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Extreme long waves over a varying bathymetry

Published online by Cambridge University Press:  17 September 2019

James G. Herterich Open the ORCID record for James G. Herterich [Opens in a new window]  and
Frédéric Dias Open the ORCID record for Frédéric Dias [Opens in a new window]
James G. Herterich*
Affiliation:
School of Mathematics and Statistics, University College Dublin, Dublin 4, Ireland Earth Institute, University College Dublin, Dublin 4, Ireland
Frédéric Dias
Affiliation:
School of Mathematics and Statistics, University College Dublin, Dublin 4, Ireland Earth Institute, University College Dublin, Dublin 4, Ireland CMLA, ENS Paris–Saclay, CNRS, Université Paris–Saclay, 94235 Cachan, France
*
†Email address for correspondence: [email protected]
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Abstract

Recent modelling work has shown that abrupt bathymetric transitions can produce dramatic amplifications of long waves, under the influence of both nonlinearity and dispersion. Here, the evolution of wave packets towards a vertical wall over a varying bathymetry is investigated with a one-dimensional conformal-mapping spectral code. In this system, wave breaking, runup and reflection, wave interference and bathymetric effects are highlighted. Wave breaking is examined with respect to geometric, kinematic and energetic conditions, with consistent results. The breaking strength is characterized for spilling and plunging based on initial wave period and amplitude. Non-breaking waves are amplified by reflection, interference and the bathymetry leading to large runups. In a typical example inspired by a real-world bathymetry, the maximum runup amplification approaches a factor of 12 – large enough for a 3 m amplitude wave to overtop a 30 m cliff.

JFM classification

Waves/Free-surface Flows: Surface gravity waves Waves/Free-surface Flows: Wave breaking
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 878 , 10 November 2019 , pp. 481 - 501
Copyright
© 2019 Cambridge University Press 

