An ‘entraining plume’ model of a spilling breaker

It is proposed that a spilling breaker can be regarded as a turbulent gravity current riding down the forward slope of a wave, and can be treated using methods which have been successful in other contexts. The whitecap retains its identity because it is lighter than the water below, owing to the trapping of air bubbles, and information from laboratory experiments is used to estimate the density of the air-water mixture in different circumstances. Entrainment of water from below, at a rate E(Ri0,) which is a function of the overall Richardson number Ri0, has two opposing effects. It provides increasing mass and buoyancy fluxes which can produce an accelerating flow and it also gives rise to a drag, because the entrained fluid has upslope momentum (inco-ordinates moving with the wave crest). A similarity solution is obtained under the assumptions that the flow is steady in time, and that the slope and the density difference remain constant. In this solution, the thickness of the whitecap is proportional to the distances measured from the crest of the wave. The tangential velocity is proportional to s½. Since the velocity in a Stokes 120° angle is also proportional to s½ this implies that such a flow can start from a small disturbance with zero flux, and propagate with constant acceleration. An important consequence of the analysis is that solutions of this kind are possible only when the slope and the density difference between the whitecap and the water below are sufficiently large; otherwise the upward drag dominates, and a self-sustaining flow cannot form. For a slope of 30°, near the crest of the breaking wave, the theory predicts that a density difference greater than 8% is required to sustain a steady motion, at which point the downslope velocity is 12% of the opposing velocity at the wave surface. A 'starting plume’ model of the advancing front of the breaker is also discussed, which suggests that this too will be accelerating uniformly, but will have a velocity somewhat less than that in the flow behind. A comparison with the laboratory observations of Kjeldsen & Olsen verifies several features of the model, including the order of magnitude of the relative velocities in the whitecaps and the wave beneath. It also reveals the intermittent nature of the flow, which is here explained as due to the intermittent rounding of the wave crest due to damping of the wave by the whitecap on the forward face.