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Unsteady exact solutions of the flow equations for three‐dimensional spherical atmospheres
Author(s) -
Staniforth A.,
White A. A.
Publication year - 2008
Publication title -
quarterly journal of the royal meteorological society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.744
H-Index - 143
eISSN - 1477-870X
pISSN - 0035-9009
DOI - 10.1002/qj.300
Subject(s) - euler equations , geopotential , spherical coordinate system , primitive equations , hydrostatic equilibrium , rotation (mathematics) , centrifugal force , inertial frame of reference , flow (mathematics) , shallow water equations , coordinate system , physics , volute , classical mechanics , euler angles , rotating reference frame , mathematical analysis , mathematics , mechanics , geometry , partial differential equation , geophysics , quantum mechanics , impeller , numerical partial differential equations
Abstract 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 Läuter et al. that inspired this study. ©Crown Copyright 2008. Reproduced with the permission of the Controller of HMSO and the Queen's Printer for Scotland. Published by John Wiley & Sons, Ltd.