Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-27T09:04:47.815Z Has data issue: false hasContentIssue false
  • English
  • Français

Laboratory experiments and simulations for solitary internal waves with trapped cores

Published online by Cambridge University Press:  19 September 2014

Paolo Luzzatto-Fegiz  and
Karl R. Helfrich
Paolo Luzzatto-Fegiz*
Affiliation:
Churchill College, University of Cambridge, Storey’s Way, Cambridge CB3 0DS, UK Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK
Karl R. Helfrich
Affiliation:
Department of Physical Oceanography, Woods Hole Oceanographic Institution, 266 Woods Hole Road, Woods Hole, MA 02543-1050, USA
*
†Email address for correspondence: P.Luzzatto-Fegiz@damtp.cam.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

We perform simultaneous coplanar measurements of velocity and density in solitary internal waves with trapped cores, as well as viscous numerical simulations. Our set-up comprises a thin stratified layer (approximately 15 % of the overall fluid depth) overlaying a deep homogeneous layer. We consider waves propagating near a free surface, as well as near a rigid no-slip lid. In the free-surface case, all trapped-core waves exhibit a strong shear instability. We propose that Marangoni effects are responsible for this instability, and use our velocity measurements to perform quantitative calculations supporting this hypothesis. These surface-tension effects appear to be difficult to avoid at the experimental scale. By contrast, our experiments with a no-slip lid yield robust waves with large cores. In order to consider larger-amplitude waves, we complement our experiments with viscous numerical simulations, employing a longer virtual tank. Where overlap exists, our experiments and simulations are in good agreement. In order to provide a robust definition of the trapped core, we propose bounding it as a Lagrangian coherent structure (instead of using a closed streamline, as has been done traditionally). This construction is less sensitive to small errors in the velocity field, and to small three-dimensional effects. In order to retain only flows near equilibrium, we introduce a steadiness criterion, based on the rate of change of the density in the core. We use this criterion to successfully select within our experiments and simulations a family of quasi-steady robust flows that exhibit good collapse in their properties. The core circulation is small (at most, around 10 % of the baroclinic wave circulation). The core density is essentially uniform; the standard deviation of the density, in the core region, is less than 4 % of the full density range. We also calculate the circulation, kinetic energy and available potential energy of these waves. We find that these results are consistent with predictions from Dubreil-Jacotin–Long theory for waves with a uniform-density irrotational core, except for an offset, which we suggest is associated with viscous effects. Finally, by computing Richardson-number fields, and performing a temporal stability analysis based on the Taylor–Goldstein equation, we show that our results are consistent with empirical stability criteria in the literature.

JFM classification

Geophysical and Geological Flows: Internal waves Waves/Free-surface Flows: Solitary waves
Type
Papers
Information
Journal of Fluid Mechanics , Volume 757 , 25 October 2014 , pp. 354 - 380
Copyright
© 2014 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

