Turbulence, waves and mixing at shear-free density interfaces. Part 1. A theoretical model

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Abstract
This paper presents a theoretical model of turbulence and mixing at a shear-free stable density interface. In one case (single-sided stirring) the interface separates a layer of fluid of density ρ in turbulent motion, with r.m.s. velocity uH and lengthscale LH, from a non-turbulent layer with density ρ+Δρ, while in the second case (double-sided stirring) the lower layer is also in turbulent motion. In both cases, the external Richardson number Ri=ΔbLH/ u2H (where Δb is the buoyancy jump across the interface) is assumed to be large. Based on the hypotheses that the effect of the interface on the turbulence is as if it were suddenly imposed (which is equivalent to generating irrotational motions) and that linear waves are generated in the interface, the techniques of rapid distortion theory are used to analyse the linear aspects of the distortion of turbulence and of the interfacial motions. New physical concepts are introduced to account for the nonlinear aspects.To describe the spectra and variations of the r.m.s. fluctuations of velocity and displacements, a statistically steady linear model is used for frequencies above a critical frequency ωr/μc, where ωr(=Δb/2uH) is the maximum resonant frequency and μcc. The model predictions of the vertical r.m.s. interfacial velocity, the interfacial wave amplitude and the velocity spectra agree closely with new and published experimental results.An exact unsteady inviscid linear analysis is used to derive the growth rate of the full spectrum, which asymptotically 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 wave cycle, is used to estimate the boundary entrainment velocity for the single-sided stirring case and the flux entrainment velocity for the double-sided stirring case, by making the assumption that the ratio of buoyancy flux to dissipation rate in forced stratified layers is constant with Ri and has the same value as in other stratified turbulent flows. The calculations are in good agreement with laboratory measurements 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 compared to that of internal waves excited by turbulence.

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