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Inertial waves in a differentially rotating spherical shell

Published online by Cambridge University Press:  19 February 2013

C. Baruteau  and
M. Rieutord
C. Baruteau*
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK
M. Rieutord
Affiliation:
Institut de Recherche en Astrophysique et Planétologie, CNRS et Université de Toulouse, 14 avenue E. Belin, 31400 Toulouse, France
*
†Email address for correspondence: C.Baruteau@damtp.cam.ac.uk
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Abstract

We investigate the properties of small-amplitude inertial waves propagating in a differentially rotating incompressible fluid contained in a spherical shell. For cylindrical and shellular rotation profiles and in the inviscid limit, inertial waves obey a second-order partial differential equation of mixed type. Two kinds of inertial modes therefore exist, depending on whether the hyperbolic domain where characteristics propagate covers the whole shell or not. The occurrence of these two kinds of inertial modes is examined, and we show that the range of frequencies at which inertial waves may propagate is broader than with solid-body rotation. Using high-resolution calculations based on a spectral method, we show that, as with solid-body rotation, singular modes with thin shear layers following short-period attractors still exist with differential rotation. They exist even in the case of a full sphere. In the limit of vanishing viscosities, the width of the shear layers seems to weakly depend on the global background shear, showing a scaling in ${E}^{1/ 3} $ with the Ekman number $E$, as in the solid-body rotation case. There also exist modes with thin detached layers of width scaling with ${E}^{1/ 2} $ as Ekman boundary layers. The behaviour of inertial waves with a corotation resonance within the shell is also considered. For cylindrical rotation, waves get dramatically absorbed at corotation. In contrast, for shellular rotation, waves may cross a critical layer without visible absorption, and such modes can be unstable for small enough Ekman numbers.

JFM classification

Geophysical and Geological Flows: Geophysical and Geological Flows Geophysical and Geological Flows: Rotating flows Geophysical and Geological Flows: Waves in rotating fluids
Type
Papers
Information
Journal of Fluid Mechanics , Volume 719 , 25 March 2013 , pp. 47 - 81
Copyright
©2013 Cambridge University Press

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