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Reflections and focusing of inertial waves in a librating cube with the rotation axis oblique to its faces

Published online by Cambridge University Press:  27 May 2020

Ke Wu Open the ORCID record for Ke Wu [Opens in a new window] ,
Bruno D. Welfert Open the ORCID record for Bruno D. Welfert [Opens in a new window]  and
Juan M. Lopez Open the ORCID record for Juan M. Lopez [Opens in a new window]
Ke Wu
Affiliation:
School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287, USA
Bruno D. Welfert
Affiliation:
School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287, USA
Juan M. Lopez*
Affiliation:
School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287, USA
*
†Email address for correspondence: [email protected]
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Abstract

The response to librational forcing of a cube in rapid rotation about a diagonal axis is explored. In this orientation, the faces of the cube are all oblique to the rotation axis. The system supports inertial waves, which predominantly comprise beams emitted from the edges and vertices of the cube. Which ones emit and the resulting complicated pattern of three-dimensional reflections and subsequent focusing depend on the libration frequency. Direct numerical simulations of the Navier–Stokes flows with no-slip boundary conditions at low Ekman number ($\text{E}=10^{-7}$) and small libration amplitude ($\unicode[STIX]{x1D716}=10^{-7}$) exhibit complicated spatio-temporal structure that is remarkably well described by considerations of the inviscid reflections of wavebeams over the whole range of libration frequencies from zero to twice the mean rotation rate of the cube.

JFM classification

Geophysical and Geological Flows: Rotating flows
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 896 , 10 August 2020 , A5
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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Footnotes

‡

Present address: Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA

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Wu et al. supplementary movie 1

Surface enstrophy over the range of half-frequency indicated from EBA and DNS; the DNS results are at phase 0 of the librational forcing.
Download Wu et al. supplementary movie 1(Video)
Video 4.5 MB

Wu et al. supplementary movie 2

Animations over one libration period of the enstrophy from DNS in three internal planes and on the surface at selected half-frequencies.

Download Wu et al. supplementary movie 2(Video)
Video 3.2 MB

Wu et al. supplementary movie 3

Enstrophy over the range of half-frequency indicated from EBA and DNS in three internal planes; the DNS results are at phase 0 of the librational forcing.

Download Wu et al. supplementary movie 3(Video)
Video 7.1 MB