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Breaking of shoaling internal solitary waves

Published online by Cambridge University Press:  15 July 2010

PAYAM AGHSAEE ,
LEON BOEGMAN  and
KEVIN G. LAMB
PAYAM AGHSAEE
Affiliation:
Department of Civil Engineering, Queen's University, Kingston, Ontario K7L 3N6, Canada
LEON BOEGMAN*
Affiliation:
Department of Civil Engineering, Queen's University, Kingston, Ontario K7L 3N6, Canada
KEVIN G. LAMB
Affiliation:
Department of Applied Mathematics, University of Waterloo, Ontario N2L 3G1, Canada
*
†Email address for correspondence: leon.boegman@civil.queensu.ca
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Abstract

The breaking of fully nonlinear internal solitary waves of depression shoaling upon a uniformly sloping boundary in a smoothed two-layer density field was investigated using high-resolution two-dimensional simulations. Our simulations were limited to narrow-crested waves, which are more common than broad-crested waves in geophysical flows. The simulations were performed for a wide range of boundary slopes S ∈ [0.01, 0.3] and wave slopes extending the parameter range to weaker slopes than considered in previous laboratory and numerical studies. Over steep slopes (S ≥ 0.1), three distinct breaking processes were observed: surging, plunging and collapsing breakers which are associated with reflection, convective instability and boundary-layer separation, respectively. Over mild slopes (S ≤ 0.05), nonlinearity varies gradually and the wave fissions into a train of waves of elevation as it passes through the turning point where solitary waves reverse polarity. The dynamics of each breaker type were investigated and the predominance of a particular mechanism was associated with a relative developmental time scale. The breaking location was modelled as a function of wave amplitude (a), characteristic wave length and the isopycnal length along the slope. The breaker type was characterized in wave slope (Sw = a/Lw, where Lw is a measure of half of the wavelength) versus S space, and the reflection coefficient (R), modelled as a function of the internal Iribarren number, was in agreement with other studies. The effects of grid resolution and wave Reynolds number (Rew) on R, boundary-layer separation and the evolution of global instability were studied. High Reynolds numbers (Rew ~ 104) were found to trigger a global instability, which modifies the breaking process relative to the lower Rew case, but not necessarily the breaking location, and results in a ~ 10 % increase in R, relative to the Rew ~ 103 case.

JFM classification

Geophysical and Geological Flows: Internal waves Geophysical and Geological Flows: Stratified flows Geophysical and Geological Flows: Topographic effects
Type
Papers
Information
Journal of Fluid Mechanics , Volume 659 , 25 September 2010 , pp. 289 - 317
Copyright
Copyright © Cambridge University Press 2010

