Skip to main content Accessibility help
Hostname: page-component-65dc7cd545-8rn5k Total loading time: 0.313 Render date: 2021-07-24T05:00:53.406Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": true, "newCiteModal": false, "newCitedByModal": true, "newEcommerce": true, "newUsageEvents": true }

Mean flow generation by three-dimensional nonlinear internal wave beams

Published online by Cambridge University Press:  07 February 2019

F. Beckebanze
Mathematical Institute, Utrecht University, P.O. Box 80010, 3508 TA Utrecht, The Netherlands
K. J. Raja
Laboratoire des Écoulements Géophysiques et Industriels, Université Grenoble Alpes, Grenoble, CS 40700, France
L. R. M. Maas
Institute for Marine and Atmospheric Research Utrecht (IMAU), Utrecht University, Princetonplein 5, 3584 CC Utrecht, The Netherlands
E-mail address:


We study the generation of resonantly growing mean flow by weakly nonlinear internal wave beams. With a perturbational expansion, we construct analytic solutions for three-dimensional internal wave beams, exact up to first-order accuracy in the viscosity parameter. We specifically focus on the subtleties of wave beam generation by oscillating boundaries, such as wave makers in laboratory set-ups. The exact solutions to the linearized equations allow us to derive an analytic expression for the mean vertical vorticity production term, which induces a horizontal mean flow. Whereas mean flow generation associated with viscous beam attenuation – known as streaming – has been described before, we are the first to also include a peculiar inviscid mean flow generation in the vicinity of the oscillating wall, resulting from line vortices at the lateral edges of the oscillating boundary. Our theoretical expression for the mean vertical vorticity production is in good agreement with earlier laboratory experiments, for which the previously unrecognized inviscid mean flow generation mechanism turns out to be significant.

JFM Papers
© 2019 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below.


