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Lookup NU author(s): Emeritus Professor Robin Johnson
This work is licensed under a Creative Commons Attribution 4.0 International License (CC BY 4.0).
© The Author(s) 2026. Starting from the general equations (in a rotating frame) for a compressible, viscous fluid, coupled to an equation of state and the first law of thermodynamics, we present a derivation of a general set of governing equations based solely on the thin-shell approximation. This uses a single small parameter (ε), measuring the thinness of the shell, keeping all other parameters fixed as ε→0. The resulting equations retain the essentials of the spherical geometry, but are inviscid (at leading order) and allow an arbitrary variation of density with height. This system is rewritten, producing a form that is suitable for seeking solutions which can be accessed (in principle) without further approximation. Arguments are presented which show that a solution exists which recovers the structure and properties of the hexagon that surrounds Saturn’s northern pole. The elucidation of some of the details requires the use of a numerical approach, because the resulting system of equations does not possess a suitable solution expressible in closed form. The hexagon structure is obtained, as well as a representation of the random-looking flows that sit outside the hexagon. The associated properties of the jet stream within the hexagon, and its temperature field, are described, including some requirements that the internal heat source must satisfy in order to maintain the hexagonal structure. All these results are compared with the available data, demonstrating an encouraging level of agreement. A critique of this work, and its shortcomings, are discussed, together with suggestions for future study.
Author(s): Johnson RS
Publication type: Article
Publication status: Published
Journal: Monatshefte fur Mathematik
Year: 2026
Pages: Epub ahead of print
Online publication date: 05/06/2026
Acceptance date: 03/03/2026
Date deposited: 22/06/2026
ISSN (print): 0026-9255
ISSN (electronic): 1436-5081
Publisher: Springer Nature
URL: https://doi.org/10.1007/s00605-026-02181-7
DOI: 10.1007/s00605-026-02181-7
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