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Internal wave generation by oscillation of a sphere, with application to internal tides

Published online by Cambridge University Press:  25 November 2010

B. VOISIN ,
E. V. ERMANYUK  and
J.-B. FLÓR
B. VOISIN*
Affiliation:
Laboratoire des Écoulements Géophysiques et Industriels, CNRS–Université de Grenoble, BP 53, 38041 Grenoble, France
E. V. ERMANYUK
Affiliation:
Lavrentyev Institute of Hydrodynamics, Siberian Division of the Russian Academy of Science, Prospekt Lavrentyev 15, Novosibirsk 630090, Russia
J.-B. FLÓR
Affiliation:
Laboratoire des Écoulements Géophysiques et Industriels, CNRS–Université de Grenoble, BP 53, 38041 Grenoble, France
*
†Email address for correspondence: [email protected]
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Abstract

A joint theoretical and experimental study is performed on the generation of internal gravity waves by an oscillating sphere, as a paradigm of the generation of internal tides by barotropic tidal flow over three-dimensional supercritical topography. The theory is linear and three-dimensional, applies both near and far from the sphere, and takes into account viscosity and the unsteadiness arising from the interference with transients generated at the start-up. The waves propagate in conical beams, evolving with distance from a bimodal to unimodal wave profile. In the near field, the profile is asymmetric with its major peak towards the axis of the cones. The experiments involve horizontal oscillations and develop a cross-correlation technique for the measurement of the deformation of fluorescent dye planes to sub-pixel accuracy. At an oscillation amplitude of one fifth of the radius of the sphere, the waves are linear and the agreement between experiment and theory is excellent. As the amplitude increases to half the radius, nonlinear effects cause the wave amplitude to saturate at a value that is 20% lower than its linear estimate. Application of the theory to the conversion rate of barotropic tidal energy into internal tides confirms the expected scaling for flat topography, and shows its transformation for hemispherical topography. In the ocean, viscous and unsteady effects have an essentially local role, in keeping the wave amplitude finite at the edges of the beams, and otherwise dissipate energy on such large distances that they hardly induce any decay.

JFM classification

Geophysical and Geological Flows: Baroclinic flows Geophysical and Geological Flows: Internal waves Geophysical and Geological Flows: Topographic effects
Type
Papers
Information
Journal of Fluid Mechanics , Volume 666 , 10 January 2011 , pp. 308 - 357
Copyright
Copyright © Cambridge University Press 2010

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References

REFERENCES

Appleby, J. C. & Crighton, D. G. 1986 Non-Boussinesq effects in the diffraction of internal waves from an oscillating cylinder. Q. J. Mech. Appl. Math. 39, 209–231.CrossRefGoogle Scholar
Appleby, J. C. & Crighton, D. G. 1987 Internal gravity waves generated by oscillations of a sphere. J. Fluid Mech. 183, 439–450.CrossRefGoogle Scholar
Baines, P. G. & Fang, X.-H. 1985 Internal tide generation at a continental shelf/slope junction: a comparison between theory and a laboratory experiment. Dyn. Atmos. Oceans 9, 297–314.CrossRefGoogle Scholar
Balmforth, N. J. & Peacock, T. 2009 Tidal conversion by supercritical topography. J. Phys. Oceanogr. 39, 1965–1974.CrossRefGoogle Scholar
Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.Google Scholar
Bell, T. H. 1975 a Lee waves in stratified flows with simple harmonic time dependence. J. Fluid Mech. 67, 705–722.CrossRefGoogle Scholar
