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Energy budget in internal wave attractor experiments

Published online by Cambridge University Press:  15 October 2019

Géraldine Davis*
Affiliation:
Univ Lyon, ENS de Lyon, Univ Claude Bernard, CNRS, Laboratoire de Physique, F-69342 Lyon, France
Thierry Dauxois*
Affiliation:
Univ Lyon, ENS de Lyon, Univ Claude Bernard, CNRS, Laboratoire de Physique, F-69342 Lyon, France
Timothée Jamin*
Affiliation:
Univ Lyon, ENS de Lyon, Univ Claude Bernard, CNRS, Laboratoire de Physique, F-69342 Lyon, France
Sylvain Joubaud*
Affiliation:
Univ Lyon, ENS de Lyon, Univ Claude Bernard, CNRS, Laboratoire de Physique, F-69342 Lyon, France

Abstract

The current paper presents an experimental study of the energy budget of a two-dimensional internal wave attractor in a trapezoidal domain filled with uniformly stratified fluid. The injected energy flux and the dissipation rate are simultaneously measured from a two-dimensional, two-component, experimental velocity field. The pressure perturbation field needed to quantify the injected energy is determined from the linear inviscid theory. The dissipation rate in the bulk of the domain is directly computed from the measurements, while the energy sink occurring in the boundary layers is estimated using the theoretical expression for the velocity field in the boundary layers, derived recently by Beckebanze et al. (J. Fluid Mech., vol. 841, 2018, pp. 614–635). In the linear regime, we show that the energy budget is closed, in the steady state and also in the transient regime, by taking into account the bulk dissipation and, more importantly, the dissipation in the boundary layers, without any adjustable parameters. The dependence of the different sources on the thickness of the experimental set-up is also discussed. In the nonlinear regime, the analysis is extended by estimating the dissipation due to the secondary waves generated by triadic resonant instabilities, showing the importance of the energy transfer from large scales to small scales. The method tested here on internal wave attractors can be generalized straightforwardly to any quasi-two-dimensional stratified flow.