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References

Akrish, G., Rabinovitch, O. & Agnon, Y. 2016a Extreme run-up events on a vertical wall due to nonlinear evolution of incident wave groups. J. Fluid Mech. 797, 644–664.Google Scholar
Akrish, G., Schwartz, R., Rabinovitch, O. & Agnon, Y. 2016b Impact of extreme waves on a vertical wall. Nat. Hazards 84 (2), 637–653.Google Scholar
Babanin, A. V., Chalikov, D., Young, I. R. & Savelyev, I. 2010 Numerical and laboratory investigation of breaking of steep two-dimensional waves in deep water. J. Fluid Mech. 644, 433–463.Google Scholar
Banner, M. L., Barthelemy, X., Fedele, F., Allis, M., Benetazzo, A., Dias, F. & Peirson, W. L. 2014 Linking reduced breaking crest speeds to unsteady nonlinear water wave group behavior. Phys. Rev. Lett. 112 (11), 114502.Google Scholar
Barthelemy, X., Banner, M. L., Peirson, W. L., Fedele, F., Allis, M. & Dias, F. 2018 On a unified breaking onset threshold for gravity waves in deep and intermediate depth water. J. Fluid Mech. 841, 463–488.Google Scholar
Benoit, M., Dias, F., Herterich, J. G. & Scolan, Y.-M. 2018 A discriminating test-case for the simulation of the propagation and run-up of tsunami-type wave trains. In 16th Journées de l’Hydrodynamique, Marseille, 27–29 November 2018; http://website.ec-nantes.fr/actesjh/images/16JH/16JH-actes.htm.Google Scholar
Bolles, C. T., Speer, K. & Moore, M. N. J. 2019 Anomalous wave statistics induced by abrupt depth change. Phys. Rev. Fluids 4 (1), 011801.Google Scholar
Brennan, J., Clancy, C., Harrington, J., Cox, R. & Dias, F. 2017 Analysis of the pressure at a vertical barrier due to extreme wave run-up over variable bathymetry. Theor. Appl. Mech. Lett. 7 (5), 269–275.Google Scholar
Brennan, J., Viotti, C. & Dias, F. 2014 Pressure fluctuations on a vertical wall during extreme run-up cycles. In ASME 2014 33rd International Conference on Ocean, Offshore and Arctic Engineering, p. V04AT02A038. American Society of Mechanical Engineers.Google Scholar
Brühl, M. & Oumeraci, H. 2016 Analysis of long-period cosine-wave dispersion in very shallow water using nonlinear Fourier transform based on KdV equation. Appl. Ocean Res. 61, 81–91.Google Scholar
Carbone, F., Dutykh, D., Dudley, J. M. & Dias, F. 2013 Extreme wave runup on a vertical cliff. Geophys. Res. Lett. 40 (12), 3138–3143.Google Scholar
Chambarel, J., Kharif, C. & Touboul, J. 2009 Head-on collision of two solitary waves and residual falling jet formation. Nonlinear Process. Geophys. 16, 111–122.Google Scholar
Chen, Y. Y., Kharif, C., Yang, J. H., Hsu, H. C., Touboul, J. & Chambarel, J. 2015 An experimental study of steep solitary wave reflection at a vertical wall. Eur. J. Mech. (B/Fluids) 49, 20–28.Google Scholar
Choi, W. & Camassa, R. 1999 Exact evolution equations for surface waves. J. Engng Mech. 125 (7), 756–760.Google Scholar
Cooker, M. J. & Peregrine, D. H. 1995 Pressure-impulse theory for liquid impact problems. J. Fluid Mech. 297, 193–214.Google Scholar
Craig, W. 2006 Surface water waves and tsunamis. J. Dyn. Differ. Equ. 18 (3), 525–549.Google Scholar
Cyril, J. & Galvin, C. J. 1972 Wave breaking in shallow water. In Waves on Beaches and Resulting Sediment Transport, pp. 413–456. Elsevier.Google Scholar
Derakhti, M., Banner, M. L. & Kirby, J. T. 2018 Predicting the breaking strength of gravity water waves in deep and intermediate depth. J. Fluid Mech. 848, R2 1–11.Google Scholar
Dias, F. & Ghidaglia, J. M. 2018 Slamming: recent progress in the evaluation of impact pressures. Annu. Rev. Fluid Mech. 50, 243–273.Google Scholar
Drazen, D. A., Melville, W. K. & Lenain, L. 2008 Inertial scaling of dissipation in unsteady breaking waves. J. Fluid Mech. 611, 307–332.Google Scholar
Duncan, J. H. 1983 The breaking and non-breaking wave resistance of a two-dimensional hydrofoil. J. Fluid Mech. 126, 507–520.Google Scholar
Fedele, F. 2014 Geometric phases of water waves. Europhys. Lett. 107 (6), 69001.Google Scholar