Bell, T. H. 1975 b Topographically generated internal waves in the open ocean. J. Geophys. Res. 80, 320–327.CrossRefGoogle Scholar
Bonnier, M., Bonneton, P. & Eiff, O. 1998 Far-wake of a sphere in a stably stratified fluid: characterization of the vortex structures. Appl. Sci. Res. 59, 269–281.CrossRefGoogle Scholar
Bühler, O. & Muller, C. J. 2007 Instability and focusing of internal tides in the deep ocean. J. Fluid Mech. 588, 1–28.CrossRefGoogle Scholar
Cerasoli, C. P. 1978 Experiments on buoyant-parcel motion and the generation of internal gravity waves. J. Fluid Mech. 86, 247–271.CrossRefGoogle Scholar
Chashechkin, Y. D. 2007 Visualization of singular components of periodic motions in a continuously stratified fluid. J. Vis. 10, 17–20.CrossRefGoogle Scholar
Chashechkin, Y. D., Kistovich, Y. V. & Smirnov, S. A. 2001 Linear generation theory of 2D and 3D periodic internal waves in a viscous stratified fluid. Environmetrics 12, 57–80.3.0.CO;2-1>CrossRefGoogle Scholar
Chashechkin, Y. D. & Prikhod'ko, Y. V. 2007 Regular and singular flow components for stimulated and free oscillations of a sphere in continuously stratified liquid. Dokl. Phys. 52, 261–265.CrossRefGoogle Scholar
Clark, H. A. & Sutherland, B. R. 2009 Schlieren measurements of internal waves in non-Boussinesq fluids. Exp. Fluids 47, 183–193.CrossRefGoogle Scholar
Dalziel, S. B., Hughes, G. O. & Sutherland, B. R. 2000 Whole-field density measurements by ‘synthetic schlieren’. Exp. Fluids 28, 322–335.CrossRefGoogle Scholar
Davis, A. M. J. & Llewellyn Smith, S. G. 2010 Tangential oscillations of a circular disk in a viscous stratified fluid. J. Fluid Mech. 656, 342–359.CrossRefGoogle Scholar
Décamp, S., Kozack, C. & Sutherland, B. R. 2008 Three-dimensional schlieren measurements using inverse tomography. Exp. Fluids 44, 747–758.CrossRefGoogle Scholar
Di Lorenzo, E., Young, W. R. & Llewellyn Smith, S. 2006 Numerical and analytical estimates of M 2 tidal conversion at steep oceanic ridges. J. Phys. Oceanogr. 36, 1072–1084.CrossRefGoogle Scholar
Echeverri, P. E, Flynn, M. R., Winters, K. B. & Peacock, T. 2009 Low-mode internal tide generation by topography: an experimental and numerical investigation. J. Fluid Mech. 636, 91–108.CrossRefGoogle Scholar
Ermanyuk, E. V. 2002 The rule of affine similitude for the force coefficients of a body oscillating in a uniformly stratified fluid. Exp. Fluids 32, 242–251.CrossRefGoogle Scholar
Ermanyuk, E. V., Flór, J.-B. & Voisin, B. 2010 Spatial structure of first and higher harmonic internal waves from a horizontally oscillating sphere. J. Fluid Mech. (submitted).CrossRefGoogle Scholar
Ermanyuk, E. V. & Gavrilov, N. V. 2002 Force on a body in a continuously stratified fluid. Part 1. Circular cylinder. J. Fluid Mech. 451, 421–443.CrossRefGoogle Scholar
Ermanyuk, E. V. & Gavrilov, N. V. 2003 Force on a body in a continuously stratified fluid. Part 2. Sphere. J. Fluid Mech. 494, 33–50.CrossRefGoogle Scholar
Ermanyuk, E. V. & Gavrilov, N. V. 2005 Duration of transient processes in the formation of internal-wave beams. Dokl. Phys. 50, 548–550.CrossRefGoogle Scholar
Ermanyuk, E. V. & Gavrilov, N. V. 2008 On internal waves generated by large-amplitude circular and rectilinear oscillations of a circular cylinder in a uniformly stratified fluid. J. Fluid Mech. 613, 329–356.CrossRefGoogle Scholar
Ferrari, R. & Wunsch, C. 2009 Ocean circulation kinetic energy: reservoirs, sources, and sinks. Annu. Rev. Fluid Mech. 41, 253–282.CrossRefGoogle Scholar
Flór, J.-B., Bush, J. W. M. & Ungarish, M. 2004 An experimental investigation of spin-up from rest of a stratified fluid. Geophys. Astrophys. Fluid Dyn. 98, 277–296.CrossRefGoogle Scholar
