Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2025-01-04T05:25:03.587Z Has data issue: false hasContentIssue false
  • English
  • Français

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]
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

‡

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

References

Beardsley, R. C. 1970 An experimental study of inertial waves in a closed cone. Stud. Appl. Maths 49, 187–196.CrossRefGoogle Scholar
Boisson, J., Lamriben, C., Maas, L. R. M., Cortet, P. P. & Moisy, F. 2012 Inertial waves and modes excited by the libration of a rotating cube. Phys. Fluids 24, 076602.CrossRefGoogle Scholar
Drijfhout, S. & Maas, L. R. M. 2007 Impact of channel geometry and rotation on the trapping of internal tides. J. Phys. Oceanogr. 37, 2740–2763.CrossRefGoogle Scholar
Eriksen, C. C. 1982 Observations of internal wave reflection off sloping bottoms. J.  Geophys. Res. 87, 525–538.CrossRefGoogle Scholar
Greenspan, H. P. 1968 The Theory of Rotating Fluids. Cambridge University Press.Google Scholar
Greenspan, H. P. 1969 On inviscid theory of rotating fluids. Stud. Appl. Maths 48, 19–28.CrossRefGoogle Scholar
Gutierrez-Castillo, P. & Lopez, J. M. 2017 Differentially rotating split-cylinder flow: responses to weak harmonic forcing in the rapid rotation regime. Phys. Rev. Fluids 2, 084802.CrossRefGoogle Scholar
Henderson, G. A. & Aldridge, K. D. 1992 A finite-element method for inertial waves in a frustum. J. Fluid Mech. 234, 317–327.CrossRefGoogle Scholar
Kelvin, Lord 1880 Vibrations of a columnar vortex. Phil. Mag. 10, 155–168.Google Scholar
Klein, M., Seelig, T., Kurgansky, M. V., Ghasemi, V. A., Borcia, I. D., Will, A., Schaller, E., Egbers, C. & Harlander, U. 2014 Inertial wave excitation and focusing in a liquid bounded by a frustum and a cylinder. J. Fluid Mech. 751, 255–297.CrossRefGoogle Scholar
Le Bars, M., Cebron, D. & Le Gal, P. 2015 Flows driven by libration, precession, and tides. Annu. Rev. Fluid Mech. 47, 163–193.CrossRefGoogle Scholar
Lopez, J. M. & Gutierrez-Castillo, P. 2016 Three-dimensional instabilities and inertial waves in a rapidly rotating split-cylinder flow. J. Fluid Mech. 800, 666–687.CrossRefGoogle Scholar
Lopez, J. M. & Marques, F. 2014 Rapidly rotating cylinder flow with an oscillating sidewall. Phys. Rev. E 89, 013013.Google ScholarPubMed
Maas, L. R. M. 2003 On the amphidromic structure of inertial waves in a rectangular parallelepiped. Fluid Dyn. Res. 33, 373–401.CrossRefGoogle Scholar
Maas, L. R. M. 2005 Wave attractors: linear yet nonlinear. Int. J. Bif. Chaos 15, 2757–2782.CrossRefGoogle Scholar
Manders, A. M. M. & Maas, L. R. 2004 On the three-dimensional structure of the inertial wave field in a rectangular basin with one sloping boundary. Fluid Dyn. Res. 35, 1–21.CrossRefGoogle Scholar
Phillips, O. M. 1963 Energy transfer in rotating fluids by reflection of inertial waves. Phys. Fluids 6, 513–520.CrossRefGoogle Scholar
Pillet, G., Maas, L. R. M. & Dauxois, T. 2019 Internal wave attractors in 3D geometries: a dynamical systems approach. Eur. J. Mech. (B/Fluids) 77, 1–16.CrossRefGoogle Scholar
Rabitti, A. & Maas, L. R. M. 2013 Meridional trapping and zonal propagation of inertial waves in a rotating fluid shell. J. Fluid Mech. 729, 445–470.CrossRefGoogle Scholar
Rabitti, A. & Maas, L. R. M. 2014 Inerial wave rays in rotating spherical fluid domains. J. Fluid Mech. 758, 621–654.CrossRefGoogle Scholar
Rubio, A., Lopez, J. M. & Marques, F. 2008 Modulated rotating convection: radially traveling concentric rolls. J. Fluid Mech. 608, 357–378.CrossRefGoogle Scholar
Rubio, A., Lopez, J. M. & Marques, F. 2009 Interacting oscillatory boundary layers and wall modes in modulated rotating convection. J. Fluid Mech. 625, 75–96.CrossRefGoogle Scholar
Wu, K., Welfert, B. D. & Lopez, J. M. 2018 Librational forcing of a rapidly rotating fluid-filled cube. J. Fluid Mech. 842, 469–494.CrossRefGoogle Scholar
Wu, K., Welfert, B. D. & Lopez, J. M. 2020 Precessing cube: resonant excitation of modes and triadic resonance. J. Fluid Mech. 887, A6.CrossRefGoogle Scholar

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