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Shoaling mode-2 internal solitary-like waves

Published online by Cambridge University Press:  02 October 2019

Magda Carr Open the ORCID record for Magda Carr [Opens in a new window] ,
Marek Stastna ,
Peter A. Davies  and
Koen J. van de Wal
Magda Carr*
Affiliation:
School of Mathematics, Statistics and Physics, Newcastle University, NE1 7RU, UK
Marek Stastna
Affiliation:
Department of Applied Mathematics, University of Waterloo, Ontario, N2L 3G1, Canada
Peter A. Davies
Affiliation:
Department of Civil Engineering, University of Dundee, DD1 4HN, UK
Koen J. van de Wal
Affiliation:
Eindhoven University of Technology, PO box 513, 5600MB Eindhoven, The Netherlands
*
†Email address for correspondence: magda.carr@ncl.ac.uk
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Abstract

The propagation of a train of mode-2 internal solitary-like waves (ISWs) over a uniformly sloping, solid topographic boundary, has been studied by means of a combined laboratory and numerical investigation. The waves are generated by a lock-release method. Features of their shoaling include (i) formation of an oscillatory tail, (ii) degeneration of the wave form, (iii) wave run up, (iv) boundary layer separation, (v) vortex formation and re-suspension at the bed and (vi) a reflected wave signal. Slope steepness, $s$, is defined to be the height of the slope divided by the slope base length. In shallow slope cases ($s\leqslant 0.07$), the wave form is destroyed by the shoaling process; the leading mode-2 ISW degenerates into a train of mode-1 waves of elevation and little boundary layer activity is seen. For steeper slopes ($s\geqslant 0.13$), boundary layer separation, vortex formation and re-suspension at the bed are observed. The boundary layer dynamics is shown (numerically) to be dependent on the Reynolds number of the flow. A reflected mode-2 wave signal and wave run up are seen for slopes of steepness $s\geqslant 0.20$. The wave run up distance is shown to be proportional to the length scale $ac^{2}/g^{\prime }h_{2}\sin \unicode[STIX]{x1D703}$ where $a,c,g^{\prime },h_{2}$ and $\unicode[STIX]{x1D703}$ are wave amplitude, wave speed, reduced gravity, pycnocline thickness and slope angle respectively.

JFM classification

Boundary Layers: Boundary layer separation Geophysical and Geological Flows: Internal waves Geophysical and Geological Flows: Stratified flows
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 879 , 25 November 2019 , pp. 604 - 632
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Copyright
© 2019 Cambridge University Press 

