LAUSR

The governing equations of a surface wave field and a coexisting roll–streak circulation typical of Langmuir circulations or submesoscale frontal circulations are derived to better describe their two-way interactions. The gradients and vertical velocities of the roll–streak circulation induce wave refraction, amplitude modulation and higher-order waves. These changes then produce wave–wave nonlinear forces and divergence of the wave-induced mass transport, both of which in turn affect the circulation. To accurately represent these processes, both a wave theory and a wave-averaged theory are developed without relying on any extrapolation, any spatiotemporal mapping or an approximation that treats the wave-induced mass divergence as being concentrated at the surface. This wave theory finds seven types of current-induced higher-order wave motions. It also determines the wave dynamics such as the governing equation of the wave action density valid in the presence of the complex circulation. The evolution of the wave action density is clearly affected by the upwelling or downwelling. The new wave-averaged theory presents the governing equations of the wave-averaged circulation which satisfies the wave-averaged mass conservation. This circulation is different from the circulation considered to satisfy the mass conservation in the Craik–Leibovich theory, and the difference becomes critical when the wave field evolves due to refraction. In this case, compared to the Craik–Leibovich theory, long waves are more important and also the rolls are more weakly forced.