A geometrical view of the shallow-atmosphere approximation, with application to the semi-Lagrangian departure point calculation


  • J. Thuburn,

    Corresponding author
    1. College of Engineering, Mathematics and Physical Sciences, University of Exeter, UK
    • College of Engineering, Mathematics and Physical Sciences, University of Exeter, Harrison Building, North Park Road, Exeter EX4 4QF, UK.
    Search for more papers by this author
  • A. A. White

    1. Met Office, Exeter, UK
    Search for more papers by this author
    • The contribution of this author was written in the course of his employment at the Met Office, UK, and is published with the permission of the Controller of HMSO and the Queen's Printer for Scotland.


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