Geological Survey Ireland2002–2016 INSS and INFOMAR bathymetry mapping. http://maps.marine.ie/infomar/, online; accessed 2016-04-18.Google Scholar
Goda, Y. 1975 New wave pressure formulae for composite breakwaters. In Coastal Engineering 1974, pp. 1702–1720. American Society of Civil Engineers.Google Scholar
Goda, Y. 2010 Reanalysis of regular and random breaking wave statistics. Coast. Engng J. 52 (01), 71–106.Google Scholar
Grilli, S. T., Svendsen, I. A. & Subramanya, R. 1997 Breaking criterion and characteristics for solitary waves on slopes. J. Waterway, Port Coast. Ocean Engng 123 (3), 102–112.Google Scholar
Grue, J. & Kolaas, J. 2017 Experimental particle paths and drift velocity in steep waves at finite water depth. J. Fluid Mech. 810, R1 1–10.Google Scholar
Grue, J., Kolaas, J. & Jensen, A. 2014 Velocity fields in breaking-limited waves on finite depth. Eur. J. Mech. (B/Fluids) 47, 97–107.Google Scholar
Hansom, J. D., Barltrop, N. D. P. & Hall, A. M. 2008 Modelling the processes of cliff-top erosion and deposition under extreme storm waves. Mar. Geol. 253 (1), 36–50.Google Scholar
Herterich, J. G., Cox, R. & Dias, F. 2018 How does wave impact generate large boulders? Modelling hydraulic fracture of cliffs and shore platforms. Mar. Geol. 399, 34–46.Google Scholar
Herterich, J. G. & Dias, F. 2017a Wave breaking and runup of long waves approaching a cliff over a variable bathymetry. Proc. IUTAM 25, 18–27.Google Scholar
Herterich, J. G. & Dias, F. 2017b Wave height to depth measures during extreme runup. In The 27th International Ocean and Polar Engineering Conference, vol. 3, pp. 1390–1394. International Society of Offshore and Polar Engineers.Google Scholar
Hwang, P. A., Xu, D. & Wu, J. 1989 Breaking of wind-generated waves: measurements and characteristics. J. Fluid Mech. 202, 177–200.Google Scholar
Iafrati, A., De Vita, F. & Verzicco, R. 2019 Effects of the wind on the breaking of modulated wave trains. Eur. J. Mech. (B/Fluids) 73, 6–23.Google Scholar
Kurnia, R. & van Groesen, E. 2014 High order Hamiltonian water wave models with wave-breaking mechanism. Coast. Engng 93, 55–70.Google Scholar
Kurnia, R. & van Groesen, E. 2017 Localization for spatial–spectral implementations of 1D analytic Boussinesq equations. Wave Motion 72, 113–132.Google Scholar
Longo, S., Clavero, M., Chiapponi, L. & Losada, M. A. 2017 Invariants of turbulence Reynolds stress and of dissipation tensors in regular breaking waves. Water 9 (11), w9110893.Google Scholar
Lugni, C., Brocchini, M. & Faltinsen, O. M. 2006 Wave impact loads: the role of the flip-through. Phys. Fluids 18 (12), 122101.Google Scholar
Mader, C. L. 1974 Numerical simulation of tsunamis. J. Phys. Oceanogr. 4 (1), 74–82.Google Scholar
Madsen, P. A. & Fuhrman, D. R. 2008 Run-up of tsunamis and long waves in terms of surf-similarity. Coast. Engng 55 (3), 209–223.Google Scholar
Madsen, P. A., Fuhrman, D. R. & Schäffer, H. A. 2008 On the solitary wave paradigm for tsunamis. J. Geophys. Res. 113 (C12), C12012.Google Scholar
Massel, S. R. 1996 On the largest wave height in water of constant depth. Ocean Engng 23 (7), 553–573.Google Scholar
Mayer, R. H. & Kriebel, D. L. 1995 Wave runup on composite-slope and concave beaches. In Coastal Engineering 1994, pp. 2325–2339. American Society of Civil Engineers.Google Scholar
Mei, C. C., Stiassnie, M. & Yue, D. K.-P. 1989 Theory and Applications of Ocean Surface Waves: Part 1: Linear Aspects. Part 2: Nonlinear Aspects. World Scientific.Google Scholar
Melville, W. K. 1996 The role of surface-wave breaking in air–sea interaction. Annu. Rev. Fluid Mech. 28 (1), 279–321.Google Scholar
Montoya, L. & Lynett, P. 2018 Tsunami versus infragravity surge: comparison of the physical character of extreme runup. Geophys. Res. Lett. 45 (23), 12–982.Google Scholar
Munk, W. H. 1949 The solitary wave theory and its application to surf problems. Ann. N.Y. Acad. Sci. 51 (3), 376–424.Google Scholar
Nelson, R. C. 1994 Depth limited design wave heights in very flat regions. Coast. Engng 23 (1–2), 43–59.Google Scholar
