Abstract. 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 is nonlinearly deformed during its propagation in the ocean which 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 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 reflection from the toe of the bottom slope. The propagation along the bottom of constant depth is described by Riemann wave, while the wave 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 run-up height which takes into account the wave front steepness is proposed.
On the wave-front shape and the advancing of the wetting front of a dam-break flood over an inclined plane of arbitrary bottom slope
