On the runup of long waves on a plane beach

[1] Using the records of free surface fluctuations at several locations during the 2011 Japan Tohoku tsunami, we first show that the leading tsunami waves in both near-field and far-field regions are small amplitude long waves. These leading waves are very different from solitary waves. We then focus on investigating the evolution and runup of non-breaking long waves on a plane beach, which is connected to a constant depth region. For this purpose, we develop a Lagrangian numerical model to solve the nonlinear shallow water equations. The Lagrangian approach tracks the moving shoreline directly without invoking any additional approximation. We also adopt and extend the analytical solutions by Synolakis (1987) and Madsen and Schaffer (2010) for runup and rundown of cnoidal waves and a train of multiple solitary waves. The analytical solutions for cnoidal waves compare well with the existing experimental data and the direct numerical results when wave amplitudes are small. However, large discrepancies appear when the incident amplitudes are finite. We also examine the relationship between the maximum runup height and the leading wave form. It is concluded that for a single wave the accelerating phase of the incident wave controls the maximum runup height. Finally, using the analytical solutions for the approximated wave forms of the leading tsunamis recorded at Iwate South station from the 2011 Tohoku Japan tsunami, we estimate the runup height.

[1]  J. Boyd.  Cnoidal Waves as Exact Sums of Repeated Solitary Waves: New Series for Elliptic Functions , 1984 .

[2]  Integral Equation Model for Wave Propagation with Bottom Frictions , 1994 .

[3]  T. Ohyama.  A BOUNDARY ELEMENT ANALYSIS FOR CNOIDAL WAVE RUN-UP , 1987 .

[4]  Kenji Satake,et al.  Tsunami source of the 2011 off the Pacific coast of Tohoku Earthquake , 2011 .

[5]  Harry Yeh,et al.  Tsunami run-up and draw-down on a plane beach , 2003, Journal of Fluid Mechanics.

[6]  C. E. Synolakis,et al.  Validation and Verification of Tsunami Numerical Models , 2008 .

[7]  P. Liu,et al.  NUMERICAL SIMULATIONS OF THE 2004 INDIAN OCEAN TSUNAMIS — COASTAL EFFECTS , 2007 .

[8]  Kinjiro Kajiura,et al.  The Leading Wave of a Tsunami , 1963 .

[9]  D. Peregrine.  Calculations of the development of an undular bore , 1966, Journal of Fluid Mechanics.

[10]  Costas E. Synolakis,et al.  Long wave runup on piecewise linear topographies , 1998, Journal of Fluid Mechanics.

[11]  Per A. Madsen,et al.  On the solitary wave paradigm for tsunamis , 2008 .

[12]  H. P. Greenspan,et al.  Water waves of finite amplitude on a sloping beach , 1958, Journal of Fluid Mechanics.

[13]  Frederic Raichlen,et al.  THE GENERATION OF LONG WAVES IN THE LABORATORY , 1980 .

[14]  Ying Li,et al.  Solitary Wave Runup on Plane Slopes , 2001 .

[15]  Utku Kanoglu,et al.  Nonlinear evolution and runup–rundown of long waves over a sloping beach , 2004, Journal of Fluid Mechanics.

[16]  G. Whitham,et al.  Linear and Nonlinear Waves , 1976 .

[17]  Geir Pedersen,et al.  Run-up of long waves on an inclined plane , 1981 .

[18]  Costas E. Synolakis,et al.  The runup of solitary waves , 1987, Journal of Fluid Mechanics.

[19]  H. Segur.  Waves in shallow water, with emphasis on the tsunami of 2004 , 2007 .

[20]  Edmund Taylor Whittaker,et al.  A Course of Modern Analysis , 2021 .

[21]  A. Gurevich,et al.  Nonstationary structure of a collisionless shock wave , 1973 .

[22]  Harvey Segur,et al.  Modelling criteria for long water waves , 1978, Journal of Fluid Mechanics.

[23]  M. Deb,et al.  The anomalous behavior of the runup of cnoidal waves , 1988 .

[24]  Per A. Madsen,et al.  Analytical solutions for tsunami runup on a plane beach: single waves, N-waves and transient waves , 2010, Journal of Fluid Mechanics.