations is introduc ed 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 a pplied to the shallow-water model encouraged the development of a baro clinic spectral LAM, now in progress.