On the vertical distribution of 〈ũw̃〉

Abstract

The aim of this paper is to present an analytical expression for the vertical distribution of the correlation between the horizontal (ũ) and vertical (w̃) wave velocity components. This quantity, 〈ũw̃〉, which appears explicitly in the time-averaged momentum balance equations, has been shown to play an important role in the vertical distribution of wave-induced currents. The proposed formulation for 〈ũw̃〉 is based on an identity that relates the effective (wave) shear stress 〈ũw̃〉 to the effective (wave) normal stresses (〈ũ〉 2 and 〈w̃〉 2) and to the vorticity of the oscillatory flow \\ ̃ gw. This general expression has been applied to simplified situations and has been shown to degenerate into other existing formulations with comparable simplifying assumptions, viz. irrotational waves in shallow water over an arbitrary bottom topography and breaking waves over a horizontal bottom. The model has also been applied to the case of waves interacting with a depth-varying current over a horizontal bottom, in which preliminary results have been obtained for a simplified situation invoking linear (small-amplitude) wave theory.