Hostname: page-component-76fb5796d-zzh7m Total loading time: 0 Render date: 2024-04-26T15:14:13.734Z Has data issue: false hasContentIssue false
  • English
  • Français

Damping of quasi-two-dimensional internal wave attractors by rigid-wall friction

Published online by Cambridge University Press:  26 February 2018

F. Beckebanze Open the ORCID record for F. Beckebanze [Opens in a new window] ,
C. Brouzet Open the ORCID record for C. Brouzet [Opens in a new window] ,
I. N. Sibgatullin  and
L. R. M. Maas Open the ORCID record for L. R. M. Maas [Opens in a new window]
F. Beckebanze*
Affiliation:
Mathematical Institute, Utrecht University, P.O. Box 80010, 3508 TA Utrecht, The Netherlands
C. Brouzet
Affiliation:
Univ Lyon, ENS de Lyon, Univ Claude Bernard, CNRS, Laboratoire de Physique, F-69342 Lyon, France
I. N. Sibgatullin
Affiliation:
Department of Mechanics and Mathematics, Moscow State University, Moscow 119191, Russia Institute for System Progamming, Moscow 109004, Russia Shirshov Institute of Oceanology, Moscow 117997, Russia
L. R. M. Maas
Affiliation:
Institute for Marine and Atmospheric Research Utrecht (IMAU), Utrecht University, Princetonplein 5, 3584 CC Utrecht, The Netherlands
*
†Email address for correspondence: f.beckebanze@uu.nl
Get access Rights & Permissions [Opens in a new window]

Abstract

The reflection of internal gravity waves at sloping boundaries leads to focusing or defocusing. In closed domains, focusing typically dominates and projects the wave energy onto ‘wave attractors’. For small-amplitude internal waves, the projection of energy onto higher wavenumbers by geometric focusing can be balanced by viscous dissipation at high wavenumbers. Contrary to what was previously suggested, viscous dissipation in interior shear layers may not be sufficient to explain the experiments on wave attractors in the classical quasi-two-dimensional trapezoidal laboratory set-ups. Applying standard boundary layer theory, we provide an elaborate description of the viscous dissipation in the interior shear layer, as well as at the rigid boundaries. Our analysis shows that even if the thin lateral Stokes boundary layers consist of no more than 1 % of the wall-to-wall distance, dissipation by lateral walls dominates at intermediate wave numbers. Our extended model for the spectrum of three-dimensional wave attractors in equilibrium closes the gap between observations and theory by Hazewinkel et al. (J. Fluid Mech., vol. 598, 2008, pp. 373–382).

JFM classification

Boundary Layers: Boundary layer structure Geophysical and Geological Flows: Geophysical and Geological Flows Geophysical and Geological Flows: Internal waves
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 841 , 25 April 2018 , pp. 614 - 635
Check for updates
Copyright
© 2018 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.)

