摘要：The run-up of random long-wave ensemble (swell, storm surge, and tsunami) on the constant-slope beach is studied in the framework of the nonlinearshallow-water theory in the approximation of non-breaking waves. If theincident wave approaches the shore from the deepest water, run-upcharacteristics can be found in two stages: in the first stage, linearequations are solved and the wave characteristics at the fixed (undisturbed)shoreline are found, and in the second stage the nonlinear dynamics of themoving shoreline is studied by means of the Riemann (nonlinear)transformation of linear solutions. In this paper, detailed results areobtained for quasi-harmonic (narrow-band) waves with random amplitude andphase. It is shown that the probabilistic characteristics of the run-upextremes can be found from the linear theory, while the same ones of themoving shoreline are from the nonlinear theory. The role of wave-breaking due to large-amplitude outliers is discussed, so that it becomes necessary to consider wave ensembles with non-Gaussian statistics within the framework of the analytical theory of non-breaking waves. The basic formulas for calculating the probabilistic characteristics of the moving shoreline and its velocity through the incident wave characteristics are given. They can be used for estimates of the flooding zone characteristics in marine natural hazards.
