Rotational waves generated by current‐topography interaction

Abstract

We study nonlinear free‐surface rotational waves generated through the interaction of a vertically sheared current with a topography. Equivalently, the waves may be generated by a pressure distribution along the free surface. A forced Korteweg–de Vries equation (fKdV) is deduced incorporating these features. The weakly nonlinear, weakly dispersive reduced model is valid for small amplitude topographies. To study the effect of gradually increasing the topography amplitude, the free surface Euler equations are formulated in the presence of a variable depth and a sheared current of constant vorticity. Under constant vorticity, the harmonic velocity component is formulated in a simplified canonical domain, through the use of a conformal mapping which flattens both the free surface as well as the bottom topography. Critical, supercritical, and subcritical Froude number regimes are considered, while the bottom amplitude is gradually increased in both the irrotational and rotational wave regimes. Solutions to the fKdV model are compared to those from the Euler equations. We show that for rotational waves the critical Froude number is shifted away from 1. New stationary solutions are found and their stability tested numerically.

Trapped solitary waves and collisions for the forced Korteweg-de Vries equation

“…Assuming that there is a constant current in the flow, the Froude number (F ) is defined by the ratio of the upstream velocity and the linear long-wave speed. In the nearly-critical regime (F ≈ 1) with obstacles of small amplitude, the free surface displacement (ζ(x, t)) is governed by forced Korteweg-de Vries equation (Wu [1987]; Milewski [2004]; Flamarion et al [2019])…”

An iterative method to compute conformal mappings and their inverses in the context of water waves over topographies

“…Analyzing the behavior of waves generated due to a current-topography interaction, Flamarion et al 13 constructed a conformal mapping to flatten simultaneously the free surface and the topography. They assumed that the topography had small amplitude and validated their numerical method comparing solutions of the Euler equations in the weakly nonlinear and weakly dispersive regime with solutions of the forced Korteweg-de Vries (fKdV) equation.…”

Gravity-capillary wave interactions generated by moving disturbances: Euler equations framework

“…The main framework used is the full Euler equations (Grimshaw & Maleewong [2013]; Hanazaki et. al [2017]; Flamarion et al [2019]). However, due to intrinsic difficulties present in the Euler equations such as nonlinearity and free boundary conditions, reduced models based on asymptotic theory have been applied as an alternative to describe this phenomenon.…”