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A theory for oceanic gyres based on Ekman flows using the thin-shell approximation with weak nonlinearity

Lookup NU author(s): Professor Robin Johnson


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© 2023, The Author(s), under exclusive licence to Springer-Verlag GmbH Austria, part of Springer Nature.Starting from the general equations for a viscous, incompressible fluid, written in rotating spherical coordinates, an asymptotic theory for steady flow is developed. This uses only the thin-shell approximation, by suitably defining the variables and parameters. The result is a consistent theory which produces an Ekman-type balance, expressed in spherical coordinates, at leading order. The correction terms, which are mainly the nonlinear contribution in the equations, can be accommodated by invoking the method of multiple scales and using a strained coordinate. The resulting leading order, with slow/weak corrections, provides the basis for a study of oceanic gyres. By choosing the velocity (and noting the vorticity) at the surface, some examples are presented. Various choices are made, for closed particle paths expressed in a simple form (using a transformation based on the Mercator projection): zero velocity and vorticity at the centre and on the periphery of the gyre; non-zero speed on the periphery; finite-strength line vortex at the centre. In addition, in one case, we describe how the slow-z variation affects the solution. This treatment of the problem shows that our extended version of the Ekman balance, valid in spherical coordinates over large regions, can be used to investigate the properties of gyres. Many analytical and numerical options are available for future study.

Publication metadata

Author(s): Johnson RS

Publication type: Article

Publication status: Published

Journal: Monatshefte fur Mathematik

Year: 2023

Volume: 202

Pages: 807-830

Print publication date: 01/12/2023

Online publication date: 13/02/2023

Acceptance date: 18/01/2023

ISSN (print): 0026-9255

ISSN (electronic): 1436-5081

Publisher: Springer


DOI: 10.1007/s00605-023-01826-1


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