Physical Oceanography, Vol. 15, No. 4, 2005
RUN-UP OF A SOLITARY TSUNAMI WAVE ON A SLOPING COAST
S. F. Dotsenko
Within the framework of the nonlinear theory of long waves, we perform the numerical analysis
of the one-dimensional run-up of solitary tsunami waves upon a plane sloping coast. We study
the dependences of the run-up heights on the parameters of waves at the entrance of the shelf
zone and on the slope of the coast. The run-up heights of tsunami waves are estimated for the
bottom topography typical of the south coast of the Crimean Peninsula.
The run-up of waves on a coast is the most important stage of the “life” of tsunami waves in the ocean from
the viewpoint of evaluation of the level of tsunami hazard for the coast. The complexity of mathematical de-
scription of this process is determined by its nonlinearity and the necessity of reconstruction of the motion of li-
quid over the dry coast. The principal results in this field of the dynamics of wave processes are obtained by us-
ing the system of equations of long surface waves. The exact analytic solutions of the one-dimensional problem
of run-up of waves and the analytic estimates of run-up heights for tsunami-type waves can be found in [1–4].
The numerical models of one- and two-dimensional run-up of long waves are proposed in [5–9] and other works.
In [2, 3, 7, 10, 11], one can find the data of experimental investigations of the run-up of waves upon the coast
and their comparison with the results of mathematical simulations.
In the present work, within the framework of the nonlinear model of long waves, we perform the numerical
analysis of one-dimensional propagation of solitary tsunami-type waves in the shelf zone followed by their run-
up upon a plane sloping coast. The results of preceding investigations are supplemented with a more detailed
analysis of the dependences of the run-up heights of tsunami waves on the parameters of tsunamis at the entrance
of the shelf zone and on the slope of the coast. The quantitative estimates of the run-up heights of solitary waves
are obtained under the conditions simulating the south coast of the Crimean Peninsula near Yalta. Earlier ,
the estimates of the run-up heights of tsunami waves for the Black-Sea coast were obtained within the frame-
work of the model of reflection of waves from a vertical wall located in the coastal zone of the Black Sea.
Mathematical Model and the Method of Solution of the Problem
We consider the process of one-dimensional propagation of a long solitary wave in the shelf zone of a basin
along the horizontal coordinate
x. The schematic diagram of the basin is presented in Fig.
1. The analyzed wave
enters the shelf zone 0
L through its open left boundary x
0, propagates in the basin of variable depth
toward the coast, runs up on the plane coast over the inclined underwater part of the bottom
making an angle
with the horizontal, and finally, is reflected from the coast. Our aim is to describe the pro-
cess of propagation of the wave approaching the coast, its run-up on the coast, and back draft from the coast and,
as a final result, estimate the maximum elevations of water on the coast (run-up heights) provided that the height
and length of the tsunami wave on the left boundary of the region x
0 are known.
Marine Hydrophysical Institute, Ukrainian Academy of Sciences, Sevastopol.
Translated from Morskoi Gidrofizicheskii Zhurnal, No.
11–18, July–August, 2005. Original article submitted May 11, 2004.
0928-5105/05/1504–0211 © 2005 Springer Science+Business Media, Inc. 211