Surfactant effects on gravity-capillary waves

Surfactants at the air–sea interface are known to alter surface wave dynamics by modifying surface tension and Marangoni stresses. In this study, we perform two-dimensional direct numerical simulations of gravity-capillary waves with insoluble surfactants using a coupled phase field and volume-of-fluid method. We consider a nonlinear equation of state for surface tension and resolve Marangoni stresses induced by surfactant concentration gradients. We explore a broad parameter space characterised by initial wave steepness ak, Bond number Bo (comparing gravity and surface tension), Reynolds number Re (comparing inertia and viscosity), and the importance of surfactant concentration and strength of the gradient, characterised by a surfactant parameter β. We analyse the impact of surfactants on wave patterns, surface roughness, wave breaking, energy dissipation and surface vorticity. Our results reveal a non-monotonic dependence of wave shape, roughness, vorticity and energy dissipation on β, which is found to be governed by Marangoni effects that peak at intermediate surfactant concentrations. Wave regime transition at high Bo is governed by an effective Bo, which accounts for the reduction in surface tension induced by surfactants. We further introduce a rescaled parameter Bo Re^-1/2 (ak)^-1 based on force balance, which collapses the transition boundaries across different Re. These findings provide a systematic understanding of surfactant-modulated wave dynamics for both laboratory and geophysical applications.