The Three-Dimensional Current and Surface Wave Equations

TLDR

Three‑dimensional surface wave equations have not been previously formulated, and the analysis assumes linear‑theory depth dependence of wave motions. This paper develops the missing 3‑D surface wave equations for ocean models. The authors derive vertically‑dependent radiation stresses, Doppler velocity, surface‑pressure forcing, and turbulence‑production terms, providing a necessary foundation for 3‑D ocean models coupling waves with currents.

Abstract

Surface wave equations appropriate to three-dimensional ocean models apparently have not been presented in the literature. It is the intent of this paper to correct that deficiency. Thus, expressions for vertically dependent radiation stresses and a definition of the Doppler velocity for a vertically dependent current field are obtained. Other quantities such as vertically dependent surface pressure forcing are derived for inclusion in the momentum and wave energy equations. The equations include terms that represent the production of turbulence energy by currents and waves. These results are a necessary precursor for three-dimensional ocean models that handle surface waves together with wind- and buoyancy-driven currents. Although the third dimension has been added here, the analysis is based on the assumption that the depth dependence of wave motions is provided by linear theory, an assumption that is the basis of much of the wave literature.