Flór, J.-B., Ungarish, M. & Bush, J. W. M. 2002 Spin-up from rest in a stratified fluid: boundary flows. J. Fluid Mech. 472, 51–82.CrossRefGoogle Scholar
Flynn, M. R., Onu, K. & Sutherland, B. R. 2003 Internal wave excitation by a vertically oscillating sphere. J. Fluid Mech. 494, 65–93.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
Garrett, C. 2003 Internal tides and ocean mixing. Science 301, 1858–1859.CrossRefGoogle ScholarPubMed
Garrett, C. & Kunze, E. 2007 Internal tide generation in the deep ocean. Annu. Rev. Fluid Mech. 39, 57–87.CrossRefGoogle Scholar
Gordon, D. & Stevenson, T. N. 1972 Viscous effects in a vertically propagating internal wave. J. Fluid Mech. 56, 629–639.CrossRefGoogle Scholar
Gorodtsov, V. A. & Teodorovich, E. V. 1982 Study of internal waves in the case of rapid horizontal motion of cylinders and spheres. Fluid Dyn. 17, 893–898.CrossRefGoogle Scholar
Gorodtsov, V. A. & Teodorovich, E. V. 1986 Energy characteristics of harmonic internal wave generators. J. Appl. Mech. Tech. Phys. 27, 523–529.CrossRefGoogle Scholar
Görtler, H. 1943 On an oscillatory phenomenon in fluids of a stable distribution of density. Z. Angew. Math. Mech. 23, 65–71 (in German).CrossRefGoogle Scholar
Gostiaux, L. & Dauxois, T. 2007 Laboratory experiments on the generation of internal tidal beams over steep slopes. Phys. Fluids 19, 028102.CrossRefGoogle Scholar
Hall, P. 1984 On the stability of the unsteady boundary layer on a cylinder oscillating transversely in a viscous fluid. J. Fluid Mech. 146, 347–367.CrossRefGoogle Scholar
Hendershott, M. C. 1969 Impulsively started oscillations in a rotating stratified fluid. J. Fluid Mech. 36, 513–527.CrossRefGoogle Scholar
Hopfinger, E. J., Flór, J.-B., Chomaz, J.-M. & Bonneton, P. 1991 Internal waves generated by a moving sphere and its wake in a stratified fluid. Exp. Fluids 11, 255–261.CrossRefGoogle Scholar
Hurley, D. G. 1969 The emission of internal waves by vibrating cylinders. J. Fluid Mech. 36, 657–672.CrossRefGoogle Scholar
Hurley, D. G. 1972 A general method for solving steady-state internal gravity wave problems. J. Fluid Mech. 56, 721–740.CrossRefGoogle Scholar
Hurley, D. G. 1997 The generation of internal waves by vibrating elliptic cylinders. Part 1. Inviscid solution. J. Fluid Mech. 351, 105–118.CrossRefGoogle Scholar
Hurley, D. G. & Hood, M. J. 2001 The generation of internal waves by vibrating elliptic cylinders. Part 3. Angular oscillations and comparison of theory with recent experimental observations. J. Fluid Mech. 433, 61–75.CrossRefGoogle Scholar
Hurley, D. G. & Keady, G. 1997 The generation of internal waves by vibrating elliptic cylinders. Part 2. Approximate viscous solution. J. Fluid Mech. 351, 119–138.CrossRefGoogle Scholar
Il'inykh, Y. S., Smirnov, S. A. & Chashechkin, Y. D. 1999 Excitation of harmonic internal waves in a viscous continuously stratified liquid. Fluid Dyn. 34, 890–895.Google Scholar
Ivanov, A. V. 1988 Visualization and measurements of density inhomogeneities in a density stratified fluid using the Moiré fringe method. Preprint No. 189, Institute of Applied Physics, Academy of Sciences of the USSR, Gorkii (in Russian).Google Scholar
Ivanov, A. V. 1989 Generation of internal waves by an oscillating source. Izv. Atmos. Ocean. Phys. 25, 61–64.Google Scholar
Ivey, G. N., Winters, K. B. & Koseff, J. R. 2008 Density stratification, turbulence, but how much mixing? Annu. Rev. Fluid Mech. 40, 169–184.CrossRefGoogle Scholar
King, B., Zhang, H. P. & Swinney, H. L. 2009 Tidal flow over three-dimensional topography in a stratified fluid. Phys. Fluids 21, 116601.CrossRefGoogle Scholar
