Hostname: page-component-848d4c4894-ndmmz Total loading time: 0 Render date: 2024-04-30T17:47:16.566Z Has data issue: false hasContentIssue false
  • English
  • Français

Nonlinear wave run-up in bays of arbitrary cross-section: generalization of the Carrier–Greenspan approach

Published online by Cambridge University Press:  30 April 2014

Alexei Rybkin ,
Efim Pelinovsky  and
Ira Didenkulova
Alexei Rybkin
Affiliation:
Department of Mathematics and Statistics, University of Alaska Fairbanks, Fairbanks, AK 99775, USA
Efim Pelinovsky
Affiliation:
Department of Nonlinear Geophysical Processes, Institute of Applied Physics, Uljanov Street 46, 603950, Nizhny Novgorod, Russia Institute for Analysis, J. Kepler University, Altenbergerstr. 69, 4040 Linz, Austria National Research University Higher School of Economics, Gr. Pechersky Street 25/12, 603950, Nizhny Novgorod, Russia Nizhny Novgorod State Technical University n.a. R. E. Alekseev, Minin Street 24, 603950, Nizhny Novgorod, Russia
Ira Didenkulova*
Affiliation:
Nizhny Novgorod State Technical University n.a. R. E. Alekseev, Minin Street 24, 603950, Nizhny Novgorod, Russia Institute of Cybernetics at Tallinn University of Technology, Akadeemia tee 21, 12618, Tallinn, Estonia MARUM – Center for Marine Environmental Sciences, University of Bremen, Leobener Str. D-28359 Bremen, Germany
*
†Email address for correspondence: ira@cs.ioc.ee
Get access Rights & Permissions [Opens in a new window]

Abstract

We present an exact analytical solution of the nonlinear shallow water theory for wave run-up in inclined channels of arbitrary cross-section, which generalizes previous studies on wave run-up for a plane beach and channels of parabolic cross-section. The solution is found using a hodograph-type transform, which extends the well-known Carrier–Greenspan transform for wave run-up on a plane beach. As a result, the nonlinear shallow water equations are reduced to a single one-dimensional linear wave equation for an auxiliary function and all physical variables can be expressed in terms of this function by purely algebraic formulas. In the special case of a U-shaped channel this equation coincides with a spherically symmetric wave equation in space, whose dimension is defined by the channel cross-section and can be fractional. As an example, the run-up of a sinusoidal wave on a beach is considered for channels of several different cross-sections and the influence of the cross-section on wave run-up characteristics is studied.

