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Axisymmetric inertial modes in a spherical shell at low Ekman numbers

Published online by Cambridge University Press:  06 April 2018

M. Rieutord Open the ORCID record for M. Rieutord [Opens in a new window]  and
L. Valdettaro
M. Rieutord*
Affiliation:
Université de Toulouse, UPS-OMP, IRAP, Toulouse, France CNRS, IRAP, 14 avenue Edouard Belin, F-31400 Toulouse, France
L. Valdettaro
Affiliation:
MOX, Dipartimento di Matematica, Politecnico di Milano, Piazza L. da Vinci, 32, 20133 Milano, Italy
*
†Email address for correspondence: mrieutord@irap.omp.eu
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Abstract

We investigate the asymptotic properties of axisymmetric inertial modes propagating in a spherical shell when viscosity tends to zero. We identify three kinds of eigenmodes whose eigenvalues follow very different laws as the Ekman number $E$ becomes very small. First are modes associated with attractors of characteristics that are made of thin shear layers closely following the periodic orbit traced by the characteristic attractor. Second are modes made of shear layers that connect the critical latitude singularities of the two hemispheres of the inner boundary of the spherical shell. Third are quasi-regular modes associated with the frequency of neutral periodic orbits of characteristics. We thoroughly analyse a subset of attractor modes for which numerical solutions point to an asymptotic law governing the eigenvalues. We show that three length scales proportional to $E^{1/6}$, $E^{1/4}$ and $E^{1/3}$ control the shape of the shear layers that are associated with these modes. These scales point out the key role of the small parameter $E^{1/12}$ in these oscillatory flows. With a simplified model of the viscous Poincaré equation, we can give an approximate analytical formula that reproduces the velocity field in such shear layers. Finally, we also present an analysis of the quasi-regular modes whose frequencies are close to $\sin (\unicode[STIX]{x03C0}/4)$ and explain why a fluid inside a spherical shell cannot respond to any periodic forcing at this frequency when viscosity vanishes.

JFM classification

Boundary Layers: Free shear layers Geophysical and Geological Flows: Rotating flows

Information

Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 844 , 10 June 2018 , pp. 597 - 634
Copyright
© 2018 Cambridge University Press 

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