Kistovich, Y. V. & Chashechkin, Y. D. 1999 a Generation of monochromatic internal waves in a viscous fluid. J. Appl. Mech. Tech. Phys. 40, 1020–1028.Google Scholar
Kistovich, Y. V. & Chashechkin, Y. D. 1999 b An exact solution of a linearized problem of the radiation of monochromatic internal waves in a viscous fluid. J. Appl. Math. Mech. 63, 587–594.CrossRefGoogle Scholar
Kunze, E. & Llewellyn Smith, S. G. 2004 The role of small-scale topography in turbulent mixing of the global ocean. Oceanography 17, 55–64.CrossRefGoogle Scholar
Lighthill, J. 1978 Waves in Fluids. Cambridge University Press.Google Scholar
Lin, Q., Boyer, D. L. & Fernando, H. J. S. 1994 Flows generated by the periodic horizontal oscillations of a sphere in a linearly stratified fluid. J. Fluid Mech. 263, 245–270.CrossRefGoogle Scholar
Llewellyn Smith, S. G. & Young, W. R. 2002 Conversion of the barotropic tide. J. Phys. Oceanogr. 32, 1554–1566.2.0.CO;2>CrossRefGoogle Scholar
Llewellyn Smith, S. G. & Young, W. R. 2003 Tidal conversion at a very steep ridge. J. Fluid Mech. 495, 175–191.CrossRefGoogle Scholar
Makarov, S. A., Neklyudov, V. I. & Chashechkin, Y. D. 1990 Spatial structure of two-dimensional monochromatic internal-wave beams in an exponentially stratified liquid. Izv. Atmos. Ocean. Phys. 26, 548–554.Google Scholar
Makhortykh, S. A. & Rybak, S. A. 1990 Effect of the near field of a point source on the generation of internal waves. Izv. Atmos. Ocean. Phys. 26, 194–198.Google Scholar
McLaren, T. I., Pierce, A. D., Fohl, T. & Murphy, B. L. 1973 An investigation of internal gravity waves generated by a buoyantly rising fluid in a stratified medium. J. Fluid Mech. 57, 229–240.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
Merzkirch, W. & Peters, F. 1992 Optical visualization of internal gravity waves in stratified fluid. Opt. Laser. Engng 16, 411–425.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, 1–16.CrossRefGoogle Scholar
Nycander, J. 2005 Generation of internal waves in the deep ocean by tides. J. Geophys. Res. 110, C10028.Google Scholar
Nycander, J. 2006 Tidal generation of internal waves from a periodic array of steep ridges. J. Fluid Mech. 567, 415–432.CrossRefGoogle Scholar
Onu, K., Flynn, M. R. & Sutherland, B. R. 2003 Schlieren measurement of axisymmetric internal wave amplitudes. Exp. Fluids 35, 24–31.CrossRefGoogle Scholar
Oster, G. 1965 Density gradients. Sci. Am. 213, 70–76.CrossRefGoogle Scholar
Oster, G. & Yamamoto, M. 1963 Density gradient techniques. Chem. Rev. 63, 257–268.CrossRefGoogle Scholar
Otto, S. R. 1992 On stability of the flow around an oscillating sphere. J. Fluid Mech. 239, 47–63.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
Peacock, T. & Weidman, P. 2005 The effect of rotation on conical wave beams in a stratified fluid. Exp. Fluids 39, 32–37.CrossRefGoogle Scholar
Peters, F. 1985 Schlieren interferometry applied to a gravity wave in a density-stratified liquid. Exp. Fluids 3, 261–269.CrossRefGoogle Scholar
Pétrélis, F., Llewellyn Smith, S. & Young, W. R. 2006 Tidal conversion at a submarine ridge. J. Phys. Oceanogr. 36, 1053–1071.CrossRefGoogle Scholar
Pozrikidis, C. 1992 Boundary Integral and Singularity Methods for Linearized Viscous Flow. Cambridge University Press.CrossRefGoogle Scholar
Pozrikidis, C. 2002 A Practical Guide to Boundary Element Methods with the Software Library BEMLIB. Chapman & Hall/CRC.CrossRefGoogle Scholar
Riley, J. J., Metcalfe, R. W. & Weissman, M. A. 1981 Direct numerical simulations of homogeneous turbulence in density-stratified fluids. AIP Conf. Proc. 76, 79–112.CrossRefGoogle Scholar