References

Beckebanze, F. & Keady, G. 2016 On functional equations leading to exact solutions for standing internal waves. Wave Motion 60, 181–195.Google Scholar
Beckebanze, F. & Maas, L. R. M. 2016 Damping of 3d internal wave attractors by lateral walls. In Proc. VIIIth Int. Symp. on Stratified Flows. San Diego, 29 August–1 September 2016, pp. 1–6. University of California at San Diego.Google Scholar
Bordes, G., Venaille, A., Joubaud, S., Odier, P. & Dauxois, T. 2012 Experimental observation of a strong mean flow induced by internal gravity waves. Phys. Fluids 24, 086602.CrossRefGoogle Scholar
Brouzet, C.2016. Internal-wave attractors: from geometrical focusing to non-linear energy cascade and mixing. PhD thesis, ENS de Lyon.Google Scholar
Brouzet, C., Ermanyuk, E. V., Joubaud, S., Pillet, G. & Dauxois, T. 2017a Internal wave attractors: different scenarios of instability. J. Fluid Mech. 811, 544–568.Google Scholar
Brouzet, C., Ermanyuk, E. V., Joubaud, S., Sibgatullin, I. & Dauxois, T. 2016a Energy cascade in internal wave attractors. Europhys. Lett. 113, 44001.CrossRefGoogle Scholar
Brouzet, C., Sibgatullin, I. N., Ermanyuk, E. V., Joubaud, S. & Dauxois, T. 2017b Scale effects in internal wave attractors. Phys. Rev. Fluids 2, 114803.CrossRefGoogle Scholar
Brouzet, C., Sibgatullin, I. N., Scolan, H., Ermanyuk, E. V. & Dauxois, T. 2016b Internal wave attractors examined using laboratory experiments and 3d numerical simulations. J. Fluid Mech. 793, 109–131.CrossRefGoogle Scholar
Bukreev, I. V. 1988 Experimental investigation of the range of applicability of the solution of Stokes’s second problem. Fluid Dyn. 23, 504–509.CrossRefGoogle Scholar
Dauxois, T., Joubaud, S., Odier, P. & Venaille, A. 2018 Instabilities of internal gravity wave beams. Annu. Rev. Fluid Mech. 50, 131–156.CrossRefGoogle Scholar
Dauxois, T. & Young, W. R. 1999 Near-critical reflection of internal waves. J. Fluid Mech. 390, 271–295.CrossRefGoogle Scholar
Dintrans, B., Rieutord, M. & Valdettaro, L. 1999 Gravito-inertial waves in a rotating stratified sphere or spherical shell. J. Fluid Mech. 398, 271–297.CrossRefGoogle Scholar
Echeverri, P., Yokossi, T., Balmforth, N. J. & Peacock, T. 2011 Tidally generated internal-wave attractors between double ridges. J. Fluid Mech. 669, 354–374.Google Scholar
Gostiaux, L., Didelle, H., Mercier, S. & Dauxois, T. 2007 A novel internal waves generator. Exp. Fluids 42 (1), 123–130.Google Scholar
Grisouard, N., Leclair, M., Gostiaux, L. & Staquet, C. 2013 Large scale energy transfer from an internal gravity wave reflecting on a simple slope. Procedia IUTAM 8, 119–128.Google Scholar
Grisouard, N., Staquet, C. & Pairaud, I. 2008 Numerical simulation of a two-dimensional internal wave attractor. J. Fluid Mech. 614, 1–14.Google Scholar
Guo, Y. & Holmes-Cerfon, M. 2016 Internal wave attractors over random, small-amplitude topography. J. Fluid Mech. 787, 148–174.CrossRefGoogle Scholar
Harlander, U. 2008 Towards an analytical understanding of internal wave attractors. Adv. Geosci. 15, 3–9.CrossRefGoogle Scholar
Hazewinkel, J., van Breevoort, P., Dalziel, S. B. & Maas, L. R. M. 2008 Observations on the wave number spectrum and evolution of an internal wave attractor. J. Fluid Mech. 598, 373–382.Google Scholar
Hazewinkel, J., Maas, L. R. M. & Dalziel, S. B. 2011 Tomographic reconstruction of internal wave patterns in a paraboloid. Exp. Fluids 50, 247–258.Google Scholar
Hazewinkel, J., Tsimitri, C., Maas, L. R. M. & Dalziel, S. B. 2010 Observations on the robustness of internal wave attractors to perturbations. Phys. Fluids 22 (10), 107102.CrossRefGoogle Scholar
Jouve, L. & Ogilvie, G. I. 2014 Direct numerical simulations of an inertial wave attractor in linear and nonlinear regimes. J. Fluid Mech. 745, 223–250.Google Scholar
King, B., Zhang, H. P. & Swinney, H. L. 2010 Tidal flow over three dimensional topography generates out of forcing plane harmonics. Geophys. Res. Lett. 37, 1–5.Google Scholar
Kistovich, Y. V. & Chashechkin, Y. D. 1995a Reflection of packets of internal waves from a rigid plane in a viscous fluid. Atmos. Ocean. Phys. 30, 718–724.Google Scholar
