  The estimate of individual wave run-up is especially important for tsunami
warning and risk assessment as it allows to evaluate the inundation area. Here
as a model of tsunami we use the long single wave of positive polarity. The
period of such wave is rather long which makes it different from the famous
Korteweg-de Vries soliton. This wave nonlinearly deforms during its propagation
in the ocean, what results in a steep wave front formation. Situations, when
waves approach the coast with a steep front are often observed during large
tsunamis, e.g. 2004 Indian Ocean and 2011 15 Tohoku tsunamis. Here we study the
nonlinear deformation and run-up of long single waves of positive polarity in
the conjoined water basin, which consists of the constant depth section and a
plane beach. The work is performed numerically and analytically in the
framework of the nonlinear shallow water theory. Analytically, wave propagation
along the constant depth section and its run-up on a beach are considered
independently without taking into account wave interaction with the toe of the
bottom slope. The propagation along the bottom of constant depth is described
by Riemann wave, while the wave 20 run-up on a plane beach is calculated using
rigorous analytical solutions of the nonlinear shallow water theory following
the Carrier-Greenspan approach. Numerically, we use the finite volume method
with the second order UNO2 reconstruction in space and the third order
Runge-Kutta scheme with locally adaptive time steps. During wave propagation
along the constant depth section, the wave becomes asymmetric with a steep wave
front. Shown, that the maximum run-up height depends on the front steepness of
the incoming wave approaching the toe of the bottom slope. The corresponding
formula for maximum 25 run-up height, which takes into account the wave front
steepness, is proposed.

Fluid Dynamics;Pattern Formation and Solitons;Atmospheric and Oceanic Physics;Computational Physics