TURBULENCE, WAVES AND MIXING AT SHEAR-FREE DENSITY INTERFACES .1. A THEORETICAL-MODEL

This paper presents a theoretical model of turbulence and mixing at a shear-free stable density interface. In one case (single-sided stirrin g) the interface separates a layer of fluid of density rho in turbulen t motion, with r.m.s. velocity mu(H) and lengthscale L-H, from a non-t urbulent layer with density rho + Delta rho, while in the second case (double-sided stirring) the lower layer is also in turbulent motion. I n both cases, the external Richardson number Ri = Delta bL(H)/u(H)(2) (where Delta b is the buoyancy jump across the interface) is assumed t o be large. Based on the hypotheses that the effect of the interface o n the turbulence is as if it were suddenly imposed (which is equivalen t to generating irrotational motions) and that linear waves are genera ted in the interface, the techniques of rapid distortion theory are us ed to analyse the linear aspects of the distortion of turbulence and o f the interfacial motions. New physical concepts are introduced to acc ount for the nonlinear aspects. To describe the spectra and variations of the r.m.s. fluctuations of velocity and displacements, a statistic ally steady linear model is used for frequencies above a critical freq uency omega(r)/mu(c), where omega(r) (= Delta b/2u(H)) is the maximum resonant frequency and mu(c) < 1. As in other nonlinear systems, obser vations below this critical frequency show the existence of long waves on the interface that can grow, break and cause mixing between the tw o fluid layer!;. A nonlinear model is constructed based on the fact th at these breaking waves have steep slopes (which determines the form o f the displacement spectrum) and on the physical argument that the ene rgy of the vertical motions of these dissipative nonlinear waves shoul d be comparable to that of the forced linear waves, which leads to an approximately constant value for the parameter mu(c). The model predic tions of the vertical r.m.s. interfacial velocity, the interfacial wav e amplitude and the velocity spectra agree closely with new and publis hed experimental results. An exact unsteady inviscid linear analysis i s used to derive the growth rate of the full spectrum, which asymptoti cally leads to the growth of resonant waves and to the energy transfer from the turbulent region to the wave motion of the stratified layer. Mean energy flux into the stratified layer, averaged over a typical w ave cycle, is used to estimate the boundary entrainment velocity for t he single-sided stirring case and the flux entrainment velocity for th e double-sided stirring case, by making the assumption that the ratio of buoyancy flux to dissipation rate in forced stratified layers is co nstant with Ri and has the same value as in other stratified turbulent flows. The calculations are in good agreement with laboratory measure ments conducted in mixing boxes and in wind tunnels. The contribution of Kelvin-Helmholtz instabilities induced by the velocity of turbulent eddies parallel to the interface is estimated to be insignificant com pared to that of internal waves excited by turbulence.