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 μc<1.
As in other nonlinear systems, observations below this critical
frequency show the existence of long waves on the interface that
can grow, break and cause mixing between
the two fluid layers. A nonlinear model is constructed based on the fact that these
breaking waves have steep slopes (which determines the form of the displacement
spectrum) and on the physical argument that the energy of the
vertical motions of these
dissipative nonlinear waves should be comparable to that of the forced linear waves,
which leads to an approximately constant value for the parameter
μc. 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.
Measuring Turbulent Motion in Planet-forming Disks with ALMA: A Detection around DM Tau and Nondetections around MWC 480 and V4046 Sgr
