Abstract
Abstract. Nonlinear effects at the bottom profile of convex shape (non-reflecting beach) are studied using asymptotic approach (nonlinear WKB approximation) and direct perturbation theory. In the asymptotic approach the nonlinearity leads to the generation of high-order harmonics in the propagating wave, which result in the wave breaking when the wave propagates shoreward, while within the perturbation theory besides wave deformation it leads to the variations in the mean sea level and wave reflection (waves do not reflect from "non-reflecting" beach in the linear theory). The nonlinear corrections (second harmonics) are calculated within both approaches and compared between each other. It is shown that for the wave propagating shoreward the nonlinear correction is smaller than the one predicted by the asymptotic approach, while for the offshore propagating wave they have a similar asymptotic. Nonlinear corrections for both waves propagating shoreward and seaward demonstrate the oscillatory character, caused by interference of the incident and reflected waves in the second-order perturbation theory, while there is no reflection in the linear approximation (first-order perturbation theory). Expressions for wave set-up and set-down along the non-reflecting beach are found and discussed.
Highlights
Propagating in the real ocean waves normally lose its energy because of numerous reflections from coasts and any inhomogeineities of the seabed (Massel, 1989; Mei, 1989; Dean and Dalrymple, 2002)
In the asymptotic approach the nonlinearity leads to the generation of high-order harmonics in the propagating wave, which result in the wave breaking when the wave propagates shoreward, while within the perturbation theory besides wave deformation it leads to the variations in the mean sea level and wave reflection
We should note that based on the analysis presented in the previous section we expect that the perturbation theory will overestimate the magnitude of the nonlinear correction by up to 20 %
Summary
Propagating in the real ocean waves normally lose its energy because of numerous reflections from coasts and any inhomogeineities of the seabed (Massel, 1989; Mei, 1989; Dean and Dalrymple, 2002). For the arbitrary varying bottom profile the study of nonlinear effects during wave propagation becomes very difficult It requires solving nonlinear PDEs with variable coefficients, while even in the first-order (linear) case the solution has a complicated form. In some special cases (“non-reflecting configurations”) the wave can propagate along rapidly changing seabed geometry without reflection up to the coast Such non-reflecting configurations have been found and studied for different bottom geometries. Amplitude of the wave, propagating along such non-reflecting configurations varies according to the Green’s law and tends to infinity at the shoreline This extreme wave amplification may be reduced by nonlinear effects, which lead to the wave breaking.
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