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The characteristics of billows generated by internal solitary waves

Published online by Cambridge University Press:  05 January 2017

Magda Carr ,
James Franklin ,
Stuart E. King ,
Peter A. Davies ,
John Grue  and
David G. Dritschel
Magda Carr*
Affiliation:
School of Mathematics and Statistics, University of St Andrews, KY16 9SS, UK
James Franklin
Affiliation:
Department of Civil Engineering, University of Dundee, DD1 4HN, UK
Stuart E. King
Affiliation:
School of Mathematics and Statistics, University of St Andrews, KY16 9SS, UK
Peter A. Davies
Affiliation:
Department of Civil Engineering, University of Dundee, DD1 4HN, UK
John Grue
Affiliation:
Department of Mathematics, University of Oslo, 0316 Oslo, Norway
David G. Dritschel
Affiliation:
School of Mathematics and Statistics, University of St Andrews, KY16 9SS, UK
*
†Email address for correspondence: magda.carr@st-andrews.ac.uk
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Abstract

The spatial and temporal development of shear-induced overturning billows associated with breaking internal solitary waves is studied by means of a combined laboratory and numerical investigation. The waves are generated in the laboratory by a lock exchange mechanism and they are simulated numerically via a contour-advective semi-Lagrangian method. The properties of individual billows (maximum height attained, time of collapse, growth rate, speed, wavelength, Thorpe scale) are determined in each case, and the billow interaction processes are studied and classified. For broad flat waves, similar characteristics are seen to those in parallel shear flow, but, for waves not at the conjugate flow limit, billow characteristics are affected by the spatially varying wave-induced shear flow. Wave steepness and wave amplitude are shown to have a crucial influence on determining the type of interaction that occurs between billows and whether billow overturning can be arrested. Examples are given in which billows (i) evolve independently of one another, (ii) pair with one another, (iii) engulf/entrain one another and (iv) fail to completely overturn. It is shown that the vertical extent a billow can attain (and the associated Thorpe scale of the billow) is dependent on wave amplitude but that its value saturates once a given amplitude is reached. It is interesting to note that this amplitude is less than the conjugate flow limit amplitude. The number of billows that form on a wave is shown to be dependent on wavelength; shorter waves support fewer but larger billows than their long-wave counterparts for a given stratification.

JFM classification

Geophysical and Geological Flows: Geophysical and Geological Flows Geophysical and Geological Flows: Internal waves
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Papers
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Journal of Fluid Mechanics , Volume 812 , 10 February 2017 , pp. 541 - 577
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© 2017 Cambridge University Press 

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Footnotes

‡

Present address: School of Mathematics, University of Edinburgh, EH9 3FD, UK.

