• geodesic;
  • instability;
  • position vector;
  • Riemannian geometry;
  • rotation matrix


The widely used shallow-atmosphere approximation is a geometrical approximation in which the metric departs from the usual Euclidean metric. This leads to a number of important consequences: shallow-atmosphere space is intrinsically curved (i.e. non-Euclidean), geodesics are not unique, the status of the centre of the Earth is uncertain, and position vectors are not well-defined. Vector semi-Lagrangian numerical models that use the shallow-atmosphere approximation must allow explicitly for the non-Euclidean geometry. During early testing of a new semi-implicit, semi-Lagrangian dynamical core, a semi-implicit (Crank–Nicolson) discretization of the vector departure point equation was found to lead to an instability in deep-atmosphere (i.e. Euclidean) geometry, but not in shallow-atmosphere geometry. The instability can be avoided by an alternative treatment in which the departure point equation is projected onto its horizontal and vertical components before discretization. Interestingly, this stable treatment of the deep-atmosphere case makes use of much of the mathematical machinery of the shallow-atmosphere departure point calculation. Copyright © 2012 Royal Meteorological Society and British Crown Copyright, the Met Office