Akylas, T. R., Grimshaw, R. H. J., Clarke, S. R. & Tabaei, A. 2007 Reflective tidal internal wave beams and local generation of solitary internal waves in the ocean thermocline. J. Fluid Mech. 593, 297–313.CrossRefGoogle Scholar
Almgren, A. S., Bell, J. B., Colella, P., Howell, L. H. & Welcome, M. L. 1998 A conservative adaptive projection method for the variable density incompressible Navier–Stokes equations. J. Comput. Phys. 142 (1), 1–46.CrossRefGoogle Scholar
Barad, M. F. & Fringer, O. B. 2010 Simulations of shear instabilities in interfacial gravity waves. J. Fluid Mech. 644, 61–95.CrossRefGoogle Scholar
Batchelor, G. K. 1956 On steady laminar flow with closed streamlines at large Reynolds number. J. Fluid Mech. 1, 177–190.CrossRefGoogle Scholar
Carr, M., Fructus, D., Grue, J., Jensen, A. & Davies, P. A. 2008 Convectively induced shear instability in large amplitude internal solitary waves. Phys. Fluids 20 (12), 126601.CrossRefGoogle Scholar
Cheung, T. K. & Little, C. G. 1990 Meteorological tower, microbarograph array, and sodar observations of solitary-like waves in the nocturnal boundary layer. J. Atmos. Sci. 47, 2516–2536.2.0.CO;2>CrossRefGoogle Scholar
Clarke, R. H., Smith, R. K. & Reid, D. G. 1981 The morning glory of the Gulf of Carpentaria: an atmospheric undular bore. Mon. Weath. Rev. 109 (8), 1726–1750.2.0.CO;2>CrossRefGoogle Scholar
Davis, R. E. & Acrivos, A. 1967 Solitary internal waves in deep water. J. Fluid Mech. 29, 593–607.CrossRefGoogle Scholar
Derzho, O. G. & Grimshaw, R. H. J. 1997 Solitary waves with a vortex core in a shallow layer of stratified fluid. Phys. Fluids 9 (11), 3378–3385.CrossRefGoogle Scholar
Diamessis, P. J. & Redekopp, L. G. 2006 Numerical investigation of solitary internal wave-induced global instability in shallow water benthic boundary layers. J. Phys. Oceanogr. 36 (5), 784–812.CrossRefGoogle Scholar
Doviak, R. J. & Christie, D. R. 1989 Thunderstorm-generated solitary waves—a wind shear hazard. J. Aircraft 26 (5), 423–431.CrossRefGoogle Scholar
Dubreil-Jacotin, M. L. 1934 Sur la détermination rigoureuse des ondes permanentes périodiques d’ampleur finie. J. Math. Pure Appl. 13, 217–291.Google Scholar
Economidou, M. & Hunt, G. R. 2008 Density stratified environments: the double-tank method. Exp. Fluids 46 (3), 453–466.CrossRefGoogle Scholar
Fructus, D., Carr, M., Grue, J., Jensen, A. & Davies, A.P. 2009 Shear-induced breaking of large internal solitary waves. J. Fluid Mech. 620, 1–29.CrossRefGoogle Scholar
Grimshaw, R. H. J. 1969 On steady recirculating flows. J. Fluid Mech. 39, 695–703.CrossRefGoogle Scholar
Grue, J., Jensen, A., Rusas, P. O. & Sveen, J. K. 2000 Breaking and broadening of internal solitary waves. J. Fluid Mech. 413, 181–217.CrossRefGoogle Scholar
Haller, G. 2005 An objective definition of a vortex. J. Fluid Mech. 525, 1–26.CrossRefGoogle Scholar
Head, M. J.1983 The use of miniature four-electrode conductivity probes for high resolution measurement of turbulent density or temperature variations in salt-stratified water flows. PhD thesis, University of California, San Diego.Google Scholar
Helfrich, K. R. & White, B. L. 2010 A model for large-amplitude internal solitary waves with trapped cores. Nonlinear Process. Geophys. 17, 303–318.CrossRefGoogle Scholar
King, S. E., Carr, M. & Dritschel, D. G. 2011 The steady-state form of large-amplitude internal solitary waves. J. Fluid Mech. 666, 477–505.CrossRefGoogle Scholar
Klymak, J. M. & Moum, J. N. 2003 Internal solitary waves of elevation advancing on a shoaling shelf. Geophys. Res. Lett. 30 (20), 2045.CrossRefGoogle Scholar