References

Alford, M. H., Gregg, M. C. & Merrifield, M. A. 2006 Structure, propagation and mixing of energetic baroclinic tides in Mamala Bay, Oahu, Hawaii. J. Phys. Oceanogr. 36 (6), 997–1018.CrossRefGoogle Scholar
Almgren, A., Camassa, R. & Tiron, R. 2012 Shear instability of internal solitary waves in Euler fluids with thin pycnoclines. J. Fluid Mech. 710, 324–361.Google Scholar
Barad, M. F. & Fringer, O. B. 2010 Simulations of shear instabilities in interfacial gravity waves. J. Fluid Mech. 644, 61–95.Google Scholar
Carpenter, J. R., Lawrence, G. A. & Smyth, W. D. 2007 Evolution and mixing of asymmetric Holmboe instabilities. J. Fluid Mech. 582, 103–132.Google Scholar
Carr, M., King, S. E. & Dritschel, D. G. 2011 Numerical simulation of shear-induced instabilities in internal solitary waves. J. Fluid Mech. 683, 263–288.Google Scholar
Caulfield, C. P., Yoshida, S. & Peltier, W. R. 1996 Secondary instability and three-dimensionalization in a laboratory accelerating shear layer with varying density differences. Dyn. Atmos. Oceans 23, 125–138.Google Scholar
Dalziel, S. B., Carr, M., Sveen, J. K. & Davies, P. A. 2007 Simultaneous synthetic schlieren and PIV measurements for internal solitary waves. Meas. Sci. Technol. 18, 533–547.Google Scholar
De Silva, I. P. D., Fernando, H. J. S., Eaton, F. & Hebert, D. 1996 Evolution of Kelvin–Helmholtz billows in nature and laboratory. Earth Planet. Sci. Lett. 143, 217–231.CrossRefGoogle Scholar
Diamessis, P. J. & Nomura, K. K. 2004 The structure and dynamics of overturns in stably stratified homogeneous turbulence. J. Fluid Mech. 499, 197–229.Google Scholar
Dillon, T. M. 1982 Vertical overturns: a comparison of Thorpe and Ozmidov length scales. J. Geophys. Res. 87 (C12), 9601–9613.Google Scholar
Dritschel, D. G. & Ambaum, M. H. P. 1997 A contour-advective semi-Lagrangian numerical algorithm for simulating fine-scale conservative dynamical fields. Q. J. R. Meteorol. Soc. 123, 1097–1130.Google Scholar
Dritschel, D. G. & Fontane, J. 2010 The combined Lagrangian advection method. J. Comput. Phys. 229, 5408–5417.Google Scholar
Egbert, G. & Ray, R. 2000 Significant dissipation of tidal energy in the deep ocean inferred from satellite altimeter data. Nature 405, 775–778.Google Scholar
Fructus, D., Carr, M., Grue, J., Jensen, A. & Davies, P. A. 2009 Shear-induced breaking of large internal solitary waves. J. Fluid Mech. 620, 1–29.CrossRefGoogle Scholar
Fukao, S., Luce, H., Megac, T. & Yamamoto, M. K. 2011 Extensive studies of large-amplitude Kelvin–Helmholtz billows in the lower atmosphere with VHF middle and upper atmosphere radar. Q. J. R. Meteorol. Soc. 137, 1019–1041.Google Scholar
Geyer, W. R., Lavery, A. C., Scully, M. E. & Trowbridge, J. H. 2010 Mixing by shear instability at high Reynolds number. Geophys. Res. Lett. 37, L22607.Google Scholar
Grue, J., Jensen, A., Rusås, P. O. & Sveen, J. K. 1999 Properties of large-amplitude internal waves. J. Fluid Mech. 380, 257–278.Google Scholar
Grue, J., Jensen, A., Rusås, P. O. & Sveen, J. K. 2000 Breaking and broadening of internal solitary waves. J. Fluid Mech. 413, 181–217.Google Scholar
van Haren, H. & Gostiaux, L. 2010 A deep-ocean Kelvin–Helmholtz billow train. Geophys. Res. Lett. 37, L030605.Google Scholar
van Haren, H. & Gostiaux, L. 2014 Characterizing turbulent overturns in CTD data. Dyn. Atmos. Oceans 66, 58–76.Google Scholar
Hazel, P. 1972 Numerical studies of the stability of inviscid stratified shear flow. J. Fluid Mech. 51 (1), 39–61.Google Scholar
Helfrich, K. R. & Melville, W. K. 2006 Long nonlinear internal waves. Annu. Rev. Fluid Mech. 38, 395–425.Google Scholar
Helmholtz, P. 1868 On discontinuous movements of fluids. Phil. Mag. 36 (244), 337–346.Google Scholar
Keller, K. H. & van Atta, C. W. 2000 An experimental investigation of the vertical temperature structure of homogeneous stratified shear turbulence. J. Fluid Mech. 425, 1–29.Google Scholar
Kelvin, Lord 1871 Hydrokinetic solutions and observations. Phil. Mag. 10, 155–168.Google Scholar
