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  • Journal of Fluid Mechanics, Volume 645
  • February 2010, pp. 27-57

Analytical solutions for tsunami runup on a plane beach: single waves, N-waves and transient waves

  • PER A. MADSEN (a1) and HEMMING A. SCHÄFFER (a1)
  • DOI: http://dx.doi.org/10.1017/S0022112009992485
  • Published online: 01 February 2010
Abstract

In the literature it has so far been common practice to consider solitary waves and N-waves (composed of solitary waves) as the appropriate model of tsunamis approaching the shoreline. Unfortunately, this approach is based on a tie between the nonlinearity and the horizontal length scale (or duration) of the wave, which is not realistic for geophysical tsunamis. To resolve this problem, we first derive analytical solutions to the nonlinear shallow-water (NSW) equations for the runup/rundown of single waves, where the duration and the wave height can be specified separately. The formulation is then extended to cover leading depression N-waves composed of a superposition of positive and negative single waves. As a result the temporal variations of the runup elevation, the associated velocity and breaking criteria are specified in terms of polylogarithmic functions. Finally, we consider incoming transient wavetrains generated by monopole and dipole disturbances in the deep ocean. The evolution of these wavetrains, while travelling a considerable distance over a constant depth, is influenced by weak dispersion and is governed by the linear Korteweg–De Vries (KdV) equation. This process is described by a convolution integral involving the Airy function. The runup on the plane sloping beach is then determined by another convolution integral involving the incoming time series at the foot of the slope. A good agreement with numerical model results is demonstrated.

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Email address for correspondence: prm@mek.dtu.dk
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M. J. Briggs , C. E. Synolakis , G. S. Harkins & D. R. Green 1995 Laboratory experiments of tsunami runup on a circular island. Pure Appl. Geophys. 144 (3/4), 569593.


I. I. Didenkulova , A. A. Kurkin & E. N. Pelinovsky 2007 Run-up of solitary waves on slopes with different profiles. Atmos. Ocean. Phys. 43 (3), 384390.

I. I. Didenkulova , N. Zahibo , A. A. Kurkin , B. V. Levin , E. N. Pelinovsky & T. Soomere 2006 Runup of nonlinearly deformed waves on a coast. Dokl. Earth Sci. 411 (8), 12411243.

D. R. Fuhrman & H. B. Bingham 2004 Numerical solutions of fully nonlinear and highly dispersive Boussinesq equations in two horizontal dimensions. Intl J. Numer. Methods Fluids 44, 231255.

D. R. Fuhrman & P. A. Madsen 2008 Simulation of nonlinear wave runup with a high-order Boussinesq model. Coast. Engng 55 (2), 139154.

E. L. Geist 1999 Local tsunamis and earthquake source parameters. Adv. Geophys. 39, 117209.

E. L. Geist & S. Yoshioka 1996 Source parameters controlling the generation and propagation of potential local tsunamis along the Cascadia Margin. Nat. Hazards 13, 151177.



L. Lewin 1991 Structural Properties of Polylogarithms. American Mathematical Society.

H. Lewy 1946 Water waves on sloping beaches. Bull. Am. Math. Soc. 52, 737755.

Y. Li & F. Raichlen 2001 Solitary wave runup on plane slopes. J. Waterway Port Coast. Ocean Engng 127 (1), 3344.




P. A. Madsen & D. R. Fuhrman 2008 Runup of tsunamis and long waves in terms of surf-similarity. Coast. Engng 55 (3), 209224.

P. A. Madsen , D. R. Fuhrman & H. A. Schäffer 2008 On the solitary wave paradigm for tsunamis. J. Geophys. Res. 113, C12012, 122.

P. A. Madsen , D. R. Fuhrman & B. Wang 2006 A Boussinesq-type method for fully nonlinear waves interacting with a rapidly varying bathymetry. Coast. Engng 53, 487504.

E. N. Pelinovsky & R. K. Mazova 1992 Exact analytical solutions of nonlinear problems of tsunami wave run-up on slopes with different profiles. Nat. Hazards 6, 227249.


C. E. Synolakis & E. N. Bernard 2006 Tsunami science before and beyond Boxing Day 2004. Phil. Trans. R. Soc. A 364, 22312265.

C. E. Synolakis , M. K. Deb & J. E. Skjelbreia 1988 The anomaleous behaviour of the runup of cnoidal waves. Phys. Fluids 31 (1), 35.

S. Tadepalli & C. E. Synolakis 1994 The run-up of N-waves on sloping beaches. Proc. R. Soc. Lond. A 445, 99112.

S. Tadepalli & C. E. Synolakis 1996 Model for the leading waves of tsunamis, Phys. Rev. Lett. 77, 21412145.


H. Yeh , P. L.-F. Liu , M. Briggs & C. E. Synolakis 1994 Propagation and amplification of tsunamis at coastal boundaries. Nature 372, 353355.

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