We consider mixing of the density field in stratified turbulence and argue that, at sufficiently high Reynolds numbers, stationary turbulence will have a mixing efficiency and closely related mixing coefficient described solely by the turbulent Froude number$Fr={\it\epsilon}_{k}/(Nu^{2})$, where${\it\epsilon}_{k}$is the kinetic energy dissipation,$u$is a turbulent horizontal velocity scale and$N$is the Brunt–Väisälä frequency. For$Fr\gg 1$, in the limit of weakly stratified turbulence, we show through a simple scaling analysis that the mixing coefficient scales as${\it\Gamma}\propto Fr^{-2}$, where${\it\Gamma}={\it\epsilon}_{p}/{\it\epsilon}_{k}$and${\it\epsilon}_{p}$is the potential energy dissipation. In the opposite limit of strongly stratified turbulence with$Fr\ll 1$, we argue that${\it\Gamma}$should reach a constant value of order unity. We carry out direct numerical simulations of forced stratified turbulence across a range of$Fr$and confirm that at high$Fr$,${\it\Gamma}\propto Fr^{-2}$, while at low$Fr$it approaches a constant value close to${\it\Gamma}=0.33$. The parametrization of${\it\Gamma}$based on$Re_{b}$due to Shihet al.(J. Fluid Mech., vol. 525, 2005, pp. 193–214) can be reinterpreted in this light because the observed variation of${\it\Gamma}$in their study as well as in datasets from recent oceanic and atmospheric measurements occurs at a Froude number of order unity, close to the transition value$Fr=0.3$found in our simulations.

We present direct numerical simulations (DNS) of unforced stratified turbulence with the objective of testing the strongly stratified turbulence theory. According to this theory the characteristic vertical scale of the turbulence is given by v ∼ u h /N, where u h is the horizontal velocity scale and N the Brunt-Väisälä frequency. Combined with the hypothesis of the energy dissipation rate scaling as ∼ u 3 h / h , this theory predicts inertial range scalings for the horizontal spectrum of horizontal kinetic energy and of potential energy, according to E(k h ) ∝ k −5/3 h . We begin by presenting a scaling analysis of the horizontal vorticity equation from which we recover the result regarding the vertical scale, v ∼ u h /N, highlighting in the process the important dynamical role of large-scale vertical shear of horizontal velocity. We then present the results from decaying DNS, which show a good agreement with aspects of the theory. In particular, the vertical Froude number is found to reach a constant plateau in time, of the formin all the runs. The derivation of the dissipation scaling ∼ u 3 h / h at low Reynolds number in the context of decaying stratified turbulence highlights that the same scaling holds at high R = ReFr 2 h 1 as well as at low R 1, which is known (see Brethouwer et al., J. Fluid Mech., vol. 585, 2007, pp. 343-368) but not sufficiently emphasized in recent literature. We find evidence in our DNS of the dissipation scaling holding at R = O(1), which we interpret as being in the viscous regime. We also find k ∼ u 3 h / h and p ∼ u 3 h / h (with = k + p ), in our high-resolution run at earlier times corresponding to R = O(10), which is in the transition between the strongly stratified and the viscous regimes. The horizontal spectrum of horizontal kinetic energy collapses in time using the scaling E h (k h ) = C 1 2/3 k k −5/3 h and the horizontal potential energy spectrum is well described byThe presence of an inertial range in the horizontal direction is confirmed by the constancy of the energy flux spectrum over narrow ranges of k h . However, the vertical energy spectrum is found to differ significantly from the expected E h (k v ) ∼ N 2 k −3 v scaling, showing that Fr v is not of order unity on a scale-by-scale basis, thus providing motivation for further investigation of the vertical structure of stratified turbulence.