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References

Aghsaee, P., Boegman, L., Diamessis, P. J. & Lamb, K. G. 2011 Boundary-layer-separation-driven vortex shedding beneath internal solitary waves of depression. J. Fluid Mech. 690, 321–344.10.1017/jfm.2011.432Google Scholar
Aghsaee, P., Boegman, L. & Lamb, K. G. 2010 Breaking of shoaling internal solitary waves. J. Fluid Mech. 659, 289–317.10.1017/S002211201000248XGoogle Scholar
Akylas, T. R. & Grimshaw, R. S. J. 1992 Solitary internal waves with oscillatory tails. J. Fluid Mech. 242, 279–298.10.1017/S0022112092002374Google Scholar
Archetti, R. & Brocchini, M. 2002 An integral swash zone model with friction: an experimental and numerical investigation. Coast. Engng 45, 89–110.10.1016/S0378-3839(02)00038-8Google Scholar
Arthur, R. S. & Fringer, O. B. 2014 The dynamics of breaking internal solitary waves on slopes. J. Fluid Mech. 761, 360–398.10.1017/jfm.2014.641Google Scholar
Boegman, L. & Ivey, G. N. 2009 Flow separation and resuspension beneath shoaling nonlinear internal waves. J. Geophys. Res. 114, C02018.10.1029/2007JC004411Google Scholar
Boegman, L., Ivey, G. N. & Imberger, J. 2005 The degeneration of internal waves in lakes with sloping topography. Limnol. Oceanogr. 50 (5), 1620–1637.10.4319/lo.2005.50.5.1620Google Scholar
Boegman, L. & Stastna, M. 2019 Sediment resuspension and transport by internal solitary waves. Annu. Rev. Fluid Mech. 51, 129–154.10.1146/annurev-fluid-122316-045049Google Scholar
Bogucki, D. J., Dickey, T. & Redekopp, L. G. 1997 Sediment resuspension and mixing by resonantly generated internal solitary waves. J. Phys. Oceanogr. 27, 1181–1196.10.1175/1520-0485(1997)027<1181:SRAMBR>2.0.CO;22.0.CO;2>Google Scholar
Bogucki, D. J., Redekopp, L. G. & Barth, J. 2005 Internal solitary waves in the Coastal Mixing and Optics 1996 experiment: multimodal structure and resuspension. J. Geophys. Res. 110, C02024.10.1029/2003JC002253Google Scholar
Bourgault, D., Galbraith, P. S. & Chavanne, C. 2016 Generation of internal solitary waves by frontally-forced intrusions in geophysical flows. Nat. Commun. 7, 13606.10.1038/ncomms13606Google Scholar
Brandt, A. & Shipley, K. R. 2014 Laboratory experiments on mass transport by large amplitude mode-2 internal solitary waves. Phys. Fluids 26, 046601.10.1063/1.4869101Google Scholar
Cacchione, D. A., Pratson, L. F. & Ogston, A. S. 2002 The shaping of continental slopes by internal tides. Science 296, 724–727.10.1126/science.1069803Google Scholar
Carr, M., Davies, P. A. & Hoebers, R. 2015 Experiments on the structure and stability of mode-2 internal solitary-like waves propagating on an offset pycnocline. Phys. Fluids 27 (4), 046602.10.1063/1.4916881Google 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, 066603.10.1063/1.2931693Google Scholar
Cheng, M.-H., Hsieh, C.-M., Hsu, J. R.-C. & Hwang, R. R. 2017 Transformation of mode-2 internal solitary wave over a pseudo slope- shelf. AIP Advances 7, 095309.10.1063/1.5000972Google Scholar
Cheng, M.-H., Hsieh, C.-M., Hwang, R. R. & Hsu, J. R.-C. 2018 Effects of initial amplitude and pycnocline thickness on the evolution of mode-2 internal solitary waves. Phys. Fluids 30, 042101.10.1063/1.5020093Google 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, 533–547.10.1088/0957-0233/18/3/001Google Scholar
Davis, R. E. & Acrivos, A. 1967 Solitary internal waves in deep water. J. Fluid Mech. 29, 593–607.10.1017/S0022112067001041Google Scholar
Deepwell, D. & Stastna, M. 2016 Mass transport by mode-2 internal solitary-like waves. Phys. Fluids 28, 056606.10.1063/1.4948544Google Scholar
Deepwell, D., Stastna, M., Carr, M. & Davies, P. A. 2017 Interaction of a mode-2 internal solitary wave with narrow isolated topography. Phys. Fluids 29, 076601.10.1063/1.4994590Google 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, 784–812.10.1175/JPO2900.1Google Scholar
