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 Shih et 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.

Direct numerical simulations of turbulent channel flows at friction Reynolds numbers (Re) of 550, 1000 and 1500 are used to analyse the turbulent production, transfer and dissipation mechanisms in the compound space of scales and wall distances by means of the Kolmogorov equation generalized to inhomogeneous anisotropic flows. Two distinct peaks of scale-energy source are identified. The first, stronger one, belongs to the near-wall cycle. Its location in the space of scales and physical space is found to scale in viscous units, while its intensity grows slowly with $\mathit{Re}$, indicating a near-wall modulation. The second source peak is found further away from the wall in the putative overlap layer, and it is separated from the near-wall source by a layer of significant scale-energy sink. The dynamics of the second outer source appears to be strongly dependent on the Reynolds number. The detailed scale-by-scale analysis of this source highlights well-defined features that are used to make the properties of the outer turbulent source independent of Reynolds number and wall distance by rescaling the problem. Overall, the present results suggest a strong connection of the observed outer scale-energy source with the presence of an outer region of turbulence production whose mechanisms are well separated from the near-wall region and whose statistical features agree with the hypothesis of an overlap layer dominated by attached eddies. Inner–outer interactions between the near-wall and outer source region in terms of scale-energy fluxes are also analysed. It is conjectured that the near-wall modulation of the statistics at increasing Reynolds number can be related to a confinement of the near-wall turbulence production due to the presence of increasingly large production scales in the outer scale-energy source region.

We study the turbulent Ekman layer at moderately high Reynolds number, $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}1600 < \mathit{Re} = \delta _{E}G/\nu < 3000$, using direct numerical simulations (DNS). Here, $\delta _{E} = \sqrt{2\nu /f}$ is the laminar Ekman layer thickness, $G$ the geostrophic wind, $\nu $ the kinematic viscosity and $f$ is the Coriolis parameter. We present results for both neutrally, moderately and strongly stably stratified conditions. For unstratified cases, large-scale roll-like structures extending from the outer region down to the wall are observed. These structures have a clear dominant frequency and could be related to periodic oscillations or instabilities developing near the low-level jet. We discuss the effect of stratification and $\mathit{Re}$ on one-point and two-point statistics. In the strongly stratified Ekman layer we observe stable co-existing large-scale laminar and turbulent patches appearing in the form of inclined bands, similar to other wall-bounded flows. For weaker stratification, continuously sustained turbulence strongly affected by buoyancy is produced. We discuss the scaling of turbulent length scales, height of the Ekman layer, friction velocity, veering angle at the wall and heat flux. The boundary-layer thickness, the friction velocity and the veering angle depend on $Lf/u_\tau $, where $u_\tau $ is the friction velocity and $L$ the Obukhov length scale, whereas the heat fluxes appear to scale with $L^+=L u_\tau /\nu $.

This work describes the derivation of an algebraic model for the Reynolds stresses and turbulent heat flux in stably stratified turbulent flows, which are mutually coupled for this type of flow. For general two-dimensional mean flows, we present a correct way of expressing the Reynolds-stress anisotropy and the (normalized) turbulent heat flux as tensorial combinations of the mean strain rate, the mean rotation rate, the mean temperature gradient and gravity. A system of linear equations is derived for the coefficients in these expansions, which can easily be solved with computer algebra software for a specific choice of the model constants. The general model is simplified in the case of parallel mean shear flows where the temperature gradient is aligned with gravity. For this case, fully explicit and coupled expressions for the Reynolds-stress tensor and heat-flux vector are given. A self-consistent derivation of this model would, however, require finding a root of a polynomial equation of sixth-order, for which no simple analytical expression exists. Therefore, the nonlinear part of the algebraic equations is modelled through an approximation that is close to the consistent formulation. By using the framework of a $K\text{{\ndash}} \omega $ model (where $K$ is turbulent kinetic energy and $\omega $ an inverse time scale) and, where needed, near-wall corrections, the model is applied to homogeneous shear flow and turbulent channel flow, both with stable stratification. For the case of homogeneous shear flow, the model predicts a critical Richardson number of 0.25 above which the turbulent kinetic energy decays to zero. The channel-flow results agree well with DNS data. Furthermore, the model is shown to be robust and approximately self-consistent. It also fulfils the requirements of realizability.

