A Combined Derivation of the Integrated and Vertically Resolved, Coupled Wave–Current Equations

Abstract

here exist different theories representing the effects of surface gravity waves on oceanic flow fields. In the past, the author has conjectured that the vertically integrated, two-dimensional fluid equations of motion put forward by Longuet-Higgins and Stewart are correct and that theories that differ from their theory cannot be entirely correct; this paper explores these differences. Longuet-Higgins and Stewart deduced vertically integrated, two-dimensional equations featuring a wave radiation stress term in the fluid dynamic, momentum equation. More recently, the author has proposed vertically dependent, three-dimensional equations that have required correction but when vertically integrated, agreed with the earlier, two-dimensional equations. This paper derives both vertically independent and vertically dependent equations from the same base and, importantly, using the same expression for pressure in the belief that the paper will contribute to the understanding and clarification of this seemingly difficult topic in ocean dynamics. An error in the classical papers by Longuet-Higgins and Stewart has been detected. Although the final phase-averaged result was correct, the error has had consequences in the development of vertically dependent equations. The prognostic equations in this paper are for the Eulerian current plus Stokes drift; toward the end of the paper these equations are contrasted with prognostic equations for the Eulerian current alone.