Rossby number expansions, slaving principles, and balance dynamics

Warn, T., Bokhove, O., Shepherd, T. G. ORCID: https://orcid.org/0000-0002-6631-9968 and Vallis, G. K. (1995) Rossby number expansions, slaving principles, and balance dynamics. Quarterly Journal of the Royal Meteorological Society, 121 (523). pp. 723-739. ISSN 1477-870X doi: 10.1002/qj.49712152313

Abstract/Summary

We consider the problem of constructing balance dynamics for rapidly rotating fluid systems. It is argued that the conventional Rossby number expansion—namely expanding all variables in a series in Rossby number—is secular for all but the simplest flows. In particular, the higher-order terms in the expansion grow exponentially on average, and for moderate values of the Rossby number the expansion is, at best, useful only for times of the order of the doubling times of the instabilities of the underlying quasi-geostrophic dynamics. Similar arguments apply in a wide class of problems involving a small parameter and sufficiently complex zeroth-order dynamics. A modified procedure is proposed which involves expanding only the fast modes of the system; this is equivalent to an asymptotic approximation of the slaving relation that relates the fast modes to the slow modes. The procedure is systematic and thus capable, at least in principle, of being carried to any order—unlike procedures based on truncations. We apply the procedure to construct higher-order balance approximations of the shallow-water equations. At the lowest order quasi-geostrophy emerges. At the next order the system incorporates gradient-wind balance, although the balance relations themselves involve only linear inversions and hence are easily applied. There is a large class of reduced systems associated with various choices for the slow variables, but the simplest ones appear to be those based on potential vorticity.

Item Type	Article
URI	https://reading-clone.eprints-hosting.org/id/eprint/32891
Identification Number/DOI	10.1002/qj.49712152313
Refereed	Yes
Divisions	No Reading authors. Back catalogue items Science > School of Mathematical, Physical and Computational Sciences > Department of Meteorology
Publisher	Royal Meteorological Society
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