Direct numerical simulations of subcritical rotating, stratified and magneto-hydrodynamic wall-bounded flows are performed in large computational domains, focusing on parameters where laminar and turbulent flow can stably coexist. In most cases, a regime of large-scale oblique laminar-turbulent patterns is identified at the onset of transition, as in the case of pure shear flows. The current study indicates that this oblique regime can be shifted up to large values of the Reynolds number by increasing the damping by the Coriolis, buoyancy or Lorentz force. We show evidence for this phenomenon in three distinct flow cases: plane Couette flow with spanwise cyclonic rotation, plane magnetohydrodynamic channel flow with a spanwise or wall-normal magnetic field, and open channel flow under stable stratification. Near-wall turbulence structures inside the turbulent patterns are invariably found to scale in terms of viscous wall units as in the fully turbulent case, while the patterns themselves remain large-scale with a trend towards shorter wavelength for increasing . Two distinct regimes are identified: at low Reynolds numbers the patterns extend from one wall to the other, while at large Reynolds number they are confined to the near-wall regions and the patterns on both channel sides are uncorrelated, the core of the flow being highly turbulent without any dominant large-scale structure.

Numerical simulations are carried out to investigate vertical fluid particle dispersion in uniformly stratified stationary turbulent flows. The results are compared with the analysis of Lindborg & Brethouwer (J. Fluid Mech., vol. 614, 2008, pp. 303–314), who derived long- and short-time relations for the mean square vertical displacement 〈δz〉 of fluid particles. Several direct numerical simulations (DNSs) with different degrees of stratification and different buoyancy Reynolds numbers are carried out to test the long-time relation 〈δz2〉 = 2ϵPt/N2. Here, ϵP is the mean dissipation of turbulent potential energy; N is the Brunt–Väisälä frequency; and t is time. The DNSs show good agreement with this relation, with a weak dependence on the buoyancy Reynolds number. Simulations with hyperviscosity are carried out to test the relation 〈δz2〉 = (1+πCPL)2ϵPt/N2, which should be valid for shorter time scales in the range N−1 ≪ t ≪ T, where T is the turbulent eddy turnover time. The results of the hyperviscosity simulations come closer to this prediction with CPL about 3 with increasing stratification. However, even in the simulation with the strongest stratification the growth of 〈δz2〉 is somewhat slower than linear in this regime. Based on the simulation results it is argued that the time scale determining the evolution of 〈δz2〉 is the eddy turnover time, T, rather than the buoyancy time scale N−1, as suggested in previous studies. The simulation results are also consistent with the prediction of Lindborg & Brethouwer (2008) that the nearly flat plateau of 〈δz2〉 observed at t ~ T should scale as 4EP/N2, where EP is the mean turbulent potential energy.

We derive a relation for the growth of the mean square of vertical displacements, δz, of fluid particles of stratified turbulence. In the case of freely decaying turbulence, we find that for large times 〈δz2〉 goes to a constant value 2(EP(0) + aE(0))/N2, where EP(0) and E(0) are the initial mean potential and total turbulent energy per unit mass, respectively, a < 1 and N is the Brunt–Väisälä frequency. In the case of stationary turbulence, we find that 〈δz2〉 = 〈δb2〉/N2 + 2εPt/N2, where εP is the mean dissipation of turbulent potential energy per unit mass and 〈δb2〉 is the Lagrangian structure function of normalized buoyancy fluctuations. The first term is the same as that obtained in the case of adiabatic fluid particle dispersion. This term goes to the finite limit 4EP/N2 as t → ∞. Assuming that the second term represents irreversible mixing, we show that the Osborn & Cox model for vertical diffusion is retained. In the case where the motion is dominated by a turbulent cascade with an eddy turnover time T ≫ N−1, rather than linear gravity waves, we suggest that there is a range of time scales, t, between N−1 and T, where 〈δb2〉 = 2πCPLεPt, where CPL is a constant of the order of unity. This means that for such motion the ratio between the adiabatic and the diabatic mean-square displacement is universal and equal to πCPL in this range. Comparing this result with observations, we make the estimate CPL ≈ 3.

We perform numerical box simulations of strongly stratified turbulence. The equations solved are the Boussinesq equations with constant Brunt–Väisälä frequency and forcing either in rotational or divergent modes, or, with another terminology, in vortical or wave modes. In both cases, we observe a forward energy cascade and inertial-range scaling of the horizontal kinetic and potential energy spectra. With forcing in rotational modes, there is approximate equipartition of kinetic energy between rotational and divergent modes in the inertial range. With forcing in divergent modes the results are sensitive to the vertical forcing wavenumber kfv. If kfv is sufficiently large the dynamics is very similar to the dynamics of the simulations which are forced in rotational modes, with approximate equipartition of kinetic energy in rotational and divergent modes in the inertial range. Frequency spectra of rotational, divergent and potential energy are calculated for individual Fourier modes. Waves are present at low horizontal wavenumbers corresponding to the largest scales in the boxes. In the inertial range, the frequency spectra exhibit no distinctive peaks in the internal wave frequency. In modes for which the vertical wavenumber is considerably larger than the horizontal wavenumber, the frequency spectra of rotational and divergent modes fall on top of each other. The simulation results indicate that the dynamics of rotational and divergent modes develop on the same time scale in stratified turbulence. We discuss the relevance of our results to atmospheric and oceanic dynamics. In particular, we review a number of observational reports indicating that stratified turbulence may be a prevalent dynamic process in the ocean at horizontal scales of the order of 10 or 100 m up to several kilometres.