Duda, T. F., Lynch, J. F., Beardsley, R. C., Ramp, S. R., Chiu, C. S., Tang, T. Y. & Yang, Y. J. 2004 Internal tide and non-linear internal wave behavior at the continental slope in the northern South China Sea. IEEE J. Ocean. Engng 29, 1105–1130.10.1109/JOE.2004.836998Google Scholar
Gavrilov, N. V. & Lyapidevski, V. Y. 2009 Symmetric solitary waves in a two-layer fluid. Dokl. Phys. 54 (11), 508–511.10.1134/S1028335809110081Google Scholar
Gavrilov, N. V. & Lyapidevski, V. Y. 2011 Large amplitude internal solitary waves over a shelf. Nat. Hazards Earth Syst. Sci. 11, 17–25.10.5194/nhess-11-17-2011Google Scholar
Grimshaw, R. 2014 Internal solitary waves. In Chapter 1 in Environmental Stratified Flows, Topics in Environmental Fluid Mechanics, vol. 3. Springer.Google Scholar
Groeskamp, S., Nauw, J. J. & Maas, L. R. M. 2011 Observations of estuarine circulation and solitary internal waves in a highly energetic tidal channel. Ocean Dyn. 61, 1767–1782.Google Scholar
Grue, J., Jensen, A., Rusås, P.-O. & Sveen, J. K. 1999 Properties of large-amplitude internal waves. J. Fluid Mech. 380, 257–278.10.1017/S0022112098003528Google Scholar
Guo, C. & Chen, X. 2012 Numerical investigation of large amplitude second mode internal solitary waves over a slope-shelf topography. Ocean Model. 42, 80–91.10.1016/j.ocemod.2011.11.003Google Scholar
Hammond, D. A. & Redekopp, L. G. 1998 Local and global instability properties of separation bubbles. Eur. J. Mech. (B/Fluids) 17, 145–164.10.1016/S0997-7546(98)80056-3Google Scholar
Helfrich, K. R. 1992 Internal solitary wave breaking and run-up on a uniform slope. J. Fluid Mech. 243, 133–154.10.1017/S0022112092002660Google Scholar
Helfrich, K. R. & Melville, W. K. 2006 Long nonlinear internal waves. Annu. Rev. Fluid Mech. 38, 395–425.10.1146/annurev.fluid.38.050304.092129Google Scholar
Horn, D. A., Imberger, J. & Ivey, G. N. 2001 The degeneration of large-scale interfacial gravity waves in lakes. J. Fluid Mech. 434, 181–207.10.1017/S0022112001003536Google Scholar
Hosegood, P. & van Haren, H. 2004 Near-bed solibores over the continental slope in the Faeroe-Shetland channel. Deep Sea Res. 51, 2943–2971.10.1016/j.dsr2.2004.09.016Google Scholar
Hughes, S. A. 2004a Estimation of wave run-up on smooth, impermeable slopes using the wave momentum flux parameter. Coast. Engng 51, 1085–1104.10.1016/j.coastaleng.2004.07.026Google Scholar
Hughes, S. A. 2004b Wave momentum flux parameter: a descriptor for nearshore waves. Coast. Engng 51, 1067–1084.10.1016/j.coastaleng.2004.07.025Google Scholar
Hyder, P., Jeans, D. R. G., Cauquil, E. & Nerzic, R. 2005 Observations and predictability of internal solitons in the northern Andaman Sea. Appl. Ocean Res. 27, 1–11.10.1016/j.apor.2005.07.001Google 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, 19–53.10.1017/S0022112085003081Google Scholar
Kao, T. W. & Pao, H.-P. 1980 Wake collapse in the thermocline and internal solitary waves. J. Fluid Mech. 97, 115–127.10.1017/S0022112080002455Google Scholar
Kundu, P. K., Cohen, I. M. & Dowling, D. R. 2015 Fluid Mechanics. Academic Press.Google Scholar
Lamb, K. G. 2014 Internal wave breaking and dissipation mechanisms on the continental slope/shelf. Annu. Rev. Fluid Mech. 46, 231–254.10.1146/annurev-fluid-011212-140701Google Scholar
Liu, A. K., Su, F.-C., Hsu, M.-K., Kuo, N.-J. & Ho, C.-R. 2013 Generation and evolution of mode-two internal waves in the South China Sea. Cont. Shelf Res. 15, 18–27.10.1016/j.csr.2013.02.009Google Scholar
MacIntyre, S. K., Flynn, M., Jellison, R. & Romero, J. R. 1999 Boundary mixing and nutrient fluxes in Mono Lake, California. Limnol. Oceanogr. 4, 512–529.10.4319/lo.1999.44.3.0512Google Scholar
MacKinnon, J. A. & Gregg, M. C. 2003 Shear and baroclinic energy flux on the summer New England shelf. J. Phys. Oceanogr. 33, 1462–1475.10.1175/1520-0485(2003)033<1462:SABEFO>2.0.CO;22.0.CO;2>Google Scholar
