Physical Oceanography, Vol.
1, May, 2011 (Ukrainian Original No.
1, January–February, 2011)
THERMOHYDRODYNAMICS OF THE OCEAN
RUNUP OF THE SURFACE WAVES OF VARIOUS SHAPES ON A SLOPING BEACH
S. F. Dotsenko
and N. K. V. Sannikova
In the long-wave approximation, we perform the numerical analysis of the plane problem of runup of
waves of various shapes on a sloping beach. We study transformations of the shape of waves flooding
the beach and in the course of their subsequent rundown. The dependence of maximum elevations and
lowerings of the sea level on the parameters of the waves approaching the beach, the depth of the shelf,
and the slope of the bottom are investigated. It is shown that the shape of waves affects the amplitude
characteristics of oscillations of the coastline. The heights of the vertical runup of waves incident on a
sloping beach can be several times higher than the amplitude of waves entering the shelf zone.
Keywords: nonlinear waves, runup of waves on the beach, long waves, plane problem, numerical mod-
The runup of surface waves on the coast is the final and most important stage of the evolution of tsunamis.
In fact, it determines the level of tsunami hazard for the sea coast. The complexity of modeling of this stage of
propagation of waves is determined by the significant nonlinearity of the process and the necessity of description
of the upward (runup of waves) and downward (rundown of waves) motion of fluid along the dry boundary of
the basin. In this field, the nonlinear models of long surface waves are used especially extensively.
The exact analytic solutions of the one-dimensional problem of runup of waves as well as the analytic esti-
mates of the vertical runup of tsunami-type waves in linear and nonlinear statements can be found in [1–7]. The
numerical models of one- and two-dimensional runup of nonlinear long waves are proposed and applied in nu-
merous works (see, e.g., [8–12]). The laboratory modeling of the runup of waves onto the coast and the compari-
son of the experimental data with the results of mathematical modeling have been carried out, e.g., in [2, 13–15].
In what follows, within the framework of the nonlinear theory of long waves without dispersion, we perform
the numerical analysis of one-dimensional propagation of sign-preserving and alternating waves over the shelf
of constant depth with subsequent runup on a plane sloping beach. The results of previous investigations are
supplemented with a more detailed analysis of the dependences of the vertical runup of tsunami waves on the
shape and parameters of waves entering the shelf zone, depth of the shelf, and the slope of the coast.
Marine Hydrophysical Institute, Ukrainian National Academy of Sciences, Sevastopol, Ukraine.
Corresponding author; e-mail: firstname.lastname@example.org.
Translated from Morskoi Gidrofizicheskii Zhurnal, No.
3–14, January–February, 2011. Original article submitted September 14,
2009; revision submitted October 12, 2009.
0928–5105/11/2101–0001 © 2011 Springer Science+Business Media, Inc. 1