JFM classification

Geophysical and Geological Flows: Coastal engineering Waves/Free-surface Flows: Solitary waves Waves/Free-surface Flows: Surface gravity waves
Type
Papers
Information
Journal of Fluid Mechanics , Volume 748 , 10 June 2014 , pp. 416 - 432
Copyright
© 2014 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Antuono, M. & Brocchini, M. 2007 The boundary value problem for the nonlinear shallow water equation. Stud. Appl. Maths 119, 71–91.Google Scholar
Antuono, M. & Brocchini, M. 2008 Maximum run-up, breaking conditions and dynamical forces in the swash zone: a boundary value approach. Coast. Engng 55, 732–740.Google Scholar
Antuono, M. & Brocchini, M. 2010 Solving the nonlinear shallow-water equations in physical space. J. Fluid Mech. 643, 207–232.Google Scholar
Brocchini, M. 1998 The run-up of weakly-two-dimensional solitary pulses. Nonlinear Process. Geophys. 5, 27–38.Google Scholar
Brocchini, M. & Gentile, R. 2001 Modelling the run-up of significant wave groups. Cont. Shelf Res. 21, 1533–1550.Google Scholar
Brocchini, M. & Peregrine, D. H. 1996 Integral flow properties of the swash zone and averaging. J. Fluid Mech. 317, 241–273.Google Scholar
Cally, P. S. 2012 Alfvén reflection and reverberation in the solar atmosphere. Solar Phys. 280 (1), 33–50.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 run-up and draw-down on a plane beach. J. Fluid Mech. 475, 79–99.Google Scholar
Choi, B. H., Hong, S. J. & Pelinovsky, E. 2006 Distribution of runup heights of the December 26, 2004 tsunami in the Indian Ocean. Geophys. Res. Lett. 33 (13), L13601; doi: 10.1029/2006GL025867.Google Scholar
Choi, B. H., Min, B. I., Pelinovsky, E., Tsuji, Y. & Kim, K. O. 2012 Comparable analysis of the distribution functions of runup heights of the 1896, 1933 and 2011 Japanese tsunamis in the Sanriku area. Nat. Hazards Earth Syst. Sci. 12, 1463–1467.CrossRefGoogle Scholar
Choi, B. H., Pelinovsky, E., Kim, D. C. & Didenkulova, I. 2008 Two- and three-dimensional computation of solitary wave runup on non-plane beach. Nonlinear Process. Geophys. 15, 489–502.Google Scholar
Courant, R. & Hilbert, D. 1989 Methods of Mathematical Physics. vol. I. Wiley.Google Scholar
Denissenko, P., Didenkulova, I., Pelinovsky, E. & Pearson, J. 2011 Influence of the nonlinearity on statistical characteristics of long wave runup. Nonlinear Process. Geophys. 18, 967–975.Google Scholar
Didenkulova, I. & Pelinovsky, E. 2008 Run-up of long waves on a beach: the influence of the incident wave form. Oceanology 48 (1), 1–6.CrossRefGoogle Scholar
Didenkulova, I. 2009 New trends in the analytical theory of long sea wave runup. In Applied Wave Mathematics: Selected Topics in Solids, Fluids, and Mathematical Methods (ed. Quak, E. & Soomere, T.), pp. 265–296. Springer.Google Scholar
Didenkulova, I., Kurkin, A. & Pelinovsky, E. 2007 Run-up of solitary waves on slopes with different profiles. Izv. Atmos. Ocean. Phys. 43 (3), 384–390.Google Scholar
Didenkulova, I. & Pelinovsky, E. 2009 Non-dispersive traveling waves in inclined shallow water channels. Phys. Lett. A 373 (42), 3883–3887.Google Scholar
Didenkulova, I. & Pelinovsky, E. 2011a Nonlinear wave evolution and runup in an inclined channel of a parabolic cross-section. Phys. Fluids 23 (8), 086602.Google Scholar
Didenkulova, I. & Pelinovsky, E. 2011b Rogue waves in nonlinear hyperbolic systems (shallow-water framework). Nonlinearity 24, R1–R18.Google Scholar
Didenkulova, I., Pelinovsky, E. & Sergeeva, A. 2011 Statistical characteristics of long waves nearshore. Coast. Engng 58 (1), 94–202.Google Scholar
Didenkulova, I., Pelinovsky, E. & Soomere, T. 2008 Run-up characteristics of tsunami waves of ‘unknown’ shapes. Pure Appl. Geophys. 165, 2249–2264.CrossRefGoogle Scholar
Dobrokhotov, S. Yu. & Tirozzi, B. 2010 Localized solutions of one-dimensional nonlinear shallow-water equations with velocity $c=(x)^{1/2}$ . Usp. Mat. Nauk 65, 77–180.Google Scholar
Dutykh, D., Labart, C. & Mitsotakis, D. 2011 Long wave runup on random beaches. Phys. Rev. Lett. 107, 184504.Google Scholar
Ezersky, A., Abcha, N. & Pelinovsky, E. 2013 Physical simulation of resonant wave run-up on a beach. Nonlinear Process. Geophys. 20, 35–40.CrossRefGoogle Scholar
Fritz, H. M., Phillips, D. A., Okayasu, A., Shimozono, T., Liu, H. J., Mohammed, F., Skanavis, V., Synolakis, C. E. & Takahashi, T. 2012 The 2011 Japan tsunami current velocity measurements from survivor videos at Kesennuma Bay using LiDAR. Geophys. Res. Lett. 39, L00G23.Google Scholar
Huang, N. E., Shen, Z. & Long, S. 1999 A new view of nonlinear water waves: the Hilbert spectrum. Annu. Rev. Fluid Mech. 31, 417–457.Google Scholar
Kânoğlu, U. 2004 Nonlinear evolution and runup-drawdown 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
Madsen, P. A. & Fuhrman, D. R. 2008 Run-up of tsunamis and periodic long waves in terms of surf-similarity. Coast. Engng 55, 209–223.Google Scholar
Morse, P. M. & Feshbach, H. 1953 Methods of Theoretical Physics. 2 vols. McGraw-Hill.Google Scholar
Okada, Y. 1985 Surface deformation due to shear and tensile faults in a half-space. Bull. Seismol. Soc. Am. 75, 1135–1154.Google Scholar
Pedersen, G. & Gjevik, B. 1983 Runup of solitary waves. J. Fluid Mech. 142, 283–299.Google Scholar
Pelinovsky, E. & Mazova, R. 1992 Exact analytical solutions of nonlinear problems of tsunami wave run-up on slopes with different profiles. Nat. Hazards 6, 227–249.Google Scholar
Renzi, E. & Sammarco, P. 2012 The influence of landslide shape and continental shelf on landslide generated tsunamis along a plane beach. Nat. Hazards Earth Syst. Sci. 12, 1503–1520.Google Scholar
Rudenko, O. V. & Soluyan, S. I. 1977 Theoretical Foundations of Nonlinear Acoustics. Consultants Bureau.Google Scholar
Sammarco, P. & Renzi, E. 2008 Landslide tsunamis propagating along a plane beach. J. Fluid Mech. 598, 107–119.CrossRefGoogle Scholar
Spielvogel, L. O. 1975 Runup of single waves on a sloping beach. J. Fluid Mech. 74, 685–694.Google Scholar
Stefanakis, T., Dias, F. & Dutykh, D. 2011 Local runup amplification by resonant wave interactions. Phys. Rev. Lett. 107, 124502.Google Scholar
Synolakis, C. E. 1987 The runup of solitary waves. J. Fluid Mech. 185, 523–545.Google Scholar
Synolakis, C. E. 1991 Tsunami runup on steep slopes: How good linear theory really is? Nat. Hazards 4, 221–234.Google Scholar
Synolakis, C. E. & Bernard, E. N. 2006 Tsunami science before and beyond Boxing Day 2004. Phil. Trans. R. Soc. A 364 (1845), 2231–2265.Google Scholar
Synolakis, C. E., Deb, M. K. & Skjelbreia, J. E. 1988 The anomalous behavior of the run-up of cnoidal waves. Phys. Fluids 31 (1), 3–5.Google Scholar
Tadepalli, S. & Synolakis, C. E. 1994 The runup of N-waves. 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
Zahibo, N., Pelinovsky, E., Golinko, V. & Osipenko, N. 2006 Tsunami wave runup on coasts of narrow bays. Intl J. Fluid Mech. Res. 33, 106–118.Google Scholar