Extended mild-slope equation for surface waves interacting with a vertically sheared current

Propagation of water waves in coastal zones is mainly affected by the influence of currents and bathymetry variations. Models describing wave propagation in coastal zones are often based on the numerical solution of the Mild Slope equation (Kirby, 1984). In this work, an extension of this equation is derived, taking into account the linear variation of the current with depth, which results in a constant horizontal vorticity, slowly varying horizontally, within the background current field. The present approach is based on the asymptotic expansion of the depth-integrated lagrangian, assuming the linear variation of the background current with depth. With the aid of selected examples the role of this horizontal vorticity, associated with the assumed background current velocity profile, is then illustrated and emphasized, demonstrating its effect on the propagation of water waves in coastal areas.