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Zonal flow in a resonant precessing cylinder

Published online by Cambridge University Press:  30 July 2021

Donglai Gao ,
Patrice Meunier Open the ORCID record for Patrice Meunier [Opens in a new window] ,
Stéphane Le Dizès Open the ORCID record for Stéphane Le Dizès [Opens in a new window]  and
Christophe Eloy Open the ORCID record for Christophe Eloy [Opens in a new window]
Donglai Gao
Affiliation:
Aix Marseille Univ, CNRS, Centrale Marseille, IRPHE, 13013Marseille, France Joint laboratory of wind tunnel and wave flume, Harbin Institute of Technology, Harbin, PR China
Patrice Meunier
Affiliation:
Aix Marseille Univ, CNRS, Centrale Marseille, IRPHE, 13013Marseille, France
Stéphane Le Dizès
Affiliation:
Aix Marseille Univ, CNRS, Centrale Marseille, IRPHE, 13013Marseille, France
Christophe Eloy*
Affiliation:
Aix Marseille Univ, CNRS, Centrale Marseille, IRPHE, 13013Marseille, France
*
†Email address for correspondence: celoy@ec-m.fr
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Abstract

A cylinder undergoes precession when it rotates around its axis and this axis itself rotates around another axis. In a precessing cylinder full of fluid, a steady and axisymmetric component of the azimuthal flow is generally present. This component is called a zonal flow. Although zonal flows have been often observed in experiments and numerical simulations, their origin has eluded theoretical approaches so far. Here, we develop an asymptotic analysis to calculate the zonal flow forced in a resonant precessing cylinder, that is when the harmonic response is dominated by a single Kelvin mode. We find that the zonal flow originates from three different sources: (1) the nonlinear interaction of the inviscid Kelvin mode with its viscous correction; (2) the steady and axisymmetric response to the nonlinear interaction of the Kelvin mode with itself; and (3) the nonlinear interactions in the end boundary layers. In a precessing cylinder, two additional sources arise due to the equatorial Coriolis force and the forced shear flow; however, they cancel exactly. The study thus generalises to any Kelvin mode, forced by precession or any other mechanism. The present theoretical predictions of the zonal flow are confirmed by comparison with numerical simulations and experimental results. We also show numerically that the zonal flow is always retrograde in a resonant precessing cylinder ($m=1$) or when it results from resonant Kelvin modes of azimuthal wavenumbers $m=2$, $3$ and, presumably, higher.

JFM classification

Geophysical and Geological Flows: Rotating flows Instability: Nonlinear instability
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 923 , 25 September 2021 , A29
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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