The dynamics of Lagrangian mean flow induced by three-dimensional internal waves under viscous conditions

We derive equations for three-dimensional internal wave beams propagating over a uniform slope in a uniformly stratified fluid. Using small-amplitude expansions, linear solutions for internal waves are obtained under weakly viscous conditions. Furthermore, a set of equations is constructed for the Lagrangian mean flow induced by the weakly nonlinear internal waves, providing the corresponding Lagrangian mean flow solutions within the boundary layer. The momentum equations of the Lagrangian mean velocity show that the Lagrangian mean flow is driven by the internal wave-induced body force, with its barotropic component related to the pressure gradient force, and its baroclinic component influenced by both viscosity and buoyancy. The Lagrangian-averaged buoyancy equation demonstrates that only the horizontal velocity for the Lagrangian mean flow exists throughout a vertically stable stratification region. This study emphasises the potential role of the Lagrangian mean flow in transporting time-averaged potential vorticity and solves for the Lagrangian mean flow in the inviscid region via mean potential vorticity conservation. The main results of the internal waves and the Lagrangian mean flow are visualised, revealing that the range of the boundary layer is related to the Reynolds number and that the intensity of the Lagrangian mean flow within the boundary layer is affected by the incidence angle and the reflection obliqueness. Theoretical analysis is provided to explain these phenomena.