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Harmonics from a magic carpet

Published online by Cambridge University Press:  28 January 2021

Thomas E. Dobra Open the ORCID record for Thomas E. Dobra [Opens in a new window] ,
Andrew G. W. Lawrie Open the ORCID record for Andrew G. W. Lawrie [Opens in a new window]  and
Stuart B. Dalziel Open the ORCID record for Stuart B. Dalziel [Opens in a new window]
Thomas E. Dobra
Affiliation:
Hele-Shaw Laboratory, University of Bristol, University Walk, BristolBS8 1TR, UK Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, CambridgeCB3 0WA, UK
Andrew G. W. Lawrie*
Affiliation:
Hele-Shaw Laboratory, University of Bristol, University Walk, BristolBS8 1TR, UK
Stuart B. Dalziel
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, CambridgeCB3 0WA, UK
*
†Email address for correspondence: [email protected]
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Abstract

We present a novel theoretical framework for the emission and absorption of two-dimensional internal waves in a density stratified medium. Our approach uses a weakly nonlinear perturbation expansion of a streamfunction field that exposes the harmonic structure emitted from a flexible boundary of infinite extent. We report the discovery of a special symmetry in polychromatic waves that share a common horizontal component of phase velocity. Under these conditions, there can be no wave–wave interactions in the domain interior, and therefore all harmonic generation is from the boundary. By activating polychromatic waves on this same flexible surface, we then consider the equivalent inverse problems of emission of a prescribed harmonic signature and absorption of wave energy from a given flow field. Specialising to monochromatic waves, to calculate the amplitudes and phases of the harmonics generated by a monochromatic boundary displacement and to find the explicit form of the absorbing boundary condition for a monochromatic internal wave, we present algorithms that refine lengthy algebraic processes down to a set of executable instructions valid for arbitrary order in the small parameter of the expansion. Finally, we compare our theoretical predictions up to third order with a sophisticated, digitally controlled experimental realisation that we call a ‘magic carpet’, and we find that harmonic analysis of the flow field convincingly supports our theory.

JFM classification

Geophysical and Geological Flows: Internal waves
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 911 , 25 March 2021 , A29
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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References

