Unsteady exact solutions of the flow equations for three-dimensional spherical atmospheres

Time-dependent, closed-form solutions of the 3D Euler equations describing motion relative to a uniformly rotating coordinate frame are derived. The spherical geopotential approximation is applied but not the shallow-atmosphere and hydrostatic approximations. The solutions correspond to cyclostrophically and hydrostatically balanced vortices that are steady in inertial space and whose symmetry axes do not coincide with the rotation axis of the coordinate frame. The inertial-frame flow velocities are readily transformed to a precisely spherical rotating coordinate system in which the 3D Euler equations contain centrifugal as well as Coriolis terms. In this form the solutions may be used to test numerical models formulated in spherical coordinates under the spherical geopotential approximation, so long as the centrifugal terms are explicitly included as forcing terms. The development is repeated for the hydrostatic and non-hydrostatic primitive equations (with the shallow-atmosphere approximation) and for the shallow-water equations. In the latter case, the required explicit centrifugal force may be provided by a zonally symmetric addition to the free surface height, with an identically equal orographic elevation to ensure conservation of mass. The solutions are then identical to the unsteady shallow-water solutions of Lauter et al. that inspired this study. (C) Crown Copyright 2008. Reproduced with the permission of the Controller of HMSO. Published by John Wiley & Sons, Ltd.