Kistovich, Y. V. & Chashechkin, Y. D. 1995b The reflection of beams of internal gravity waves at a flat rigid surface. Z. Angew. Math. Mech. 59, 579–585.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.Google Scholar
Lamb, K. G. 2014 Internal wave breaking and dissipation mechanisms on the continental slope/shelf. Annu. Rev. Fluid Mech. 46, 231–254.Google Scholar
Maas, L. R. M. 2005 Wave attractors: linear yet nonlinear. Intl J. Bifurcation Chaos 15, 2757–2782.Google Scholar
Maas, L. R. M. 2009 Exact analytic self-similar solution of a wave attractor field. Physica D 238, 502–505.Google Scholar
Maas, L. R. M., Benielli, D., Sommeria, J. & Lam, F. P. A. 1997 Geometric focusing of internal waves. Nature 388, 557–561.CrossRefGoogle Scholar
Maas, L. R. M. & Lam, F. P. A. 1995 Geometric focusing of internal waves. J. Fluid Mech. 300, 1–41.Google Scholar
Manders, A. M. M. & Maas, L. R. M. 2003 Observations of inertial waves in a rectangular basin with one sloping boundary. J. Fluid Mech. 493, 59–88.CrossRefGoogle Scholar
Ogilvie, G. I. 2005 Wave attractors and the asymptotic dissipation rate of tidal disturbances. J. Fluid Mech. 543, 19–44.CrossRefGoogle Scholar
Rayson, M. D., Bluteau, C. E., Ivey, G. N. & Jones, N. L. 2016 Observations of high-frequency internal waves and strong turbulent mixing in a channel flow between two coral atolls. San Diego, 29 August-1 September 2016. In Proc. VIIIth Int. Symp. on Stratified Flows, University of California at San Diego.Google Scholar
Renaud, A. & Venaille, A. 2018 Boundary streaming by internal waves. J. Fluid Mech. 1–10; (in press).Google Scholar
Rieutord, M., Georgeot, B. & Valdettaro, L. 2001 Inertial waves in a rotating spherical shell: attractors and asymptotic spectrum. J. Fluid Mech. 435, 103–144.Google Scholar
Rieutord, M., Georgeot, B. & Valdettaro, L. 2002 Analysis of singular inertial modes in a spherical shell: the slender toroidal shell mode. J. Fluid Mech. 463, 345–360.Google Scholar
Schlichting, H. & Gersten, K. 2000 Boundary-Layer Theory. Springer.Google Scholar
Scolan, H., Ermanyuk, E. V. & Dauxois, T. 2013 Nonlinear fate of internal wave attractors. Phys. Rev. Lett. 110, 234501.Google Scholar
Sibgatullin, I. & Kalugin, M. 2016 High-resolution simulation of internal wave attractors and impact of interaction of high amplitude internal waves with walls on dynamics of wave attractors. In Proc. VII European Congress on Computational Methods in Applied Sciences and Engineering. Crete Island, Greece, 5–10 June 2016, pp. 1–8. National Technical University of Athens.Google Scholar
Sutherland, B. R. 2010 Internal Gravity Waves. Cambridge University Press.Google Scholar
Swart, A.2007 Internal waves and the poincaré equation. PhD thesis, Utrecht University.Google Scholar
Swart, A., Manders, A., Harlander, U. & Maas, L. R. M. 2010 Experimental observation of strong mixing due to internal wave focusing over sloping terrain. Dyn. Atmos. Oceans 50, 16–34.Google Scholar
Swart, A., Sleijpen, G. L. G., Maas, L. R. M. & Brandts, J. 2007 Numerical solution of the two-dimensional Poincaré equation. J. Comput. Appl. Maths 200, 317–341.CrossRefGoogle Scholar
Tabaei, A. & Akylas, T. R. 2003 Nonlinear internal gravity wave beams. J. Fluid Mech. 482, 141–161.Google Scholar
Tang, W. & Peacock, T. 2010 Lagrangian coherent structures and internal wave attractors. Chaos 20, 017508.CrossRefGoogle ScholarPubMed
Thomas, N. H. & Stevenson, T. N. 1973 An internal wave in a viscous ocean stratified by both salt and heat. J. Fluid Mech. 61, 301–1304.Google Scholar
Troitskaya, S. 2017 Mathematical analysis of inertial waves in rectangular basins with one sloping boundary. Stud. Appl. Maths 139 (3), 434–456.Google Scholar
Vasiliev, A. Y. & Chashechkin, Y. D. 2003 Generation of 3d periodic internal wave beams. Z. Angew. Math. Mech. 67, 397–405.CrossRefGoogle Scholar
Voisin, B. 2003 Limit states of internal wave beams. J. Fluid Mech. 496, 243–293.Google Scholar
Wang, G., Zheng, Q., Lin, M., Dai, D. & Qiao, F. 2015 Three dimensional simulation of internal wave attractors in the Luzon Strait. Acta Oceanol. Sin. 34, 14–21.Google Scholar
Wunsch, C. & Ferrari, R. 2004 Vertical mixing, energy, and the general circulation of the oceans. Annu. Rev. Fluid Mech. 36, 281–314.Google Scholar