A Comparison of Spatially Averaged Eulerian and Semi-Lagrangian Treatments of Mountains

Abstract

It has been demonstrated previously by both analysis and numerical integration that there is a serious problem incorporating orographic forcing into semi-implicit semi-Lagrangian models, since spurious resonance can develop in mountainous regions for Courant numbers larger than unity. Rivest et al. recommended using a second-order instead of a first-order semi-implicit off-centering to eliminate the spurious resonances, the former being more accurate. The present study shows by a linear one-dimensional analysis that a first-order semi-implicit off-centering can be used more effectively to eliminate the spurious resonances when combined with a spatially averaged Eulerian instead of a semi-Lagrangian treatment of mountains. The analysis reveals that the resonance is much less severe with the spatially averaged Eulerian treatment of mountains and, hence, can be suppressed with a weaker first-order off-centering. This combination could represent a valid alternative to second-order off-centering that needs extra time levels. The study also reveals that a serious truncation error is present in the neighborhood of the twin resonances when a semi-Lagrangian treatment of mountains is used. With the spatially averaged Eulerian treatment of the mountains the numerical solution filters the corresponding waves. These various points are illustrated with both barotropic and baroclinic semi-implicit semi-Lagrangian spectral models. An important feature of the baroclinic model formulation is the inclusion of topography in the basic-state solution that is used for the semi-implicit treatment of the gravity-wave-producing terms. In tests run from real data it appears that, in current three-time-level models, simply changing from the semi-Lagrangian to the spatially averaged Eulerian treatment of mountains is sufficient to significantly reduce the topographic resonance problem, permitting the use of larger time steps that produce acceptable time truncation errors without provoking the fictitious numerical amplification of short-scale waves.