A Coupled-Mode Technique for Wave-Current Interactionin Variable Bathymetry Regions

摘要

A coupled-mode technique for wave-current interaction is presented,rnwith application to the problem of wave scattering by steady currents inrnvariable bathymetry regions, and current variations on various scales.rnWe consider obliquely incident waves on a horizontally nonhomogeneousrncurrent in a variable-depth strip, which is characterizedrnby straight and parallel bottom contours. The flow associated with therncurrent is assumed to be parallel to the bottom contours (along-axisrncurrent) and it is considered to be known. In a finite subregionrncontaining the bottom irregularity we assume an arbitrary horizontalrncurrent structure. Outside this region, the current is assumed to bernuniform (or zero). At a first-order of approximation the wave flow isrnassumed to be irrotational, I.e. the vorticity of the total field is the samernwith the vorticity associated with the current. Then, restrictingrnourselves to linear, monochromatic (harmonic) waves of absoluternfrequency ω, the wave potential, including the scattering effect by therncurrent, is obtained as a solution to the modified Helmholtz equation,rnsubject to the free-surface boundary condition formulated with respectrnto the intrinsic frequency, the bottom boundary condition, and thernconditions at infinity. Based on an appropriate variational principle, inrnconjunction with a rapidly-convergent local-mode series expansion ofrnthe wave field in a finite subregion containing the current variation andrnthe bottom irregularity, a coupled-mode system is obtained that can bernconsidered as a generalization of the one derived by Athanassoulis &rnBelibassakis (1999). The present approach can be considered as anrnextension of the works by Smith (1983, 1987), and some of its mainrnfeatures are that it can be further elaborated to treat lateralrndiscontinuities (e.g. vertical vortex sheets) and more general verticalrncurrent profiles with cross-jet component, and to include the effects ofrnweak nonlinearity.