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The near-shore behaviour of shallow-water waves with localized initial conditions

Published online by Cambridge University Press:  30 October 2007

DAVID PRITCHARD  and
LAURA DICKINSON
DAVID PRITCHARD
Affiliation:
Department of Mathematics, University of Strathclyde, 26 Richmond St, Glasgow G1 1XH, [email protected]
LAURA DICKINSON
Affiliation:
Division of Civil Engineering, University of Dundee, Dundee DD1 4HN, [email protected]
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Abstract

We consider the behaviour of solutions to the nonlinear shallow-water equations which describe wave runup on a plane beach, concentrating on the behaviour at and just behind the moving shoreline. We develop regular series expansions for the hydrodynamic variables behind the shoreline, which are valid for any smooth initial condition for the waveform. We then develop asymptotic descriptions of the shoreline motion under localized initial conditions, in particular a localized Gaussian waveform: we obtain estimates for the maximum runup and drawdown of the wave, for its maximum velocities and the forces it is able to exert on objects in its path, and for the conditions under which such a wave breaks down. We show how these results may be extended to include initial velocity conditions and initial waveforms which may be approximated as the sum of several Gaussians. Finally, we relate these results tentatively to the observed behaviour of a tsunami.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 591 , 25 November 2007 , pp. 413 - 436
Copyright
Copyright © Cambridge University Press 2007

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References

REFERENCES

Antuono, M. & Brocchini, M. 2007 The boundary value problem for the nonlinear shallow water equations. Stud. Appl. Maths 119, 73–93.Google Scholar
Carrier, G. F. & Greenspan, H. P. 1958 Water waves of finite amplitude on a sloping beach. J. Fluid Mech. 4, 97–109.Google Scholar
Carrier, G. F., Wu, T. T. & Yeh, H. 2003 Tsunami runup and drawdown on a plane beach. J. Fluid Mech. 475, 79–99.Google Scholar
Cousins, W. J., Power, W. L., Palmer, N. G., Reese, S., Tejakusuma, I. & Nugrahdi, S. 2006 South Java Tsunami of 17th July 2006: reconnaissance report. Tech. Rep. 2006/33. GNS Science.Google Scholar
Dawson, A. G. & Shi, S. 2000 Tsunami deposits. Pure Appl. Geophys. 157, 875–897.Google Scholar
DCRC 2006 Modeling a tsunami generated by the July 17, 2006 earthquake south of Java, Indonesia. Disaster Control Research Centre, Tohoku university (published online at http://www.tsunami.civil.tohoku.ac.jp/hokusai2/disaster/06_Java/July17Java.html.)Google Scholar
Dickinson, L. 2005 The sedimentary signature of long waves on coasts. Master's thesis, University of Cambridge.Google Scholar
Dyer, K. R. 1986 Coastal and Estuarine Sediment Dynamics. Wiley.Google Scholar
Gradshteyn, I. S. & Ryzhik, I. M. 2000 Table of Integrals, Series and Products, 6th edn. Academic.Google Scholar
Greenspan, H. P. 1958 On the breaking of water waves of finite amplitude on a sloping beach. J. Fluid Mech. 4, 330–334.Google Scholar
Grilli, S. T., Svendsen, I. A. & Subramanya, R. 1997 Breaking criterion and characteristics for solitary waves on slopes. J. Waterway Port Coastal Ocean Engng 123 (3), 102–112.Google Scholar
Guard, P. A., Baldock, T. & Nielsen, P. 2005 General solutions for the initial runup of a breaking tsunami front. In Intl Symp. on Disaster Relief on Coasts. Monash University, Australia.Google Scholar
Heller, V., Unger, J. & Hager, W. 2005 Tsunami runup – a hydraulic perspective. J. Hydraul. Engng 131 (9), 743–747.Google Scholar
Hoefel, F. & Elgar, S. 2003 Wave-induced sediment transport and sandbar migration. Science 299, 1885–1887.Google Scholar
Jensen, A., Pedersen, G. K. & Wood, D. J. 2003 An experimental study of wave runup at a steep beach. J. Fluid Mech. 486, 161–188.Google Scholar
Kânoğlu, U. 2004 Nonlinear evolution and runup–rundown of long waves over a sloping beach. J. Fluid Mech. 513, 363–372.Google Scholar
Kânoğlu, U. & Synolakis, C. 2006 Initial value problem solution of nonlinear shallow water-wave equations. Phys. Rev. Lett. 97, 148501.Google Scholar
Kongko, W. et al. 2006 Rapid survey on tsunami Jawa 17 July 2006. Tech. Rep. UNESCO Report (published online at http://ioc3.unesco.org/itic/files/tsunami-java170706_e.pdf.)Google Scholar
Lavigne, F., Gomez, C., Giffo, M., Wassmer, P., Hoebreck, C., Mardiatno, D., Prioyono, J. & Paris, R. 2007 Field observations of the 17 July 2006 tsunami in Java. Nat. Haz. Earth Sys. Sci. 7, 177–183.Google Scholar
Luccio, P. A., Voropayev, S. I., Fernando, H. J. S., Boyer, D. L. & Houston, W. N. 1998 The motion of cobbles in the swash zone on an impermeable slope. Coastal Engng 33, 41–60.Google Scholar
Meyer, R. E. 1986 a On the shore singularity of water waves. I. The local model. Phys. Fluids 29, 3152–3163.Google Scholar
Meyer, R. E. 1986 b Regularity for a singular conservation law. Adv. Appl. Math. 7, 465–501.Google Scholar
Nott, J. 2003 Waves, coastal boulder deposits and the importance of the pre-transport setting. Earth Planet. Sci. Lett. 210, 269–276.Google Scholar
Peregrine, D. H. 1972 Equations for water waves and the approximations behind them. In Waves on Beaches and Resulting Sediment Transport (ed. Meyer, R. E.), chap. 3, pp. 95–121. Academic.Google Scholar
Polet, J. & Kanamori, H. 2000 Shallow subduction zone earthquakes and their tsunamigenic potential. Geophys. J. Intl 142, 684–702.Google Scholar
Pribadi, S., Fachrizal, I., Gunawan, I., Hermawan, I., Tsuji, Y. & Sub, S. 2006 Gempabumi dan tsunami selatan jawa barat, 17 juli 2006. Tech. Rep. BMG Jakarta, in Indonesian with English abstract. (Available online at http://aeic.bmg.go.id/file/Pangadaran_report_en.pdf.)Google Scholar
Synolakis, C. E. 1987 The runup of solitary waves. J. Fluid Mech. 185, 523–545.Google Scholar
Tadepalli, S. & Synolakis, C. E. 1994 The runup of N-waves on sloping beaches. Proc. R. Soc. Lond. A 445, 99–112.Google Scholar
Tinti, S. & Tonini, R. 2005 Analytical evolution of tsunamis induced by near-shore earthquakes on a constant-slope ocean. J. Fluid Mech. 535, 33–64.Google Scholar
Titov, V. V. & Synolakis, C. E. 1998 Numerical modelling of tidal wave runup. J. Waterway Port Coastal Ocean Engng 124 (4), 157–171.Google Scholar
Yeh, H. 2006 Maximum fluid forces in the tsunami runup zone. J. Waterway Port Coastal Ocean Engng 132 (6), 496–500.Google Scholar