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Stress-driven spin-down of a viscous fluid within a spherical shell

Published online by Cambridge University Press:  13 October 2020

D. Gagnier Open the ORCID record for D. Gagnier [Opens in a new window]  and
M. Rieutord Open the ORCID record for M. Rieutord [Opens in a new window]
D. Gagnier*
Affiliation:
IRAP, Université de Toulouse, CNRS, UPS, CNES, 14, avenue Édouard Belin, F-31400Toulouse, France Department of Astronomy, University of Geneva, Chemin des Maillettes 51, 1290Versoix, Switzerland
M. Rieutord
Affiliation:
IRAP, Université de Toulouse, CNRS, UPS, CNES, 14, avenue Édouard Belin, F-31400Toulouse, France
*
†Email address for correspondence: dgagnier@irap.omp.eu
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Abstract

We investigate the linear properties of the steady and axisymmetric stress-driven spin-down flow of a viscous fluid inside a spherical shell, both within the incompressible and anelastic approximations, and in the asymptotic limit of small viscosities. From boundary layer analysis, we derive an analytical geostrophic solution for the three-dimensional incompressible steady flow, inside and outside the cylinder $\mathcal {C}$ that is tangent to the inner shell. The Stewartson layer that lies on $\mathcal {C}$ is composed of two nested shear layers of thickness $O(E^{2/7})$ and $O(E^{1/3})$ where E is the Ekman number. We derive the lowest-order solution for the $E^{2/7}$-layer. A simple analysis of the $E^{1/3}$-layer lying along the tangent cylinder, reveals it to be the site of an upwelling flow of amplitude $O(E^{1/3})$. Despite its narrowness, this shear layer concentrates most of the global meridional kinetic energy of the spin-down flow. Furthermore, a stable stratification does not perturb the spin-down flow provided the Prandtl number is small enough. If this is not the case, the Stewartson layer disappears and meridional circulation is confined within the thermal layers. The scalings for the amplitude of the anelastic secondary flow have been found to be the same as for the incompressible flow in all three regions, at the lowest order. However, because the velocity no longer conforms the Taylor–Proudman theorem, its shape differs outside the tangent cylinder $\mathcal {C}$, that is, where differential rotation takes place. Finally, we find the settling of the steady state to be reached on a viscous time for the weakly, strongly and thermally unstratified incompressible flows. Large density variations relevant to astro- and geophysical systems, tend to slightly shorten the transient.

JFM classification

Geophysical and Geological Flows: Rotating flows Boundary Layers: Free shear layers
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 904 , 10 December 2020 , A35
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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References

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