Wave and turbulent stress transport equations for flow over surface waves

Understanding air-sea interactions is crucial for both improving weather and climate predictions and has significant implications for marine renewable energies. In the study of turbulent flows over surface waves, the necessity of decoupling the wave-induced and turbulence motion has long been recognized and commonly employed in theoretical and numerical studies. Despite developing the decomposed wave and turbulent conservation equations for mass, momentum, and energy, the further step of looking into the transport equations of wave and turbulent stress is rarely taken. This has prevented the complete resolution of the governing equations and the associated closure problem.

In this study, we developed the transport equations of wave-induced turbulent and wave stresses from the decomposed momentum equations. Further, using appropriate boundary layer scaling, the transport equations have been significantly simplified based on the assumption that the vertical length scale of the motion is small compared to the horizontal length scale. We assess these assumptions using the experimental laboratory data and present different terms within the transport equations, including advection, production, diffusion, and turbulence spatial flux. The outcomes of boundary layer scaling result in simplified transport equations that can contribute to resolving the wave boundary layer closure problem.