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

A 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.