摘要：The estimate of an individual wave run-up is especiallyimportant for tsunami warning and risk assessment, as it allows for evaluatingthe inundation area. Here, as a model of tsunamis, we use the long single waveof positive polarity. The period of such a wave is rather long, which makes itdifferent from the famous Korteweg–de Vries soliton. This wave nonlinearlydeforms during its propagation in the ocean, which results in a steep wavefront formation. Situations in which waves approach the coast with a steepfront are often observed during large tsunamis, e.g. the 2004 Indian Ocean and2011 Tohoku tsunamis. Here we study the nonlinear deformation and run-up oflong single waves of positive polarity in the conjoined water basin, whichconsists of the constant depth section and a plane beach. The work isperformed numerically and analytically in the framework of the nonlinearshallow-water theory. Analytically, wave propagation along the constantdepth section and its run up on a beach are considered independently withouttaking into account wave interaction with the toe of the bottom slope. Thepropagation along the bottom of constant depth is described by the Riemann wave,while the wave run-up on a plane beach is calculated using rigorousanalytical solutions of the nonlinear shallow-water theory following theCarrier–Greenspan approach. Numerically, we use the finite-volume methodwith the second-order UNO2 reconstruction in space and the third-orderRunge–Kutta scheme with locally adaptive time steps. During wave propagationalong the constant depth section, the wave becomes asymmetric with a steepwave front. It is shown that the maximum run-up height depends on the frontsteepness of the incoming wave approaching the toe of the bottom slope. Thecorresponding formula for maximum run-up height, which takes into accountthe wave front steepness, is proposed.
