Nonlinear long-wave deformation and runup in a basin of varying depth

Abstract

Abstract. Nonlinear transformation and runup of long waves of finite amplitude in a basin of variable depth is analyzed in the framework of 1-D nonlinear shallow-water theory. The basin depth is slowly varied far offshore and joins a plane beach near the shore. A small-amplitude linear sinusoidal incident wave is assumed. The wave dynamics far offshore can be described with the use of asymptotic methods based on two parameters: bottom slope and wave amplitude. An analytical solution allows the calculation of increasing wave height, steepness and spectral amplitudes during wave propagation from the initial wave characteristics and bottom profile. Three special types of bottom profile (beach of constant slope, and convex and concave beach profiles) are considered in detail within this approach. The wave runup on a plane beach is described in the framework of the CarrierGreenspan approach with initial data, which come from wave deformation in a basin of slowly varying depth. The dependence of the maximum runup height and the condition of a wave breaking are analyzed in relation to wave parameters in deep water.

Nonlinear wave effects at the non-reflecting beach

“…The details of the derivation of Eq. (3) can be found in (Didenkulova, 2009). Equation (3) describes the wave propagating in any onshore or offshore direction.…”

“…However, if we apply the direct perturbation theory to Eq. (3), found with an assumption of smoothly varying depth (see Didenkulova, 2009 for details), which is formally valid at distances smaller than X Br , in the second order of the perturbation theory the field will consist of two harmonics only:…”

“…The breaking distance (the distance from the coast, where the wave front becomes vertical) can also be found from Eq. (3) (Didenkulova, 2009). In the case of the non-reflecting beach Eq.…”

“…Since the energy flux is conserved in smoothly inhomogeneous medium, further we will call this case adiabatic or WKB approach. The nonlinear transformation of weakly nonlinear long waves in the basin of slowly varying depth is studied in (Varley et al, 1971;Gurtin, 1975;Caputo and Stepanyants, 2003;Didenkulova, 2009) within asymptotic nonlinear WKB approximation for different kinds of bottom profiles. In the case of the weakamplitude wave above the non-reflecting beach Eq.…”

Resonance phenomena at the long wave run-up on the coast

“…For more complicated geometry of coastal zone consisting of several pieces with different slopes, the solutions for each region of constant slope are matched (Kânoglu and Synolakis, 1998;Didenkulova, 2009). Simplified solutions in the form of a product of such elementary solutions can be given if the incident wave length is less than a bottom piece length.…”