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References

REFERENCES

Apel, J. R. 2002 Oceanic internal waves and solitons. In An Atlas of Oceanic Internal Solitary Waves, Global Ocean Associates. Prepared for Office of Naval Research (Code 322 PO), 1-40.Google Scholar
Apel, J. R., Holbrook, J. R., Liu, A. K. & Tsai, J. J. 1985 The Sulu Sea internal soliton experiment. J. Phys. Oceanogr. 15, 1625–1651.2.0.CO;2>CrossRefGoogle Scholar
Boegman, L., Imberger, J., Ivey, G. N. & Antenucci, J. P. 2003 High-frequency internal waves in large stratified lakes. Limnol. Oceanogr. 48, 895–919.CrossRefGoogle Scholar
Boegman, L. & Ivey, G. N. 2009 Flow separation and resuspension beneath shoaling nonlinear internal waves. J. Geophys. Res. 114, C02018.CrossRefGoogle Scholar
Boegman, L., Ivey, G. N. & Imberger, J. 2005 The degeneration of internal waves in lakes with sloping topography. Limnol. Oceanogr. 50, 1620–1637.CrossRefGoogle Scholar
Bogucki, D. J. & Redekopp, L. G. 1999 Global instability of separated flows: a paradigm for sediment resuspension by internal waves. Geophys. Res. Lett. 26, 1317–1320.CrossRefGoogle Scholar
Bogucki, D. J. Redekopp, L. G. & Barth, J. 2005 Internal solitary waves in the coastal mixing and optics 1996 experiment: multimodal structure and re-suspension. J. Geophys. Res. 110, C02024.CrossRefGoogle Scholar
Bourgault, D., Blokhina, M. D., Mirshak, R. & Kelley, D. E. 2007 Evolution of a shoaling internal solitary wave train. Geophys. Res. Lett. 34, L03601.CrossRefGoogle Scholar
Bourgault, D. & Kelley, D. 2003 Wave-induced boundary mixing in a partially mixed estuary. J. Marine Res. 61, 553–567.CrossRefGoogle Scholar
Bourgault, D. & Kelley, D. 2007 On the reflectance of uniform slopes for normally incident interfacial solitary waves. J. Phys. Oceanogr. 37, 1156–1162.CrossRefGoogle Scholar
Carr, M. & Davies, P. A. 2006 The motion of an internal solitary wave of depression over a fixed bottom boundary in a shallow, two-layer fluid. Phys. Fluids 18, 016601.CrossRefGoogle Scholar
Carr, M., Davies, P. A. & Shivaram, P. 2008 Experimental evidence of internal solitary wave-induced global instability in shallow water benthic boundary layers. Phys. Fluids 20, 066603.CrossRefGoogle Scholar
Dean, R. G. & Dalrymple, R. A. 1991 Water wave mechanics for engineers and scientists. In Advanced Series on Ocean Engineering (ed. Liu, L.-F.), vol 2. World Scientific.Google Scholar
Diamessis, P. & Redekopp, L. G. 2006 Numerical investigation of solitary internal wave-induced global instability in shallow water benthic boundary layers. J. Phys. Oceanogr. 36, 784–812.CrossRefGoogle Scholar
Fringer, O. B. & Street, R. L. 2003 The dynamics of breaking progressive interfacial waves. J. Fluid Mech. 494, 319–353.CrossRefGoogle 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
Grimshaw, R., Pelinovsky, E. & Talipova, T. 1998 Solitary wave transformation due to a change in polarity. Stud. Appl. Math. 101, 357–388.CrossRefGoogle Scholar
Helfrich, K. R. 1992 Internal solitary wave breaking and run-up on a uniform slope. J. Fluid Mech. 243, 133–154.CrossRefGoogle Scholar
Helfrich, K. R., Melville, W. K. & Miles, J. W. 1984 On interfacialsolitary waves over slowly varying topography. J. Fluid Mech. 149, 305–317.CrossRefGoogle Scholar
Helfrich, K. R. & Melville, W. K. 1986 On long nonlinear internal waves over slope-shelf topography. J. Fluid Mech. 167, 285–308.CrossRefGoogle Scholar
Horn, D. A., Imberger, J. & Ivey, G. N. 2001 The degeneration of large-scale interfacial gravity waves in lakes. J. Fluid Mech. 434, 181–207.CrossRefGoogle Scholar
Hosegood, P., Bonnin, J. & van Haren, H. 2004 Solibore-induced sediment resuspension in the Faeroe–Shetland Channel. Geophys. Res. Lett. 31, L09301.CrossRefGoogle Scholar
Hult, E. L., Troy, C. D. & Koseff, J. R. 2009 The breaking of interfacial waves at a submerged bathymetric ridge. J. Fluid Mech. 637, 45–71.CrossRefGoogle Scholar
Imberger, J. 1998 Flux path in a stratified lake. In Physical Processes in Lakes and Oceans, Coastal and Estuarine Studies (ed. Imberger, J.), vol. 54, pp. 1–17. American Geophysical Union.CrossRefGoogle Scholar
Jackson, C. 2007 Internal wave detection using the Moderate Resolution Imaging Spectroradiometer (MODIS). J. Geophys. Res. 112, C11012.CrossRefGoogle Scholar
