Some exact solutions of geophysical fluid‐dynamics equations for testing models in spherical and plane geometry

Abstract

AbstractA methodology is presented for constructing a family of exact axially‐symmetric solutions to various geophysical fluid‐dynamics equation sets, with the aim of facilitating the development and testing of numerical models. The construction is done first for the shallow‐water equations in spherical geometry, and then used to construct solutions for the shallow‐water equations in Cartesian geometry, the vertical‐slice Euler equations on an f–F plane, and the three‐dimensional deep‐ and shallow‐atmosphere Euler equations in spherical geometry. The solutions are steady, vortical, axially symmetric, and non‐divergent. Illustrative solutions are presented for five simple model problems: an equatorial jet, twin mid‐latitude jets, an isolated polar vortex, multiple Jovian jets, and an exponentially‐decaying vortex. It is also shown how to make a non‐hydrostatic vertical‐slice Euler‐equations model on an f–F plane emulate a shallow‐water model on an f plane by suitably setting parameter values and initial conditions. Crown Copyright 2007. Reproduced with the permission of the Controller of Her Majesty's Stationery Office. Published by John Wiley & Sons, Ltd.