1 Introduction

In Part 1 (Brocchini and Peregrine 2001) the effects of strong turbulence at a free surface are described, and a semi-quantitative classification introduced. Part 2 is also motivated by a wish to improve modelling of breaking water waves, but the analysis is applicable to the wider range of flows described in Part 1. We approach the problem from the standard Reynolds averaging of the equations of motion of a turbulent fluid. The two-phase nature of the surface region, especially when splashing occurs is recognized. We choose to derive anew the equations of motion for the two-phase flow in order to better display effects of interest and to introduce notation. Although the equations are not new, our derivation using integral formulations is new and leads to some extra terms in the transport equation for the turbulent kinetic energy. However, the main aim is to obtain a simplification leading to a form of boundary conditions that can be applied to averaged equations of motion for the main volume of water. Although we usually refer to air and water the analysis is applicable to any combination of liquid and gas which have a similar large ratio of densities and at scales where viscosity is for the most part unimportant.

If a free surface remains sufficiently smooth and non-overhanging then the surface can be described by a single variable giving its height above some datum. For this case Hasselmann (1971) is the only paper that to our knowledge gives a good account...