Stably stratified turbulence is investigated with the aim of increasing our limited understanding of the vertical structure of this type of turbulent flow. For strongly stratified turbulence there is a theoretical prediction that the energy spectra in the vertical direction of gravity are very steep, possessing the well-known form E h (k v) ∝ N 2 k −3 v , where N is the Brunt-Väisälä frequency and k v is the vertical wave number, but supporting evidence from experiments and numerical simulations is lacking. We conduct direct numerical simulation (DNS) with uniform background stratification and forcing at large scales. In order to consider the large anisotropic scales only, the vertical energy spectra are decomposed into large-scale vertical spectra E large (k v) and small-scale vertical spectra E small (k v) using a horizontal demarcation scale. We find that this approach gives results that are in close agreement with E large (k v) ∝ N 2 k −3 v for the DNS runs performed. This result holds approximately over the wave-number range k b k v k oz , where k b is the buoyancy wave number and k oz is the Ozmidov wave number, in agreement with theory. Similarly, large-scale vertical spectra of potential energy are found to be E p,large (k v) ∝ N 2 k −3 v , over a narrower range of wave numbers. The evidence supports the existence of a scale-by-scale balance between inertia and buoyancy occurring in strongly stratified turbulence at large horizontal scales. Finally, the current results are put in the context of ocean turbulence by making a comparison with measurements of vertical shear spectra made in the ocean interior.

Localized regions of turbulence, or turbulent clouds, in a stratified fluid are the subject of this study, which focuses on the edge dynamics occurring between the turbulence and the surrounding quiescent region. Through laboratory experiments and numerical simulations of stratified turbulent clouds, we confirm that the edge dynamics can be subdivided into materially driven intrusions and horizontally travelling internal wave-packets. Three-dimensional visualizations show that the internal gravity wave-packets are in fact large-scale pancake structures that grow out of the turbulent cloud into the adjacent quiescent region. The wave-packets were tracked in time, and it is found that their speed obeys the group speed relation for linear internal gravity waves. The energetics of the propagating waves, which include waveforms that are inclined with respect to the horizontal, are also considered and it is found that, after a period of two eddy turnover times, the internal gravity waves carry up to 16 % of the cloud kinetic energy into the initially quiescent region. Turbulent events in nature are often in the form of decaying turbulent clouds, and it is therefore suggested that internal gravity waves radiated from an initial cloud could play a significant role in the reorganization of energy and momentum in the atmosphere and oceans.

Internal gravity waves propagating within homogeneous stratified turbulence are the subject of the present study. A spatiotemporal analysis is carried out on the results of direct numerical simulations including a forcing term, with the aim of showing the energy content of the simulations as a function of frequency, ω, and wave-vector inclination to the horizontal, θ. Clear signatures of the dispersion relation of internal gravity waves, ω = ±N cos θ , where N is the Brunt-Väisälä frequency, are observed in all our simulations, which have low Froude number, Fr h 1, and increasing buoyancy Reynolds number up to Re b ≈ 10. Interestingly, we observe the presence of high-frequency waves with ω ∼ N and a corresponding low-frequency vortex mode, both containing a non-negligible amount of energy. These waves are large-scale waves, their energy signature being found at scales larger than the forcing scales. We also observe the growth of energy in the shear modes, constituting a horizontal mean flow, and we show that their continuous growth is due to an upscale energy transfer, from the forcing scales to larger horizontal as well as vertical scales. These shear modes are found to be responsible for Doppler shifting the frequency of the large-scale waves. When considering the wave energy across the simulations at varying Re b , such energy is seen to reduce as Re b is increased and the flow enters the strongly stratified turbulence regime. The classical wave-vortex decomposition, based on a purely spatial decomposition of instantaneous snapshots of the flow, is analyzed within the current framework and is seen to correspond relatively well to the "true" wave signal identified by the spatiotemporal analysis, at least for the large-scale waves with ω ∼ N. Distinct energy peaks in θ -ω space highlight that the waves have preferential directions of propagation, specifically θ = 45 • and θ ≈ 55 • , similar to observations in studies of wave radiation from localized regions of turbulence. This suggests that the same wave-generation mechanisms may be relevant for homogeneous and inhomogeneous stratified turbulent flows.