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

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Abstract

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.

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Acknowledgements

The author thanks a referee for some instructive comments and suggestions, which have enabled this work to be placed within a slightly more general framework, connecting it to other properties of Ekman-type flows.

Funding

The author is solely responsible for the ideas and material presented here; there was no direct financial support for this research, although it was initiated during a visit to the Faculty of Mathematics, Vienna University, organised and supported by Prof. Adrian Constantin.

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  1. School of Mathematics, Statistics and Physics, Newcastle University, Newcastle, NE1 7RU, UK

    R. S. Johnson

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  1. R. S. Johnson
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Correspondence to R. S. Johnson.

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Communicated by Michael Kunzinger.

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Johnson, R.S. A theory for oceanic gyres based on Ekman flows using the thin-shell approximation with weak nonlinearity. Monatsh Math 202, 807–830 (2023). https://doi.org/10.1007/s00605-023-01826-1

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