Treatment of vector equations in deep‐atmosphere, semi‐Lagrangian models. II: Kinematic equation

Abstract

AbstractThe rotation matrix formalism described in Part I of this study is here applied to the deep‐atmosphere form of the vector kinematic equation and the determination of semi‐Lagrangian ‘departure points’ (locations at the current time of fluid particles that arrive at model gridpoints at the forecast time). Following a brief review of published methods for determining departure points in models that incorporate the restrictive shallow‐atmosphere approximation, a certain discrete form of the kinematic equation is used to illustrate application of the rotation matrix technique in spherical coordinates in the deep‐atmosphere case. This discrete form–referred to as ‘doubly implicit’ –involves velocities at arrival gridpoints as well as at departure points but has several advantages over alternatives that involve velocities at a point or points along the intervening trajectory. A local Cartesian transformation method for finding departure points emerges from the rotation matrix approach applied in spherical polar coordinates. In spheroidal coordinates, a local transformation method also emerges, but the relevant Cartesian origins lie along the axis of symmetry of the spheroids rather than at their common centre. Modifications for the spherical polar case under the shallow‐atmosphere approximation are obtained by a constraint method; apart from order‐unity multiplicative factors, the resulting departure‐point equations are of the form expected given the shallow‐atmosphere momentum component equations found in Part I. © Crown Copyright 2010. Published by John Wiley & Sons, Ltd.