Type
JFM Papers
Copyright
© 2019 Cambridge University Press 

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References

Allshouse, M. R., Lee, F. M., Morrison, P. J. & Swinney, H. L. 2016 Internal wave pressure, velocity, and energy flux from density perturbations. Phys. Rev. Fluids 1, 014301.Google Scholar
André, Q. S., Mathis, S. & Barker, A. J. 2019 Layered semi-convection and tides in giant planet interiors. II. Tidal dissipation. Astron. Astrophys. 626, A82.Google Scholar
Beckebanze, F., Brouzet, C., Sibgatullin, I. N. & Maas, L. R. M. 2018 Damping of quasi-two-dimensional internal wave attractors by rigid-wall friction. J. Fluid Mech. 841, 614635.Google Scholar
Brouzet, C., Ermanyuk, E., Joubaud, S., Pillet, G. & Dauxois, T. 2017 Internal wave attractors: different scenarios of instability. J. Fluid Mech. 811, 544568.Google Scholar
Brouzet, C., Ermanyuk, E. V., Joubaud, S., Sibgatullin, I. N. & Dauxois, T. 2016a Energy cascade in internal wave attractors. Europhys. Lett. 113, 44001.Google 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, 109131.Google Scholar
Buhler, O. & Holmes-Cerfon, M. 2011 Decay of an internal tide due to random topography in the ocean. J. Fluid Mech. 678, 271293.Google Scholar
Campagne, A., Roumaissa, H., Redor, I., Sommeria, J., Valran, T., Viboud, S. & Mordant, N. 2018 Impact of dissipation on the energy spectrum of experimental turbulence of gravity surface waves. Phys. Rev. Fluids 3, 044801.Google Scholar
Clark, H. A. & Sutherland, B. R. 2010 Generation, propagation, and breaking of an internal wave beam. Phys. Fluids 22, 076601.Google Scholar
Dauxois, T., Joubaud, S., Odier, P. & Venaille, A. 2018 Instabilities of internal gravity wave beams. Annu. Rev. Fluid Mech. 50, 131156.Google Scholar
Dauxois, T. & Young, W. R. 1999 Near-critical refection of internal waves. J. Fluid Mech. 390, 271295.Google Scholar
Fincham, A. & Delerce, G. 2000 Advanced optimization of correlation imaging velocimetry algorithms. Exp. Fluids. 29 (S), S13S22.Google Scholar
Grisouard, N., Staquet, C. & Pairaud, I. 2008 Numerical simulation of a two-dimensional internal wave attractor. J. Fluid Mech. 614, 114.Google Scholar
van Haren, H. & Gostiaux, L. 2012 Detailed internal wave mixing observed above a deep-ocean slope. J. Mar. Res. 70, 173197.Google Scholar
Hazewinkel, J., van Breevoort, P., Dalziel, S. & Maas, L. R. M. 2008 Observations on the wavenumber spectrum and evolution of an internal wave attractor. J. Fluid Mech. 598, 373382.Google Scholar
Horne, E., Beckebanze, F., Micard, D., Odier, P., Maas, L. R. M. & Joubaud, S. 2019 Particle transport induced by internal wave beam streaming in lateral boundary layers. J. Fluid Mech. 870, 848869.Google Scholar
Ivey, G., Winters, K. & Koseff, J. 2008 Density stratification, turbulence, but how much mixing? Annu. Rev. Fluid Mech. 40 (1), 169184.Google Scholar
Jouve, L. & Ogilvie, G. I. 2014 Direct numerical simulations of an inertial wave attractor in linear and nonlinear regimes. J. Fluid Mech. 745, 223250.Google Scholar
Kundu, P. K. 1990 Fluid Mechanics. Academic Press.Google Scholar
Lamb, K. G. 2004 Nonlinear interaction among internal wave beams generated by tidal flow over supercritical topography. Geophys. Res. Lett. 31, L09313.Google Scholar
Lee, F. M., Paoletti, M. S., Swinney, H. L. & Morrison, P. J. 2014 Experimental determination of radiated internal wave power without pressure field data. Phys. Fluids 26, 046606.Google Scholar
Lee, F. M., Allshouse, M. R., Swinney, H. L. & Morrison, P. J. 2018 Internal wave energy flux from density perturbations in nonlinear stratifications. J. Fluid Mech. 856, 898920.Google Scholar
Maas, L. R. M. 2005 Wave attractors: linear yet nonlinear. Intl J. Bifurcation Chaos 15 (9), 27572782.Google Scholar
Maas, L. R. M. & Lam, F. P. A. 1995 Geometric focusing of internal waves. J. Fluid Mech. 300, 141.Google Scholar
Maas, L. R. M., Benielli, D., Sommeria, J. & Lam, F. P. A. 1997 Observations of an internal wave attractor in a confined stably stratified fluid. Nature 388, 557561.Google Scholar
Mathur, M. & Peacock, T. 2009 Internal wave beam propagation in non-uniform stratifications. J. Fluid Mech. 639, 133152.Google Scholar
Mercier, M. J., Garnier, N. B. & Dauxois, T. 2008 Refection and diffraction of internal waves analysed with the Hilbert transform. Phys. Fluids 20 (8), 086601.Google Scholar
Mercier, M. J., Martinand, D., Mathur, M., Gostiaux, L., Peacock, T. & Dauxois, T. 2010 New wave generation. J. Fluid Mech. 657, 308334.Google Scholar
Munk, W. 1966 Abyssal recipes. Deep-Sea Res. I 13, 707730.Google Scholar
Munk, W. & Wunsch, C. 1998 Abyssal recipes II: energetics of tidal and wind mixing. Deep-Sea Res. I 45 (12), 19772010.Google Scholar
Nash, J. D., Alford, M. H. & Kunze, E. 2005 Estimating internal wave energy fluxes in the ocean. J. Atmos. Ocean. Technol. 22, 15511570.Google Scholar
Nikurashin, M. & Ferrari, R. 2010 Radiation and dissipation of internal waves generated by geostrophic motions impinging on small-scale topography: application to the Southern Ocean. J. Phys. Oceanogr. 40 (9), 20252042.Google Scholar
Ogilvie, G. I. 2005 Wave attractors and the asymptotic dissipation rate of tidal disturbances. J. Fluid Mech. 543, 1944.Google Scholar
Passaggia, P.-Y., Meunier, P. & Le Dizès, S. 2014 Response of a stratified boundary layer on a tilted wall to surface undulations. J. Fluid Mech. 751, 663684.Google Scholar
Pillet, G., Ermanyuk, E. V., Maas, L., Sibgatullin, I. N. & Dauxois, T. 2018 Internal wave attractors in three-dimensional geometries: trapping by oblique reflection. J. Fluid Mech. 845, 203225.Google Scholar
Rieutord, M., Georgeot, B. & Valdettaro, L 2001 Inertial waves in a rotating spherical shell: attractors and asymptotic spectrum. J. Fluid Mech. 435, 103144.Google Scholar
Rapaka, N. R., Gayen, B. & Sarkar, S. 2013 Tidal conversion and turbulence at a model ridge: direct and large eddy simulations. J. Fluid Mech. 715, 181209.Google Scholar
Renaud, A. & Venaille, A. 2019 Boundary streaming by internal waves. J. Fluid Mech. 858, 7190.Google Scholar
Scolan, H., Ermanyuk, E. & Dauxois, T. 2013 Nonlinear fate of internal waves attractors. Phys. Rev. Lett. 110, 234501.Google Scholar
Sutherland, B. R., Achatz, U., Caulfield, C. P. & Klymak, J. M. 2019 Recent progress in modeling imbalance in the atmosphere and ocean. Phys. Rev. Fluids 4, 010501.Google Scholar
Thomas, N. H. & Stevenson, T. N. 1973 An internal wave in a viscous stratified by both salt and heat. J. Fluid Mech. 61, 301304.Google Scholar