A note on the vertical distribution of momentum transport in water waves

Abstract

Abstract In this paper, the classical problem of horizontal waveinduced momentum transport is analyzed once again. A new analytical approach has been employed to reveal the vertical variation of this transport in the Eulerian description. In mathematical terms, this variation is shown to have (after “smoothing out” the surface corrugation) the character of a generalized function (distribution) and is described by a classical function in the water depths and by an additional Dirac-delta-function component on the averaged free surface. In terms of physics, the considered variation consists of two entities: (i) a continuous distribution of the mean momentum transport flux density (tensorial radiation pressure) over the entire water column, and (ii) an additional momentum transport flux concentrated on the mean free surface level (tensorial radiation surface pressure). Simple analytical formulae describing this variation have been derived. This allowed a conventional expression to be derived, describing the depth-integrated excess of horizontal momentum flux due to the presence of waves (the so-called “radiation stress”), confirming to some extent the correctness of the whole analysis carried out. The results obtained may be important to the ocean dynamics, especially in view of their possible application in the field of hydrodynamics of wave-dominated coastal zones.