Abstract
Abstract. 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 nonlinear shallow-water theory in the approximation of non-breaking waves. If the incident wave approaches the shore from the deepest water, run-up characteristics can be found in two stages: in the first stage, linear equations are solved and the wave characteristics at the fixed (undisturbed) shoreline are found, and in the second stage the nonlinear dynamics of the moving shoreline is studied by means of the Riemann (nonlinear) transformation of linear solutions. In this paper, detailed results are obtained for quasi-harmonic (narrow-band) waves with random amplitude and phase. It is shown that the probabilistic characteristics of the run-up extremes can be found from the linear theory, while the same ones of the moving 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.
Highlights
The flooded area size, the water flow depth, and its speed on the coast, and the coastal topography characteristics determine the consequences of marine natural disasters on the coast
We study the run-up of irregular narrow-band waves with a random envelope on a beach of a constant slope
75 % of historical tsunami waves climbed onto the coast with no breaking (Mazova et al, 1983)
Summary
The flooded area size, the water flow depth, and its speed on the coast, and the coastal topography characteristics determine the consequences of marine natural disasters on the coast. The nonlinear shallow-water equations for the bottom geometry of this kind are linearized by using the hodograph (Legendre) transformations This step makes it possible to obtain a number of exact solutions describing the run-up on the coast. The statistical wave theory can be applied to such records and nonlinear shallow-water equations in the random function class can be solved. This approach was used to describe the statistical moments of the long-wave run-up characteristics in Didenkulova et al (2008, 2010, 2011).
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