Kao, T. W., Pan, F. S. & Renouard, D. 1985 Internal solitons on the pycnocline: generation, propagation and shoaling and breaking over a slope. J. Fluid Mech. 159, 19–53.CrossRefGoogle Scholar
Ledwell, J. R. & Hickey, B. M. 1995 Evidence for enhanced boundary mixing for the Santa Monica basin. J. Geophys. Res. 100 (20), 20665–20679.CrossRefGoogle Scholar
Lamb, K. G. 1994 Numerical experiments of internal wave generation by strong tidal flow across a finite amplitude bank edge. J. Geophy. Res. 99, 843–864.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., Boegman, L. & Ivey, G. N. 2005 Numerical simulations of shoaling internal solitary waves in tilting tank experiments. In Proceedings of the 9th European Workshop on Physical Processes in Natural Waters, pp. 31–38. Lancaster University.Google Scholar
Lamb, K. G. 2007 Energy and pseudoenergy flux in the internal wave field generated by tidal flow over topography. Cont. Shelf Res. 27, 1208–1232.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, 559–580.CrossRefGoogle Scholar
Lamb, K. G. & Wan, B. 1998 Conjugate flows and flat solitary waves for a continuously stratified fluid. Phys. Fluids. 10, 2061–2079.CrossRefGoogle Scholar
Legg, S. & Adcroft, A. 2003 Internal wave breaking at concave and convex continental slopes. J. Phys. Oceanogr. 33, 2224–2246.2.0.CO;2>CrossRefGoogle Scholar
Lorke, A. 2007 Boundary mixing in the thermocline of a large lake. J. Geophys. Res. 112, C09019.CrossRefGoogle Scholar
Marti, C. L. & Imberger, J. 2004 Exchange processes between littoral and pelagic waters in a stratified lake. Mećanica Comput. 23, 1261–1274.Google Scholar
Michallet, H. & Ivey, G. N. 1999 Experiments on mixing due to internal solitary waves breaking on uniform slopes. J. Geophys. Res. 104, 13467–13477.CrossRefGoogle 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
Orr, M. H., & Mignerey, P. C. 2003 Nonlinear internal waves in the South China Sea: observation of the conversion of depression internal waves to elevation internal waves. J. Geophys. Res. 108, C3, 3064.CrossRefGoogle Scholar
Osborne, A. R., & Burch, T. L. 1980 Internal solitons in the Andaman Sea. Science 208, 451–60.CrossRefGoogle ScholarPubMed
Peregrine, D. H. 1983 Breaking waves on beaches. Annu. Rev. Fluid Mech. 15, 149–178.CrossRefGoogle Scholar
Shroyer, E. L., Moum, J. N. & Nash, J. D. 2009 Observations of polarity reversal in shoaling nonlinear internal waves. J. Phys. Oceanogr. 39, 691–701.CrossRefGoogle Scholar
Smyth, W. D. & Peltier, W. R. 1990 Three-dimensional primary instabilities of a stratified dissipative, parallel flow. Geophys. Astrophys. Fluid Dyn. 52, 249–261.CrossRefGoogle Scholar
Stastna, M. & Lamb, K. G. 2008 Sediment resuspension mechanisms associated with internal waves in coastal waters. J. Geophys. Res. 113, C10016.CrossRefGoogle Scholar
Sveen, J. K., Guo, Y., Davies, P. A. & Grue, J. 2002 On the breaking of internal solitary waves at a ridge. J. Fluid Mech. 469, 161–188.CrossRefGoogle Scholar
Talipova, T. G., Pelinovskii, E. N. & Grimshaw, R. 1997 Transformation of a soliton at appoint of zero nonlinearity. Am. Inst. Phys. JETP Lett. 65, 120–125.CrossRefGoogle Scholar
Thorpe, S. A. 1998 Some dynamical effects of internal waves and the sloping sides of lakes. In Physical Processes in Lakes and Oceans, Coastal and Estuarine Studies (ed. Imberger, J.), vol. 54, pp. 441–460. American Geophysical Union.Google Scholar
Troy, C. D. & Koseff, J. R. 2005 The instability and breaking of long internal waves. J. Fluid Mech. 543, 107–136.CrossRefGoogle Scholar
Venayagamoorthy, S. K. & Fringer, O. B. 2007 On the formation and propagation of nonlinear internal boluses across a shelf break. J. Fluid Mech. 577, 137–159.CrossRefGoogle Scholar
Vlasenko, V. & Hutter, K. 2002 Numerical experiments on the breaking of solitary internal waves over a slope-shelf topography. J. Phys. Oceanogr. 32 (6), 1779–1793.2.0.CO;2>CrossRefGoogle Scholar
Vlasenko, V. & Stashchuk, N. 2007 Three-dimensional shoaling of large-amplitude internal waves. J. Geophys. Res. 112, C11018.CrossRefGoogle Scholar
Wüest, A., Piepke, G. & Van Senden, D. C. 2000 Turbulent kinetic energy balance as a tool for estimating vertical diffusivity in wind-forced stratified waters. Limnol. Oceanogr. 45, 1388–1400.CrossRefGoogle Scholar
Zhao, Z., Klemas, V. V., Zheng, Q. & Yan, X-H. 2003 Satellite observation of internal solitary waves converting polarity. Geophys. Res. Lett. 30, 19.CrossRefGoogle Scholar