Ning, D., Li, X. & Zhang, C. 2018 Nonlinear simulation of wave train impact on a vertical seawall. Water 10 (8), 986.Google Scholar
Peregrine, D. H. 2003 Water–wave impact on walls. Annu. Rev. Fluid Mech. 35 (1), 23–43.Google Scholar
Perlin, M., Choi, W. & Tian, Z. 2013 Breaking waves in deep and intermediate waters. Annu. Rev. Fluid Mech. 45, 115–145.Google Scholar
Phillips, O. M. 1985 Spectral and statistical properties of the equilibrium range in wind-generated gravity waves. J. Fluid Mech. 156, 505–531.Google Scholar
Riedel, H. P. & Byrne, A. P. 1987 Random breaking waves – horizontal seabed. In Coastal Engineering 1986, pp. 903–908. American Society of Civil Engineers.Google Scholar
Ryu, Y. & Chang, K.-A. 2008 Green water void fraction due to breaking wave impinging and overtopping. Exp. Fluids 45 (5), 883–898.Google Scholar
Ryu, Y., Chang, K.-A. & Mercier, R. 2007 Runup and green water velocities due to breaking wave impinging and overtopping. Exp. Fluids 43 (4), 555–567.Google Scholar
Saket, A., Peirson, W. L., Banner, M. L. & Allis, M. J. 2018 On the influence of wave breaking on the height limits of two-dimensional wave groups propagating in uniform intermediate depth water. Coast. Engng 133, 159–165.Google Scholar
Saket, A., Peirson, W. L., Banner, M. L., Barthelemy, X. & Allis, M. J. 2017 On the threshold for wave breaking of two-dimensional deep water wave groups in the absence and presence of wind. J. Fluid Mech. 811, 642–658.Google Scholar
Saville, T. Jr 1957 Wave run-up on composite slopes. Coast. Engng Proc. 1 (6), 41.Google Scholar
Schwendeman, M. S. & Thomson, J. 2017 Sharp-crested breaking surface waves observed from a ship-based stereo video system. J. Phys. Oceanogr. 47 (4), 775–792.Google Scholar
Seiffert, B. R. & Ducrozet, G. 2018 Simulation of breaking waves using the high-order spectral method with laboratory experiments: wave-breaking energy dissipation. Ocean Dyn. 68, 65–89.Google Scholar
Seiffert, B. R., Ducrozet, G. & Bonnefoy, F. 2017 Simulation of breaking waves using the high-order spectral method with laboratory experiments: wave-breaking onset. Ocean Model. 119, 94–104.Google Scholar
Stansell, P. & MacFarlane, C. 2002 Experimental investigation of wave breaking criteria based on wave phase speeds. J. Phys. Oceanogr. 32 (5), 1269–1283.Google Scholar
Toffoli, A., Babanin, A., Onorato, M. & Waseda, T. 2010 Maximum steepness of oceanic waves: field and laboratory experiments. Geophys. Res. Lett. 37 (5), L05603.Google Scholar
Trillo, S., Deng, G., Biondini, G., Klein, M., Clauss, G. F., Chabchoub, A. & Onorato, M. 2016 Experimental observation and theoretical description of multisoliton fission in shallow water. Phys. Rev. Lett. 117 (14), 144102.Google Scholar
Viotti, C., Carbone, F. & Dias, F. 2014 Conditions for extreme wave runup on a vertical barrier by nonlinear dispersion. J. Fluid Mech. 748, 768–788.Google Scholar
Viotti, C. & Dias, F. 2014 Extreme waves induced by strong depth transitions: fully nonlinear results. Phys. Fluids 26 (5), 051705.Google Scholar
Viotti, C., Dutykh, D. & Dias, F. 2014 The conformal-mapping method for surface gravity waves in the presence of variable bathymetry and mean current. Proc. IUTAM 11, 110–118.Google Scholar
Viotti, C., Dutykh, D., Dudley, J. M. & Dias, F. 2013 Emergence of coherent wave groups in deep-water random sea. Phys. Rev. E 87 (6), 063001.Google Scholar
Weggel, J. R. 1973 Maximum breaker height for design. In Coastal Engineering 1972, pp. 419–432.Google Scholar
Wu, C. H. & Nepf, H. M. 2002 Breaking criteria and energy losses for three-dimensional wave breaking. J. Geophys. Res. 107 (C10), 3177.Google Scholar
Wu, Y. H. & Tian, J.-W. 2000 Mathematical analysis of long-wave breaking on open channels with bottom friction. Ocean Engng 27 (2), 187–201.Google Scholar
Xu, D., Hwang, P. A. & Wu, J. 1986 Breaking of wind-generated waves. J. Phys. Oceanogr. 16 (12), 2172–2178.Google Scholar