The Evolution of Wind Waves in Shallow Water over Variable Topography and a Background Current: Korteweg–de Vries Framework

The Korteweg–de Vries (KdV) equation is widely known as a canonical model for weakly nonlinear and weakly dispersive waves, notably and historically for water waves in shallow depths. Being integrable, it has a rich solution set of interacting solitary and periodic waves. Recently, we extended it with several forcing/friction terms to describe the evolution of wind-driven water wave packets in shallow water. The outcome is a modified KdV–Burgers equation, whose relevant solutions are principally solitary wave trains forming a soliton gas. In this article that is extended further by allowing the water depth to be slowly spatially varying, and introducing a basic horizontal current, also slowly spatially varying. The outcome is a modified KdV–Burgers equation with spatially slowly varying coefficients. We adapt the Whitham modulation theory for a slowly varying solitary wave train, allowing for the prediction of wave amplitude growth/decay due to a combination of the slowly varying background and the forcing/friction terms. Numerical simulations using a Fourier spectral method are performed to exhibit and validate the modulation theory.