Treatment of vector equations in deep-atmosphere, semi-Lagrangian models. I: Momentum equation

✍ Scribed by A. Staniforth; A. A. White; N. Wood

• Publisher: John Wiley and Sons
• Year: 2010
• Tongue: English
• Weight: 178 KB
• Volume: 136
• Category: Article
• ISSN: 0035-9009
• DOI: 10.1002/qj.562

✦ Synopsis

Abstract

Application of the semi‐Lagrangian method to the vector momentum equation in orthogonal curvilinear systems is considered, with emphasis on spherical and spheroidal coordinates for deep‐atmosphere models (in which the shallow‐atmosphere approximation is not made). In spherical coordinates, a certain rotation matrix allows vector components at semi‐Lagrangian departure points to be expressed in the required component forms at the corresponding arrival points. This rotation matrix also allows the unit vector triad at the arrival point to be expressed in terms of the unit vector triad at the departure point. The resulting formalisation of the momentum equation applies to timesteps of arbitrary length, and in the limit of infinitesimal timestep it delivers the familiar Eulerian components. Extension of the rotation matrix technique to more general orthogonal curvilinear systems is remarkably straightforward, thus demonstrating the versatility of the semi‐Lagrangian method. Explicit forms are given for systems having longitude as one coordinate and general orthogonal coordinates in the meridional plane, and in particular these forms are applicable to both confocal oblate spheroidal and similar oblate spheroidal systems. A specific factorisation of the rotation matrix in the spherical polar case involves three matrices, one of which represents rotation of the unit vector triad along a great circle arc; the non‐Euclidean, shallow‐atmosphere approximation of the total rotation matrix results when the great circle rotation matrix is replaced by the identity matrix (but the other two matrices involved in the factorisation are kept intact). This shallow‐atmosphere rotation matrix agrees with that used by ECMWF and Météo‐France. © Crown Copyright 2010. Published by John Wiley & Sons, Ltd.