Andrews, D. G. & McIntyre, M. E. 1978 On wave-action and its relatives. J. Fluid Mech. 89 (4), 647664.CrossRefGoogle Scholar
Bhatia, H., Norgard, G., Pascucci, V. & Bremer, P.-T. 2013 The Helmholtz–Hodge decomposition – a survey. IEEE Trans. Vis. Comput. Graphics 19 (8), 13861404.CrossRefGoogle ScholarPubMed
Binson, J. 1997 Chaotic mixing by internal inertia-gravity waves. Phys. Fluids 9 (8), 945.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 (4), 086602.CrossRefGoogle Scholar
Bretherton, F. P. 1969 On the mean motion induced by internal gravity waves. J. Fluid Mech. 36 (4), 785803.CrossRefGoogle Scholar
Brouzet, C., Sibgatullin, I. N., Scolan, H., Ermanyuk, E. V. & Dauxois, T. 2016 Internal wave attractors examined using laboratory experiments and 3D numerical simulations. J. Fluid Mech. 793, 109131.CrossRefGoogle Scholar
Bühler, O. 2010 Waves and Mean Flows. Cambridge University Press.Google Scholar
Bühler, O., Kuan, M. & Tabak, E. G. 2017 Anisotropic Helmholtz and wave–vortex decomposition of one-dimensional spectra. J. Fluid Mech. 815, 361387.CrossRefGoogle Scholar
Bühler, O. & McIntyre, M. E. 2005 Wave capture and wave–vortex duality. J. Fluid Mech. 534, 6795.CrossRefGoogle Scholar
Dauxois, T., Joubaud, S., Odier, P. & Venaille, A. 2018 Instabilities of internal gravity wave beams. Annu. Rev. Fluid Mech. 50, 128.CrossRefGoogle Scholar
Fan, B., Kataoka, T. & Akylas, T. R. 2018 On the interaction of an internal wavepacket with its induced mean flow and the role of streaming. J. Fluid Mech. 838, R1.CrossRefGoogle Scholar
Garrett, C. & Kunze, E. 2007 Internal tide generation in deep ocean. Annu. Rev. Fluid Mech. 87, 5787.CrossRefGoogle Scholar
Gostiaux, L., Didelle, H., Mercier, S. & Dauxois, T. 2007 A novel internal waves generator. Exp. Fluids 42 (1), 123130.CrossRefGoogle Scholar
Grisouard, N.2010 Réflexions et réfractions non-linéaires d’ondes de gravité internes. PhD thesis, University Grenoble-Alpes.Google Scholar
Grisouard, N. & Bühler, O. 2012 Forcing of oceanic mean flows by dissipating internal tides. J. Fluid Mech. 708, 250278.CrossRefGoogle 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, 119128.CrossRefGoogle Scholar
Hoskins, B. 1997 A potential vorticity view of synoptic development. Meteorol. Appl. 4, 325334.CrossRefGoogle Scholar
Kataoka, T. & Akylas, T. R. 2015 On three-dimensional internal gravity wave beams and induced large-scale mean flows. J. Fluid Mech. 769, 621634.CrossRefGoogle Scholar
Kataoka, T., Ghaemsaidi, S. J., Holzenberger, N., Peacock, T. & Akylas, T. R. 2017 Tilting at wave beams: a new perspective on the St. Andrew’s Cross. J. Fluid Mech. 830, 660680.CrossRefGoogle 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, 15.CrossRefGoogle Scholar
Kistovich, Y. V. & Chashechkin, Y. D. 2001 Mass transport and the force of a beam of two-dimensional periodic internal waves. Z. Angew. Math. Mech. J. Appl. Math. Mech. 65 (2), 237242.CrossRefGoogle Scholar
Krol, M. S. 1991 On the averaging method in nearly time-periodic advection–diffusion problems. SIAM J. Appl. Maths 51 (6), 16221637.CrossRefGoogle Scholar
Lamb, K. G. 2014 Internal wave breaking and dissipation mechanisms on the continental slope/shelf. Annu. Rev. Fluid Mech. 46, 231254.CrossRefGoogle Scholar
Lighthill, J. 1978 Acoustic streaming. J. Sound Vib. 61, 391418.CrossRefGoogle Scholar
Longuet-Higgins, M. S. & Stewart, R. W. 1964 Radiation stresses in water waves; a physical discussion, with applications. Deep Sea Res. 11, 529562.Google Scholar
McEwan, A. C. 1973 Interactions between internal gravity wave and their traumatic effect on a continuous stratification. Boundary-Layer Meteorol. 5, 159175.CrossRefGoogle Scholar
Mercier, M. J., Garnier, N. B. & Dauxois, T. 2008 Reflection and diffraction of internal waves analyzed with the Hilbert transform. Phys. Fluids 20 (8), 086601.CrossRefGoogle Scholar
Mercier, M. J., Martinand, D., Mathur, M., Gostiaux, L., Peacock, T. & Dauxois, T. 2010 New wave generation. J. Fluid Mech. 657, 308334.CrossRefGoogle Scholar
Ou, H. W. & Maas, L. 1986 Tidal-induced buoyancy flux and mean transverse circulation. Cont. Shelf Res. 5, 611628.CrossRefGoogle Scholar
Pillet, G., Ermanyuk, E. V., Maas, L. R. M., Sibgatullin, I. N. & Dauxois, T. 2018 Internal wave attractors in three-dimensional geometries: trapping by oblique reflection. J. Fluid Mech. 203225.CrossRefGoogle Scholar
Raja, K. J.2018 Internal waves and mean flow in the presence of topography. PhD thesis, University Grenoble-Alpes.Google Scholar
Riley, N. 2001 Steady streaming. Annu. Rev. Fluid Mech. 33, 4365.CrossRefGoogle Scholar
Schlichting, H. & Gersten, K. 2000 Boundary-Layer Theory. Springer.CrossRefGoogle Scholar
Semin, B., Facchini, G., Pétrélis, F. & Fauve, S. 2016 Generation of a mean flow by an internal wave. Phys. Fluids 28 (9), 096601.CrossRefGoogle 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, Greece, 5–10 June 2016. National Technical University of Athens.Google Scholar
Staquet, C. & Riley, J. J. 1989 On the velocity field associated with potential vorticity. Dyn. Atmos. Oceans 14, 93123.CrossRefGoogle Scholar
Staquet, C. & Sommeria, J. 2002 Internal gravity waves: from instabilities to turbulence. Annu. Rev. Fluid Mech. 34, 559593.CrossRefGoogle Scholar
Sutherland, B. R. 2006 Internal wave instability: Wave–wave versus wave-induced mean flow interactions. Phys. Fluids 18 (7), 074107.CrossRefGoogle Scholar
Sutherland, B. R. 2010 Internal Gravity Waves. Cambridge University Press.CrossRefGoogle Scholar
Tabaei, A. & Akylas, T. R. 2003 Nonlinear internal gravity wave beams. J. Fluid Mech. 482, 141161.CrossRefGoogle Scholar
Tabaei, A. & Akylas, T. R. 2007 Resonant long–short wave interactions in an unbounded rotating stratified fluid. Stud. Appl. Maths 119 (3), 271296.CrossRefGoogle Scholar
Tabaei, A., Akylas, T. R. & Lamb, K. G. 2005 Nonlinear effects in reflecting and colliding internal wave beams. J. Fluid Mech. 526, 217243.CrossRefGoogle Scholar
Thomas, N. H. & Stevenson, T. N. 1973 An internal wave in a viscous ocean stratified by both salt and heat. J. Fluid Mech. 61, 301304.CrossRefGoogle Scholar
Thorpe, S. A. 1987 On the reflection of a train of finite-amplitude internal waves from a uniform slope. J. Fluid Mech. 178, 279302.CrossRefGoogle Scholar
van den Bremer, T. S.2014 The induced mean flow of surface, internal and interfacial gravity wave groups. PhD thesis, University of Oxford.Google Scholar
van den Bremer, T. S. & Sutherland, B. R. 2018 The wave-induced flow of internal gravity wavepackets with arbitrary aspect ratio. J. Fluid Mech. 834, 385408.CrossRefGoogle Scholar
Voisin, B. 2003 Limit states of internal wave beams. J. Fluid Mech. 496, 243293.CrossRefGoogle Scholar
Wagner, G. L. & Young, W. R. 2015 Available potential vorticity and wave-averaged quasi-geostrophic flow. J. Fluid Mech. 785, 401424.CrossRefGoogle Scholar
Wunsch, C. 1971 Note on some Reynolds stress effects of internal waves on slopes. Deep-Sea Res. 18, 583591.Google Scholar
Wunsch, C. & Ferrari, R. 2004 Vertical mixing, energy, and the general circulation of the oceans. Annu. Rev. Fluid Mech. 36, 281314.CrossRefGoogle Scholar
Zhou, Q. & Diamessis, P. J. 2015 Lagrangian flows within reflecting internal waves at a horizontal free-slip surface. Phys. Fluids 27 (12), 126601.CrossRefGoogle Scholar
Cited by

Send article to Kindle

To send this article to your Kindle, first ensure is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the or variations. ‘’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Mean flow generation by three-dimensional nonlinear internal wave beams
Available formats

Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

Mean flow generation by three-dimensional nonlinear internal wave beams
Available formats

Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

Mean flow generation by three-dimensional nonlinear internal wave beams
Available formats

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *