On the Formation of Whitecaps by a Threshold Mechanism. Part I:. Basic Formalism

Abstract

In this paper and the two which follow we seek to evaluate the hypothesis that deep water whitecapping is predictable in terms of a threshold mechanism involving the vertical acceleration; i.e., that whitecapping occurs in regions of fluid where the surface wave motions require the downward acceleration to exceed some dynamical threshold. Our point of view is macroscopic and wave-oriented. We do not seek to analyze or describe in any detail motions within a whitecap itself; rather we concentrate on assessing the geometrical and geometro-statistical consequences of the threshold assumption and on comparing these consequences with observation. This first paper develops the descriptive framework for he investigation and examines several statistical predictions of a threshold model. The concept of a breaking variable is introduced and its statistics are related to the directional wave spectrum. The surface geometry of whitecap events is discussed and three classes of geometrical moments are defined. The statistics of these moments are related to the statistics of the breaking variable (and to the directional wave spectrum). Finally, several lower order probability densities for the breaking variable are computed for a JONSWAP spectrum. The succeeding papers deal with the computation of moment statistics by Monte Carlo simulation of the breaking variable field, and with a field experiment in which whitecap photographs were obtained simultaneously with array measurements of waves.