Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-17T14:16:38.303Z Has data issue: false hasContentIssue false
  • English
  • Français

Long-time dynamics of internal wave streaming

Published online by Cambridge University Press:  17 November 2020

Timothée Jamin Open the ORCID record for Timothée Jamin [Opens in a new window] ,
Takeshi Kataoka Open the ORCID record for Takeshi Kataoka [Opens in a new window] ,
Thierry Dauxois Open the ORCID record for Thierry Dauxois [Opens in a new window]  and
T. R. Akylas Open the ORCID record for T. R. Akylas [Opens in a new window]
Timothée Jamin
Affiliation:
Laboratoire de Physique, Univ Lyon, ENS de Lyon, Univ Claude Bernard, CNRS, Lyon, France
Takeshi Kataoka
Affiliation:
Department of Mechanical Engineering, Graduate School of Engineering, Kobe University, Rokkodai, Nada, Kobe657-8501, Japan
Thierry Dauxois
Affiliation:
Laboratoire de Physique, Univ Lyon, ENS de Lyon, Univ Claude Bernard, CNRS, Lyon, France
T. R. Akylas*
Affiliation:
Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA02139, USA
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The mean flow induced by a three-dimensional propagating internal gravity wave beam in a uniformly stratified fluid is studied experimentally and theoretically. Previous related work concentrated on the early stage of mean-flow generation, dominated by the phenomenon of streaming – a horizontal mean flow that grows linearly in time – due to resonant production of mean potential vorticity in the vicinity of the beam. The focus here, by contrast, is on the long-time mean-flow evolution. Experimental observations in a stratified fluid tank for times up to $t=120T_0$, where $T_0$ is the beam period, reveal that the induced mean flow undergoes three distinct stages: (i) resonant growth of streaming in the beam vicinity; (ii) saturation of streaming and onset of horizontal advection; and (iii) establishment of a quasi-steady state where the mean flow is highly elongated and stretches in the along-tank horizontal direction. To capture (i)–(iii), the theoretical model of Fan et al. (J. Fluid Mech., vol. 838, 2018, R1) is extended by accounting for the effects of horizontal advection and viscous diffusion of mean potential vorticity. The predictions of the proposed model, over the entire mean-flow evolution, are in excellent agreement with the experimental observations as well as numerical simulations based on the full Navier–Stokes equations.

JFM classification

Geophysical and Geological Flows: Internal waves Geophysical and Geological Flows: Stratified flows
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 907 , 25 January 2021 , A2
Check for updates
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.)

References

REFERENCES

Benney, D. J. & Roskes, G. J. 1969 Wave instabilities. Stud. Appl. Maths 48, 377–385.CrossRefGoogle Scholar
Bordes, G. 2012 Interactions non-linéaires d'ondes et tourbillons en milieu stratifié ou tournant. PhD thesis, Ecole normale supérieure de Lyon.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
Bretherton, F. P. 1969 On the mean motion induced by internal gravity waves. J. Fluid Mech. 36, 785–803.CrossRefGoogle Scholar
Bühler, O. 2000 On the vorticity transport due to dissipating or breaking waves in shallow-water flow. J. Fluid Mech. 407, 235–263.CrossRefGoogle Scholar
Bühler, O. 2014 Waves and Mean Flows. Cambridge University Press.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
Fan, B. 2017 On three-dimensional internal wavepackets, beams, and mean flows in a stratified fluid. Master's thesis, Massachusetts Institute of Technology.Google 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
Fincham, A. & Delerce, G. 2000 Advanced optimization of correlation imaging velocimetry algorithms. Exp. Fluids 42, S13–S22.CrossRefGoogle Scholar
Fovell, R., Durran, D. & Holton, J. R. 1992 Numerical simulations of convectively generated stratospheric gravity waves. J. Atmos. Sci. 49, 1427–1442.2.0.CO;2>CrossRefGoogle Scholar
Grimshaw, R. 1979 Mean flows induced by internal gravity wave packets propagating in a shear flow. Phil. Trans. R. Soc. Lond. A 292, 391–417.Google Scholar
Grimshaw, R. H. J. 1977 The modulation of an internal gravity-wave packet, and the resonance with the mean motion. Stud. Appl. Maths 56, 241–266.CrossRefGoogle Scholar
Grisouard, N. & Bühler, O. 2012 Forcing of oceanic mean flows by dissipating internal tides. J. Fluid Mech. 708, 250–278.CrossRefGoogle Scholar
Johnston, T. M. S., Rudnick, D. L., Carter, G. S., Todd, R. E. & Cole, S. T. 2011 Internal tidal beams and mixing near monterey bay. J. Geophys. Res. 116, C03017.CrossRefGoogle Scholar
Kataoka, T. & Akylas, T. R. 2013 Stability of internal gravity wave beams to three-dimensional modulations. J. Fluid Mech. 736, 67–90.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, 621–634.CrossRefGoogle Scholar
Kataoka, T. & Akylas, T. R. 2016 Three-dimensional instability of internal gravity wave beams. In Proceedings of the VIII International Symposium on Stratified Flows, San Diego, 29 August–1 September 2016. University of California, San Diego.Google Scholar
Kataoka, T. & Akylas, T. R. 2020 Viscous reflection of internal waves from a slope. Phys. Rev. Fluids 5, 014803.CrossRefGoogle Scholar
Lamb, K. G. 2004 Nonlinear interaction among internal wave beams generated by tidal flow over supercritical topography. Geophys. Res. Lett. 31, L09313.CrossRefGoogle Scholar
Lighthill, M. J. 1978 Waves in Fluids. Cambridge University Press.Google Scholar
McIntyre, M. E. & Norton, W. A. 1990 Dissipative wave-mean interactions and the transport of vorticity or potential vorticity. J. Fluid Mech. 212, 403–435.CrossRefGoogle Scholar
Mercier, M. J., Martinand, D., Mathur, M., Gostiaux, L., Peacock, T. & Dauxois, T. 2010 New wave generation. J. Fluid Mech. 657, 308–334.CrossRefGoogle Scholar
Peacock, T., Echeverri, P. & Balmforth, N. J. 2008 An experimental investigation of internal tide generation by two-dimensional topography. J. Phys. Oceanogr. 38, 235–242.CrossRefGoogle Scholar
Shrira, V. I. 1981 On the propagation of a three-dimensional packet of weakly non-linear internal gravity waves. Intl J. Non-Linear Mech. 16, 129–138.CrossRefGoogle Scholar
Tabaei, A. & Akylas, T. R. 2003 Nonlinear internal gravity wave beams. J. Fluid Mech. 482, 141–161.CrossRefGoogle Scholar
Tabaei, A. & Akylas, T. R. 2007 Resonant long–short wave interactions in an unbounded rotating stratified fluid. Stud. Appl. Maths 119, 271–296.CrossRefGoogle Scholar