Riley, N. 2001 Steady streaming. Annu. Rev. Fluid Mech. 33, 43–65.CrossRefGoogle Scholar
Roddier, F. 1978 Distributions et Transformation de Fourier. McGraw-Hill.Google Scholar
Saint-Exupéry, A. de 1946 Le Petit Prince. Gallimard.Google Scholar
Sakai, S. 1990 Visualization of internal gravity wave by Moiré method. Trans. Vis. Soc. Japan 10, 65–68 (in Japanese).Google Scholar
Sarma, L. V. K. V. & Krishna, D. V. 1972 Oscillation of axisymmetric bodies in a stratified fluid. Zastosow. Matem. 13, 109–121.Google Scholar
Simakov, S. T. 1994 Formation of singularities of limiting amplitude in a density stratified fluid disturbed by an extended monochromatic forcing. Wave Motion 19, 11–27.CrossRefGoogle Scholar
St. Laurent, L. & Garrett, C. 2002 The role of internal tides in mixing the deep ocean. J. Phys. Oceanogr. 32, 2882–2899.2.0.CO;2>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
Sutherland, B. R., Flynn, M. R. & Onu, K. 2003 Schlieren visualisation and measurement of axisymmetric disturbances. Nonlinear Process. Geophys. 10, 303–309.CrossRefGoogle Scholar
Sutherland, B. R., Hughes, G. O., Dalziel, S. B. & Linden, P. F. 2000 Internal waves revisited. Dyn. Atmos. Oceans 31, 209–232.CrossRefGoogle Scholar
Sutherland, B. R. & Linden, P. F. 2002 Internal wave excitation by a vertically oscillating elliptical cylinder. Phys. Fluids 14, 721–731.CrossRefGoogle Scholar
Thomas, L. P., Marino, B. M. & Dalziel, S. B. 2009 Synthetic schlieren: determination of the density gradient generated by internal waves propagating in a stratified fluid. J. Phys. Conf. Ser. 166, 012007.CrossRefGoogle Scholar
Thomas, N. H. & Stevenson, T. N. 1972 A similarity solution for viscous internal waves. J. Fluid Mech. 54, 495–506.CrossRefGoogle Scholar
Vlasenko, V., Stashchuk, N. & Hutter, K. 2005 Baroclinic Tides: Theoretical Modeling and Observational Evidence. Cambridge University Press.CrossRefGoogle Scholar
Voisin, B. 1991 Internal wave generation in uniformly stratified fluids. Part 1. Green's function and point sources. J. Fluid Mech. 231, 439–480.CrossRefGoogle Scholar
Voisin, B. 2003 Limit states of internal wave beams. J. Fluid Mech. 496, 243–293.CrossRefGoogle Scholar
Voisin, B. 2007 Lee waves from a sphere in a stratified flow. J. Fluid Mech. 574, 273–315.CrossRefGoogle Scholar
Voisin, B. 2010 a Oscillating bodies in density-stratified fluids. Part 1. Wave field. In preparation.Google Scholar
Voisin, B. 2010 b Oscillating bodies in density-stratified fluids. Part 2. Added mass. In preparation.Google Scholar
Voisin, B. 2010 c Transition regimes of internal wave beams. In preparation.Google Scholar
Westerweel, J. 1997 Fundamentals of digital particle image velocimetry. Meas. Sci. Technol. 8, 1379–1392.CrossRefGoogle Scholar
Wu, J. 1969 Mixed region collapse with internal wave generation in a density-stratified medium. J. Fluid Mech. 35, 531–544.CrossRefGoogle Scholar
Wunsch, C. & Ferrari, R. 2004 Vertical mixing, energy, and the general circulation of the oceans. Annu. Rev. Fluid Mech. 36, 281–314.CrossRefGoogle Scholar
Xu, Y., Boyer, D. L., Fernando, H. J. S. & Zhang, X. 1997 Motion fields generated by the oscillatory motion of a circular cylinder in a linearly stratified fluid. Expl Thermal Fluid Sci. 14, 277–296.CrossRefGoogle Scholar
Zhang, H. P., King, B. & Swinney, H. L. 2007 Experimental study of internal gravity waves generated by supercritical topography. Phys. Fluids 19, 096602.CrossRefGoogle Scholar
Zhang, H. P., King, B. & Swinney, H. L. 2008 Resonant generation of internal waves on a model continental slope. Phys. Rev. Lett. 100, 244504.CrossRefGoogle Scholar