Magalhaes, J. M. & da Silva, J. C. B. 2018 Internal solitary waves in the andaman sea: new insights from sar imagery. Remote Sens. 10, 861–877.10.3390/rs10060861Google Scholar
Maxworthy, T. 1980 On the formation of nonlinear internal waves by the gravitational collapse of mixed regions in two and three dimensions. J. Fluid Mech. 96, 47–64.10.1017/S0022112080002017Google 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), 522–538.10.1016/j.fluiddyn.2006.03.002Google Scholar
Nakayama, K., Sato, T., Shimizu, K. & Boegman, L. 2019 Classification of internal solitary wave breaking over a slope. Phys. Rev. Fluids. 4, 014801.10.1103/PhysRevFluids.4.014801Google Scholar
Nash, J. D. & Moum, J. N. 2005 River plumes as a source of large-amplitude internal waves in the coastal ocean. Nature 437, 400–403.10.1038/nature03936Google Scholar
Olsthoorn, J., Baglaenko, A. & Stastna, M. 2013 Analysis of asymmetries in propagating mode-2 waves. Nonlin. Processes Geophys. 20, 59–69.10.5194/npg-20-59-2013Google Scholar
Osborne, A. R. & Burch, T. L. 1980 Internal solitons in the Andaman Sea. Science 208, 451–460.10.1126/science.208.4443.451Google Scholar
Salloum, M., Knio, O. M. & Brandt, A. 2012 Numerical simulation of mass transport in internal solitary waves. Phys. Fluids 24, 016602.10.1063/1.3676771Google Scholar
Schmidt, N. P.1998 Generation, propagation and dissipation of second mode internal solitary waves. PhD thesis, University of Canterbury.Google Scholar
Shroyer, E. L., Moum, J. N. & Nash, J. D. 2010 Mode 2 waves on the continental shelf: ephemeral components of the nonlinear internal wavefield. J. Geophys. Res. 115, C07001.10.1029/2009JC005605Google Scholar
Stamp, A. P. & Jacka, M. 1995 Deep-water internal solitary waves. J. Fluid Mech. 305, 347–371.10.1017/S0022112095004654Google Scholar
Stastna, M. & Lamb, K. G. 2002 Large fully nonlinear internal solitary waves: the effect of background current. Phys. Fluids 14 (9), 2987–2999.10.1063/1.1496510Google Scholar
Stastna, M. & Peltier, W. R. 2005 On the resonant generation of large-amplitude internal solitary and solitary-like waves. J. Fluid Mech. 543, 267–292.10.1017/S002211200500652XGoogle Scholar
Subich, C. J., Lamb, K. G. & Stastna, M. 2013 Simulation of the Navier–Stokes equations in three dimensions with a spectral collocation method. Intl J. Numer. Meth. Fluids 73, 103–129.10.1002/fld.3788Google Scholar
Sutherland, B. R., Barrett, K. J. & Ivey, G. N. 2013 Shoaling internal solitary waves. J. Geophys. Res. 118, 4111–4124.Google Scholar
Terez, D. E. & Knio, O. M. 1998 Numerical simulations of large-amplitude internal solitary waves. J. Fluid Mech. 362, 53–82.10.1017/S0022112098008799Google Scholar
Terletska, K., Jung, K. T., Talipova, T., Maderich, V., Brovchenko, I. & Grimshaw, R. 2016 Internal breather-like wave generation by second mode solitary wave interaction with a step. Phys. Fluids 28, 116602.10.1063/1.4967203Google Scholar
Van Gastel, P., Ivey, G. N., Meuleners, M. J., Antenucci, J. P. & Fringer, O. 2009 The variability of the large-amplitude internal wave field on the Australian North West Shelf. Cont. Shelf Res. 29, 1373–1383.10.1016/j.csr.2009.02.006Google Scholar
Yang, Y. J., Fang, Y. C., Chang, M-H., Ramp, S. R., Kao, C-C. & Tang, T. Y. 2009 Observations of second baroclinic mode internal solitary waves on the continental slope of the northern South China Sea. J. Geophys. Res. 114, C1003.10.1029/2009JC005318Google Scholar
Yang, Y. J., Fang, Y. C., Tang, T. Y. & Ramp, S. R. 2010 Convex and concave types of second baroclinic mode internal solitary waves. Nonlin. Processes Geophys. 17, 605–614.10.5194/npg-17-605-2010Google Scholar
Yuan, C., Grimshaw, R. & Johnson, E. 2018 The evolution of second mode internal solitary waves over variable topography. J. Fluid Mech. 836, 238–259.10.1017/jfm.2017.812Google Scholar