King, S. E., Carr, M. & Dritschel, D. G. 2010 The steady state form of large amplitude internal solitary waves. J. Fluid Mech. 666, 477–505.CrossRefGoogle Scholar
Koop, C. G. & Browand, F. K. 1979 Instability and turbulence in a stratified fluid with shear. J. Fluid Mech. 93, 135–159.Google Scholar
Lamb, K. G. 2014 Internal wave breaking and dissipation mechanisms on the continental slope/shelf. Annu. Rev. Fluid Mech. 46, 231–254.CrossRefGoogle Scholar
Lamb, K. G. & Farmer, D. 2011 Instabilities in an internal solitary-like wave on the Oregon Shelf. J. Phys. Oceanogr. 41, 67–87.Google Scholar
Lamb, K. G. & Wan, B. 1998 Conjugate flows and flat solitary waves for a continuously stratified fluid. Phys. Fluids 10 (8), 2061–2079.Google Scholar
Legg, S. 2009 Improving oceanic overflow representation in climate models: the gravity current entrainment climate process team. Bull. Am. Meteorol. Soc. 90, 657–670.Google Scholar
Lien, R.-C., Henyey, F. & Ma, B. 2014 Large-amplitude internal solitary waves observed in the Northern South China sea: properties and energetics. J. Phys. Oceanogr. 44, 1095–1115.Google Scholar
Luce, H., Mega, T., Yamamoto, M. K., Yamamoto, M., Hashiguchi, H., Fukao, S., Nishi, N., Tajiri, T. & Nakazato, M. 2010 Observations of Kelvin–Helmholtz instability at a cloud base with the middle and upper atmosphere (MU) and weather radars. J. Geophys. Res. 115, D19116.Google Scholar
MacKinnon, J. A. & Gregg, M. C. 2005 Spring mixing: turbulence and internal waves during restratification on the New England shelf. J. Phys. Oceanogr. 35 (12), 2425–2443.Google Scholar
Mashayek, A. & Peltier, W. R. 2011 Three-dimensionalization of the stratified mixing layer at high Reynolds number. Phys. Fluids 23, 111701.Google Scholar
Mater, B. D., Schaad, S. M. & Venayagamoorthy, S. K. 2013 Relevance of the Thorpe length scale in stably stratified turbulence. Phys. Fluids 25, 076604.Google Scholar
Moum, J. N., Farmer, D. M., Smyth, W. D., Armi, L. & Vagle, S. 2003 Structure and generation of turbulence at interfaces strained by internal solitary waves propagating shoreward over the continental shelf. J. Phys. Oceanogr. 33, 2093–2112.2.0.CO;2>CrossRefGoogle Scholar
Munro, R. J. & Davies, P. A. 2009 The flow generated in a continuously stratified rotating fluid by the differential rotation of a plane horizontal disc. Fluid Dyn. Res. 38 (8), 522–538.Google Scholar
Ostrovsky, L. A. & Grue, J. 2003 Evolution equations for strongly nonlinear internal waves. Phys. Fluids 15, 2934–2948.Google Scholar
Pawlak, G. & Armi, L. 1998 Vortex dynamics in a spatially accelerating shear layer. J. Fluid Mech. 376, 1–35.Google Scholar
Peltier, W. R. & Caulfield, C. P. 2003 Mixing efficiency in stratified shear flows. Annu. Rev. Fluid Mech. 35, 135–167.Google Scholar
Shroyer, E. L., Moum, J. N. & Nash, J. D. 2011 Nonlinear internal waves over New Jersey’s continental shelf. J. Geophys. Res. 116, C03022.Google Scholar
Smyth, W. D. & Moum, J. N. 2012 Ocean mixing by Kelvin–Helmholtz instability. Oceanography 25 (2), 140–149.Google Scholar
Smyth, W. D., Moum, J. N. & Caldwell, D. R. 2001 The efficiency of mixing in turbulent patches: inferences from direct simulations and microstructure observations. J. Phys. Oceanogr. 31, 1969–1992.Google Scholar
Thorpe, S. A. 1973 Experiments on instability and turbulence in a stratified shear flow. J. Fluid Mech. 61, 731–751.Google Scholar
Thorpe, S. A. 1977 Turbulence and mixing in a Scottish loch. Phil. Trans. R. Soc. Lond. A 286 (1334), 125–181.Google Scholar
Thorpe, S. A. 2004 Recent developments in the study of ocean turbulence. Annu. Rev. Earth Planet. Sci. 32, 91–109.Google Scholar
Vlasenko, V., Stashchuk, N. & Hutter, K. 2005 Baroclinic Tides: Theoretical Modeling and Observational Evidence. Cambridge University Press.Google Scholar
Zhang, S. & Alford, M. H. 2015 Instabilities in nonlinear internal waves on the Washington continental shelf. J. Geophys. Res. Oceans 120, 5272–5283.Google Scholar