2026 Gallet Surface Gravity Wave Mean Flow Part I

We investigate deep-water surface gravity waves propagating above a background flow with comparable spatial scales, emphasizing a regime where the flow is slow in comparison to wave velocity. Our objective is to introduce an ‘equivalent solvability condition’ method to derive reduced equations while ensuring the leading-order solution and first solvability condition align with the original system upon multiple-timescale expansion. The proposed method transforms the complete 3D problem into a 2D reduced equation for the wave field. We derive these reduced equations for both 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 resembles a Schrödinger equation that only incorporates the near-surface vorticity of the background flow, demonstrating that the effects of near-surface horizontal flow divergence are subdominant. Our findings not only reduce the spatial dimensions of the problem but also characterize the wave field's evolution over the slow advective timescale of the background flow, thereby ameliorating the need for resolving the rapid oscillation of the waves. We validate the capabilities of these reduced equations through an analytical solution for the weak scattering of a wave packet and numerical solutions for the substantial scattering of a wave packet in chaotic flow configurations.