Livermore, PW and Hollerbach, R (2012) Successive elimination of shear layers by a hierarchy of constraints in inviscid spherical-shell flows. Journal of Mathematical Physics, 53 (7). 073104. ISSN 0022-2488
Abstract
In a rotating spherical shell, an inviscid inertia-free flow driven by an arbitrary body force will have cylindrical components that are either discontinuous across, or singular on, the tangent cylinder, the cylinder tangent to the inner core and parallel to the rotation axis. We investigate this problem analytically, and show that there is an infinite hierarchy of constraints on this body force which, if satisfied, sequentially remove discontinuities or singularities in flow derivatives of progressively higher order. By splitting the solution into its equatorial symmetry classes, we are able to provide analytic expressions for the constraints and demonstrate certain inter-relations between them. We show numerically that viscosity smoothes any singularity in the azimuthal flow component into a shear layer, comprising inner and outer layers, either side of the tangent cylinder, of width O(E2/7) and O(E1/4), respectively, where E is the Ekman number. The shear appears to scale as O(E-1/3) in the equatorially symmetric case, although in a more complex fashion when considering equatorial antisymmetry, and attains a maximum value in either the inner or outer sublayers depending on equatorial symmetry. In the low-viscosity magnetohydrodynamic system of the Earth's core, magnetic tension within the fluid resists discontinuities in the flow and may dynamically adjust the body force in order that a moderate number of the constraints are satisfied. We speculate that it is violations of these constraints that excites torsional oscillations, magnetohydrodynamic waves that are observed to emanate from the tangent cylinder.
Metadata
Item Type: | Article |
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Authors/Creators: |
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Copyright, Publisher and Additional Information: | © 2012, American Institute of Physics. Reproduced in accordance with the publisher's self-archiving policy. |
Dates: |
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Institution: | The University of Leeds |
Academic Units: | The University of Leeds > Faculty of Environment (Leeds) > School of Earth and Environment (Leeds) The University of Leeds > Faculty of Engineering & Physical Sciences (Leeds) > School of Mathematics (Leeds) > Applied Mathematics (Leeds) |
Depositing User: | Symplectic Publications |
Date Deposited: | 04 Oct 2013 12:35 |
Last Modified: | 15 Jan 2018 20:44 |
Published Version: | http://dx.doi.org/10.1063/1.4736990 |
Status: | Published |
Publisher: | American Institute of Physics |
Identification Number: | 10.1063/1.4736990 |
Related URLs: | |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:76564 |