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Shoaling mode-2 internal solitary-like waves

  • Magda Carr (a1), Marek Stastna (a2), Peter A. Davies (a3) and Koen J. van de Wal (a4)

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.

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Corresponding author

Email address for correspondence: magda.carr@ncl.ac.uk

References

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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, 321344.10.1017/jfm.2011.432
Aghsaee, P., Boegman, L. & Lamb, K. G. 2010 Breaking of shoaling internal solitary waves. J. Fluid Mech. 659, 289317.
Akylas, T. R. & Grimshaw, R. S. J. 1992 Solitary internal waves with oscillatory tails. J. Fluid Mech. 242, 279298.
Archetti, R. & Brocchini, M. 2002 An integral swash zone model with friction: an experimental and numerical investigation. Coast. Engng 45, 89110.
Arthur, R. S. & Fringer, O. B. 2014 The dynamics of breaking internal solitary waves on slopes. J. Fluid Mech. 761, 360398.10.1017/jfm.2014.641
Boegman, L. & Ivey, G. N. 2009 Flow separation and resuspension beneath shoaling nonlinear internal waves. J. Geophys. Res. 114, C02018.10.1029/2007JC004411
Boegman, L., Ivey, G. N. & Imberger, J. 2005 The degeneration of internal waves in lakes with sloping topography. Limnol. Oceanogr. 50 (5), 16201637.10.4319/lo.2005.50.5.1620
Boegman, L. & Stastna, M. 2019 Sediment resuspension and transport by internal solitary waves. Annu. Rev. Fluid Mech. 51, 129154.10.1146/annurev-fluid-122316-045049
Bogucki, D. J., Dickey, T. & Redekopp, L. G. 1997 Sediment resuspension and mixing by resonantly generated internal solitary waves. J. Phys. Oceanogr. 27, 11811196.
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.
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/ncomms13606
Brandt, A. & Shipley, K. R. 2014 Laboratory experiments on mass transport by large amplitude mode-2 internal solitary waves. Phys. Fluids 26, 046601.10.1063/1.4869101
Cacchione, D. A., Pratson, L. F. & Ogston, A. S. 2002 The shaping of continental slopes by internal tides. Science 296, 724727.
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.
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, 066603.10.1063/1.2931693
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.
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.
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, 533547.10.1088/0957-0233/18/3/001
Davis, R. E. & Acrivos, A. 1967 Solitary internal waves in deep water. J. Fluid Mech. 29, 593607.
Deepwell, D. & Stastna, M. 2016 Mass transport by mode-2 internal solitary-like waves. Phys. Fluids 28, 056606.10.1063/1.4948544
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, 076601.10.1063/1.4994590
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, 784812.
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, 11051130.10.1109/JOE.2004.836998
Gavrilov, N. V. & Lyapidevski, V. Y. 2009 Symmetric solitary waves in a two-layer fluid. Dokl. Phys. 54 (11), 508511.
Gavrilov, N. V. & Lyapidevski, V. Y. 2011 Large amplitude internal solitary waves over a shelf. Nat. Hazards Earth Syst. Sci. 11, 1725.
Grimshaw, R. 2014 Internal solitary waves. In Chapter 1 in Environmental Stratified Flows, Topics in Environmental Fluid Mechanics, vol. 3. Springer.
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, 17671782.
Grue, J., Jensen, A., Rusås, P.-O. & Sveen, J. K. 1999 Properties of large-amplitude internal waves. J. Fluid Mech. 380, 257278.
Guo, C. & Chen, X. 2012 Numerical investigation of large amplitude second mode internal solitary waves over a slope-shelf topography. Ocean Model. 42, 8091.
Hammond, D. A. & Redekopp, L. G. 1998 Local and global instability properties of separation bubbles. Eur. J. Mech. (B/Fluids) 17, 145164.10.1016/S0997-7546(98)80056-3
Helfrich, K. R. 1992 Internal solitary wave breaking and run-up on a uniform slope. J. Fluid Mech. 243, 133154.
