Surface gravity wave-mean flow interaction with comparable spatial scales. Part I: reduced wave equations
We consider deep-water surface gravity waves propagating above a background flow whose spatial scale is comparable to the wavelength, focusing on the regime where the flow is slow compared to the group velocity of the waves. We introduce an "equivalent solvability condition" method to construct reduced equations, demanding that, upon multiple-timescale expansion, the reduced equations share the same leading-order solution and first solvability condition as the original system. This approach turns the full 3D problem into a 2D reduced equation for the wave field. We derive such reduced equations for broad-band waves above a depth-invariant background flow, and for narrow-band waves above a fully 3D background flow. In the latter case the reduced equation takes the form of a Schrodinger equation involving the near-surface vorticity of the background flow only, with the impact of the near-surface horizontal flow divergence shown to be subdominant. Beyond the reduction in spatial dimensionality, the latter reduced equation describes the evolution of the wave field over the slow advective timescale of the background flow, thereby eliminating the computational burden of time-resolving the fast wave period. We illustrate the capabilities of the reduced equations through an analytical solution for the weak scattering of a wave packet by a patch of organized flow, followed by numerical solutions for stronger scattering of a wave packet by a patch of disorganized flow.