Kinematics of Gravity–Capillary Waves under Coupled Turbulent Air–Water Boundary Layers

We perform Direct Numerical Simulations (DNS) of a broadband gravity-capillary wave spectrum forced by turbulent wind conditions coupled with a developing underwater current transitioning from a viscous to a turbulent boundary layer. Utilizing the open-source solver Basilisk, we solve the full two-phase air-water Navier-Stokes equations with adaptive mesh refinement, capturing surface tension effects and using geometric Volume-of-Fluid interface reconstruction. Our simulations cover a wide range of scales, from millimeter-scale capillary ripples to meter-scale gravity waves.

Employing space-time Fourier analysis, we examine in detail the propagation, nonlinear interactions, growth, and decay of the wave spectrum across various wind-wave regimes, including changes in wind intensity and initial wave steepness. Our analysis identifies bound harmonic waves and quantifies the Doppler shift induced by depth-varying current profiles. We obtain a generalized nonlinear dispersion relation that effectively captures these complex interactions, providing new physical insights into wave kinematics and energy transfer mechanisms in fully coupled wind-wave-current systems.