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A three‐dimensional generalization of Eliassen's balanced vortex equations derived from Hamilton's principle
Author(s) -
Craig George C.
Publication year - 1991
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.49711749902
Subject(s) - primitive equations , geopotential , geostrophic wind , potential vorticity , classical mechanics , vorticity , rossby number , mathematical analysis , noether's theorem , mathematics , vortex , equations of motion , eulerian path , conservation law , physics , hydrostatic equilibrium , simultaneous equations , mechanics , differential equation , turbulence , lagrangian , quantum mechanics , geophysics
A new set of equations is derived for nearly circular flow in gradient balance. For precisely axisymmetric motion, the system reduces to the well‐known balanced vortex equations of Eliassen. The derivation is based on the assumption that the radial component of velocity is small in comparison to the azimuthal component. By applying this approximation to Hamilton's principle for a continuum of fluid parcels, while preserving the time and particle‐labelling symmetries of the primitive equations, it is ensured that the resulting system possesses conservation laws for energy and potential vorticity. In potential radius coordinates, the Lagrangian equations of motion take the form of the geostrophic and hydrostatic balance conditions. The system is also presented in Eulerian form, and a practical integration scheme, based on a linear elliptic equation for geopotential tendency, is described. Finally, the set is rederived by a conventional scale analysis in order to determine constraints on the diabatic forcings which are required for consistency with the approximation.