REFERENCES

Bourget, B., Dauxois, T., Joubaud, S. & Odier, P. 2013 Experimental study of parametric subharmonic instability for internal plane waves. J. Fluid Mech. 723, 1–20.CrossRefGoogle Scholar
Clark, H. A. & Sutherland, B. R. 2010 Generation, propagation, and breaking of an internal wave beam. Phys. Fluids 22 (7), 1–16.CrossRefGoogle Scholar
d'Alembert, 1747 Recherches sur la courbe que forme une corde tenduë mise en vibration. Hist. l'académie R. des Sci. belles lettres Berlin 3, 214–219.Google Scholar
Dalziel, S. B., Hughes, G. O. & Sutherland, B. R. 1998 Synthetic schlieren. In Proceedings of the 8th International Symposium on Flow Visualization (ed. G. M. Carlomagno & I. Grant), paper 062.Google Scholar
Dalziel, S. B., Hughes, G. O. & Sutherland, B. R. 2000 Whole-field density measurements by ‘synthetic schlieren’. Exp. Fluids 28 (4), 322–335.CrossRefGoogle Scholar
Dalziel, S. B., Patterson, M. D., Caulfield, C. P. & Le Brun, S. 2011 The structure of low-Froude-number lee waves over an isolated obstacle. J. Fluid Mech. 689, 3–31.CrossRefGoogle Scholar
Dalziel Research Partners 2018 DigiFlow vv3.6.0–4.2.0. http://www.dalzielresearch.com/digiflow/.Google Scholar
Dobra, T. E. 2018 Nonlinear interactions of internal gravity waves. PhD thesis, University of Bristol.Google Scholar
Dobra, T. E., Lawrie, A. G. W. & Dalziel, S. B. 2019 The magic carpet: an arbitrary spectrum wave maker for internal waves. Exp. Fluids 60 (11), 172.CrossRefGoogle Scholar
Egbert, G. D. & Ray, R. D. 2001 Estimates of $M_{2}$ tidal energy dissipation from TOPEX/Poseidon altimeter data. J. Geophys. Res. Ocean 106 (C10), 22475–22502.CrossRefGoogle Scholar
Ermanyuk, E. V. 2000 The use of impulse response functions for evaluation of added mass and damping coefficient of a circular cylinder oscillating in linearly stratified fluid. Exp. Fluids 28 (2), 152–159.CrossRefGoogle Scholar
Ermanyuk, E. V., Flór, J. B. & Voisin, B. 2011 Spatial structure of first and higher harmonic internal waves from a horizontally oscillating sphere. J. Fluid Mech. 671, 364–383.CrossRefGoogle Scholar
Ermanyuk, E. V., Shmakova, N. D. & Flór, J. B. 2017 Internal wave focusing by a horizontally oscillating torus. J. Fluid Mech. 813, 695–715.CrossRefGoogle Scholar
Fortuin, J. M. H. 1960 Theory and application of two supplementary methods of constructing density gradient columns. J. Polym. Sci. 44, 505–515.CrossRefGoogle Scholar
Gostiaux, L., Didelle, H., Mercier, S. & Dauxois, T. 2007 A novel internal waves generator. Exp. Fluids 42 (1), 123–130.CrossRefGoogle Scholar
Gradshteyn, I. S. & Ryzhik, I. M. 2014 Table of Integrals, Series, and Products, 8th edn. Academic Press.Google Scholar
van Haren, H., Maas, L. & van Aken, H. 2002 On the nature of internal wave spectra near a continental slope. Geophys. Res. Lett. 29 (12), 57.CrossRefGoogle Scholar
Hurley, D. G. 1972 A general method for solving steady-state internal gravity wave problems. J. Fluid Mech. 56 (4), 721–740.CrossRefGoogle Scholar
McEwan, A. D. 1973 Interactions between internal gravity waves and their traumatic effect on a continuous stratification. Boundary-Layer Meteorol. 5 (1–2), 159–175.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, 086601.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
Mowbray, D. E. & Rarity, B. S. H. 1967 A theoretical and experimental investigation of the phase configuration of internal waves of small amplitude in a density stratified liquid. J. Fluid Mech. 28 (1), 1–16.CrossRefGoogle Scholar
Nikurashin, M. & Ferrari, R. 2013 Overturning circulation driven by breaking internal waves in the deep ocean. Geophys. Res. Lett. 40 (12), 3133–3137.CrossRefGoogle Scholar
Oster, G. 1965 Density gradients. Sci. Am. 213 (2), 70–76.CrossRefGoogle Scholar
Pollard, R. T. 1970 On the generation by winds of inertial waves in the ocean. Deep Sea Res. Oceanogr. Abstr. 17 (4), 795–812.CrossRefGoogle Scholar
Scorer, R. S. 1949 Theory of waves in the lee of mountains. Q. J. R. Meteorol. Soc. 75 (323), 41–56.CrossRefGoogle Scholar
Smith, S. & Crockett, J. 2014 Experiments on nonlinear harmonic wave generation from colliding internal wave beams. Exp. Therm. Fluid Sci. 54, 93–101.CrossRefGoogle Scholar
Sutherland, B. R. 2010 Internal Gravity Waves. Cambridge University Press.CrossRefGoogle Scholar
Sutherland, B. R. 2016 Excitation of superharmonics by internal modes in non-uniformly stratified fluid. J. Fluid Mech. 793, 335–352.CrossRefGoogle Scholar
Sutherland, B. R., Dalziel, S. B., Hughes, G. O. & Linden, P. F. 1999 Visualization and measurement of internal waves by ‘synthetic schlieren’. Part 1. Vertically oscillating cylinder. J. Fluid Mech. 390, 93–126.CrossRefGoogle Scholar
Sveen, J. K. & Dalziel, S. B. 2005 A dynamic masking technique for combined measurements of PIV and synthetic schlieren applied to internal gravity waves. Meas. Sci. Technol. 16 (10), 1954–1960.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. & Lamb, K. G. 2005 Nonlinear effects in reflecting and colliding internal wave beams. J. Fluid Mech. 526, 217–243.CrossRefGoogle Scholar