Kropfli, R. A., Ostrovski, L. A., Stanton, T. P., Skirta, E. A., Keane, A. N. & Irisov, V. 1999 Relationships between strong internal waves in the coastal zone and their radar and radiometric signatures. J. Geophys. Res. 104, 3133–3148.CrossRefGoogle Scholar
Lamb, K. G. 2002 A numerical investigation of solitary internal waves with trapped cores formed via shoaling. J. Fluid Mech. 451, 109–144.CrossRefGoogle Scholar
Lamb, K. G. & Farmer, D. 2011 Instabilities in an internal solitary-like wave on the Oregon Shelf. J. Phys. Oceanogr. 41 (1), 67–87.CrossRefGoogle Scholar
Lamb, K. G. & Nguyen, V. T. 2009 Calculating energy flux in internal solitary waves with an application to reflectance. J. Phys. Oceanogr. 39 (3), 559–580.CrossRefGoogle Scholar
Law, A. W. K. & Wang, H. W. 2000 Measurement of mixing processes with combined digital particle image velocimetry and planar laser induced fluorescence. Exp. Therm. Fluid Sci. 22, 213–229.CrossRefGoogle Scholar
Leal, L. G. 2007 Advanced Transport Phenomena: Fluid Mechanics and Convective Transport Processes, Cambridge Series in Chemical Engineering. Cambridge University Press.CrossRefGoogle Scholar
Lien, R. C., D’Asaro, E. A., Henyey, F., Chang, M., Tang, T. & Yang, Y. 2012 Trapped core formation within a shoaling nonlinear internal wave. J. Phys. Oceanogr. 42 (4), 511–525.CrossRefGoogle Scholar
Long, R. 1953 Some aspects of the flow of stratified fluids: I. A theoretical investigation. Tellus 5 (1), 42–58.CrossRefGoogle Scholar
Melton, L. A. & Lipp, C. W. 2003 Criteria for quantitative PLIF experiments using high-power lasers. Exp. Fluids 35 (4), 310–316.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
Moum, J. N., Farmer, D. M., Shroyer, E. L., Smyth, W. D. & Armi, L. 2007a Dissipative losses in nonlinear internal waves propagating across the continental shelf. J. Phys. Oceanogr. 37 (7), 1989–1995.CrossRefGoogle Scholar
Moum, J. N., Klymak, J. M., Nash, J. D., Perlin, A. & Smyth, W. D. 2007b Energy transport by nonlinear internal waves. J. Phys. Oceanogr. 37 (7), 1968–1988.CrossRefGoogle Scholar
Rao, M. P., Castracane, P., Casadio, S., Fuá, D. & Fiocco, G. 2004 Observations of atmospheric solitary waves in the urban boundary layer. Boundary-Layer Meteorol. 111 (1), 85–108.CrossRefGoogle Scholar
Rosen, M. J. & Kunjappu, J. T. 2012 Surfactants and Interfacial Phenomena. Wiley.CrossRefGoogle Scholar
Salloum, M., Knio, O. M. & Brandt, A. 2012 Numerical simulation of mass transport in internal solitary waves. Phys. Fluids 24 (1), 016602.CrossRefGoogle Scholar
Schlichting, H. 1979 Boundary Layer Theory. McGraw-Hill.Google Scholar
Scotti, A. & Pineda, J. 2004 Observation of very large and steep internal waves of elevation near the Massachusetts coast. Geophys. Res. Lett. 31 (22), L22307.CrossRefGoogle Scholar
Shadden, S. C., Dabiri, J. O. & Marsden, J. E. 2006 Lagrangian analysis of fluid transport in empirical vortex ring flows. Phys. Fluids 18, 047105.CrossRefGoogle Scholar
Shreffler, J. H. & Binkowski, F. S. 1981 Observations of pressure jump lines in the Midwest, 10–12 August 1976. Mon. Weather Rev. 109 (8), 1713–1725.2.0.CO;2>CrossRefGoogle Scholar
Smyth, W. D., Moum, J. N. & Nash, J. D. 2011 Narrowband oscillations in the upper equatorial ocean. Part II: properties of shear instabilities. J. Phys. Oceanogr. 41 (3), 412–428.CrossRefGoogle Scholar
Sutherland, B. R., Barrett, K. J. & Ivey, G. N. 2013 Shoaling internal solitary waves. J. Geophys. Res. Oceans 118 (9), 4111–4124.CrossRefGoogle Scholar
Troy, C. D. & Koseff, J. R. 2005 The instability and breaking of long internal waves. J. Fluid Mech. 543, 107–136.CrossRefGoogle Scholar
Wu, M. & Gharib, M. 2002 Experimental studies on the shape and path of small air bubbles rising in clean water. Phys. Fluids 14 (7), L49–L52.CrossRefGoogle Scholar