Helfrich, K. R. & Melville, W. K. 2006 Long nonlinear internal waves. Annu. Rev. Fluid Mech. 38, 395425.
Horn, D. A., Imberger, J. & Ivey, G. N. 2001 The degeneration of large-scale interfacial gravity waves in lakes. J. Fluid Mech. 434, 181207.
Hosegood, P. & van Haren, H. 2004 Near-bed solibores over the continental slope in the Faeroe-Shetland channel. Deep Sea Res. 51, 29432971.
Hughes, S. A. 2004a Estimation of wave run-up on smooth, impermeable slopes using the wave momentum flux parameter. Coast. Engng 51, 10851104.10.1016/j.coastaleng.2004.07.026
Hughes, S. A. 2004b Wave momentum flux parameter: a descriptor for nearshore waves. Coast. Engng 51, 10671084.
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, 111.
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, 1953.10.1017/S0022112085003081
Kao, T. W. & Pao, H.-P. 1980 Wake collapse in the thermocline and internal solitary waves. J. Fluid Mech. 97, 115127.
Kundu, P. K., Cohen, I. M. & Dowling, D. R. 2015 Fluid Mechanics. Academic Press.
Lamb, K. G. 2014 Internal wave breaking and dissipation mechanisms on the continental slope/shelf. Annu. Rev. Fluid Mech. 46, 231254.
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, 1827.
MacIntyre, S. K., Flynn, M., Jellison, R. & Romero, J. R. 1999 Boundary mixing and nutrient fluxes in Mono Lake, California. Limnol. Oceanogr. 4, 512529.10.4319/lo.1999.44.3.0512
MacKinnon, J. A. & Gregg, M. C. 2003 Shear and baroclinic energy flux on the summer New England shelf. J. Phys. Oceanogr. 33, 14621475.
Magalhaes, J. M. & da Silva, J. C. B. 2018 Internal solitary waves in the andaman sea: new insights from sar imagery. Remote Sens. 10, 861877.
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, 4764.
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), 522538.
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.4.014801
Nash, J. D. & Moum, J. N. 2005 River plumes as a source of large-amplitude internal waves in the coastal ocean. Nature 437, 400403.
Olsthoorn, J., Baglaenko, A. & Stastna, M. 2013 Analysis of asymmetries in propagating mode-2 waves. Nonlin. Processes Geophys. 20, 5969.
Osborne, A. R. & Burch, T. L. 1980 Internal solitons in the Andaman Sea. Science 208, 451460.
Salloum, M., Knio, O. M. & Brandt, A. 2012 Numerical simulation of mass transport in internal solitary waves. Phys. Fluids 24, 016602.10.1063/1.3676771
Schmidt, N. P.1998 Generation, propagation and dissipation of second mode internal solitary waves. PhD thesis, University of Canterbury.
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.
Stamp, A. P. & Jacka, M. 1995 Deep-water internal solitary waves. J. Fluid Mech. 305, 347371.10.1017/S0022112095004654
Stastna, M. & Lamb, K. G. 2002 Large fully nonlinear internal solitary waves: the effect of background current. Phys. Fluids 14 (9), 29872999.
Stastna, M. & Peltier, W. R. 2005 On the resonant generation of large-amplitude internal solitary and solitary-like waves. J. Fluid Mech. 543, 267292.10.1017/S002211200500652X
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, 103129.
Sutherland, B. R., Barrett, K. J. & Ivey, G. N. 2013 Shoaling internal solitary waves. J. Geophys. Res. 118, 41114124.
Terez, D. E. & Knio, O. M. 1998 Numerical simulations of large-amplitude internal solitary waves. J. Fluid Mech. 362, 5382.10.1017/S0022112098008799
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.
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, 13731383.10.1016/j.csr.2009.02.006
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, C1003.10.1029/2009JC005318
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, 605614.10.5194/npg-17-605-2010
Yuan, C., Grimshaw, R. & Johnson, E. 2018 The evolution of second mode internal solitary waves over variable topography. J. Fluid Mech. 836, 238259.
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Shoaling mode-2 internal solitary-like waves

  • Magda Carr (a1), Marek Stastna (a2), Peter A. Davies (a3) and Koen J. van de Wal (a4)

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