Development and validation of a three dimensional wave-current interaction formulation

Abstract

This study aims at developing a new set of equations of mean motion in the presence of surface waves, which is practically applicable from deep water to the coastal zone, estuaries, and outflow areas. The Generalized Lagrangian Mean method is employed to derive a set of Quasi-Eulerian mean three-dimensional equations of motion, where effects of surface waves are included through source terms. The obtained equations are expressed to the second-order of wave amplitude. Whereas the classical Eulerian-mean equations of motion are only applicable below the wave trough, the new set of equations is valid until the mean water surface even in the presence of finite-amplitude surface waves. Both conservative and non-conservative waves are under consideration, especially in the presence of a strong ambient current. A concept of three-dimensional wave radiation stress is introduced to express the effects of surface waves on the currents. It is an extension of the classical radiation stress concept. Especially, the relationship between three-dimensional wave radiation stress and vortex force representations is investigated in detail in conditions of both conservative and nonconservative waves. Through that relationship, comparisons between the new set of equations and other sets of equations implemented in recent well-known numerical models are given. It is useful for selecting a suitable numerical ocean model to simulate the mean current in a specific condition of waves combined with currents.
A two-dimensional numerical model (2DV model) is developed to validate the new set of equations of motion. The model passes the test of steady monochromatic waves propagating on a slope without dissipation (adiabatic condition). This is a primary test for equations of mean motion with a known analytical solution. In addition to this, experimental data for the interaction between random waves and currents in both non-breaking and breaking waves are employed to validate the 2DV model. As shown by this successful implementation and validation, the implementation of the new set of equations in any 3D model code is straightforward and may be expected to provide consistent results from deep water to the surfzone, in both conditions of weak and strong ambient currents.