Direct numerical simulations of stably and strongly stratified turbulent flows with Reynolds number Re ≫ 1 and horizontal Froude number Fh ≪ 1 are presented. The results are interpreted on the basis of a scaling analysis of the governing equations. The analysis suggests that there are two different strongly stratified regimes according to the parameter . When , viscous forces are unimportant and lv scales as lv ∼ U/N (U is a characteristic horizontal velocity and N is the Brunt–Väisälä frequency) so that the dynamics of the flow is inherently three-dimensional but strongly anisotropic. When , vertical viscous shearing is important so that (lh is a characteristic horizontal length scale). The parameter is further shown to be related to the buoyancy Reynolds number and proportional to (lO/η)4/3, where lO is the Ozmidov length scale and η the Kolmogorov length scale. This implies that there are simultaneously two distinct ranges in strongly stratified turbulence when : the scales larger than lO are strongly influenced by the stratification while those between lO and η are weakly affected by stratification. The direct numerical simulations with forced large-scale horizontal two-dimensional motions and uniform stratification cover a wide Re and Fh range and support the main parameter controlling strongly stratified turbulence being . The numerical results are in good agreement with the scaling laws for the vertical length scale. Thin horizontal layers are observed independently of the value of but they tend to be smooth for < 1, while for > 1 small-scale three-dimensional turbulent disturbances are increasingly superimposed. The dissipation of kinetic energy is mostly due to vertical shearing for < 1 but tends to isotropy as increases above unity. When < 1, the horizontal and vertical energy spectra are very steep while, when > 1, the horizontal spectra of kinetic and potential energy exhibit an approximate k−5/3h-power-law range and a clear forward energy cascade is observed.

The effect of rotation on a homogeneous turbulent shear flow has been studied by means of a series of direct numerical simulations with different rotation numbers. The evolution of passive scalar fields with mean gradients in each of the three orthogonal directions in the flow was investigated in order to elucidate the effect of rotation on turbulent scalar transport. Conditions of the near-wall region of a boundary layer were approached by using a rapid shear and therefore, comparisons could be made with rapid distortion theory based on the linearized equations of the flow and scalar transport. Reynolds stresses, pressure–strain correlations and two-point velocity correlations were computed and turbulent structures were visualized. It is shown that rotation has a strong influence on the time development of the turbulent kinetic energy, the anisotropy of the flow and on the turbulent structures. Furthermore, rotation significantly affects turbulent scalar transport. The transport rate of the scalar and the direction of the scalar flux vector show large variations with different rotation numbers, and a strong alignment was observed between the scalar flux and the principal axes of the Reynolds stress tensor. The ratio of the turbulent and scalar time scales is influenced by rotation as well. The predictions of the linear theory of the turbulent one-point statistics and the scalar flux agreed fairly well with direct numerical simulation (DNS) results based on the full nonlinear governing equations. Nonetheless, some clear and strong nonlinear effects are observed in a couple of cases which significantly influence the development of the turbulence and scalar transport.

This paper presents an analysis and numerical study of the relations between the small-scale velocity and scalar fields in fully developed isotropic turbulence with random forcing of the large scales and with an imposed constant mean scalar gradient. Simulations have been performed for a range of Reynolds numbers from Reλ = 22 to 130 and Schmidt numbers from Sc = 1/25 to 144.

The simulations show that for all values of Sc [ges ] 0.1 steep scalar gradients are concentrated in intermittently distributed sheet-like structures with a thickness approximately equal to the Batchelor length scale η/Sc½ with η the Kolmogorov length scale. We observe that these sheets or cliffs are preferentially aligned perpendicular to the direction of the mean scalar gradient. Due to this preferential orientation of the cliffs the small-scale scalar field is anisotropic and this is an example of direct coupling between the large- and small-scale fluctuations in a turbulent field. The numerical simulations also show that the steep cliffs are formed by straining motions that compress the scalar field along the imposed mean scalar gradient in a very short time period, proportional to the Kolmogorov time scale. This is valid for the whole range of Sc. The generation of these concentration gradients is amplified by rotation of the scalar gradient in the direction of compressive strain. The combination of high strain rate and the alignment results in a large increase of the scalar gradient and therefore in a large scalar dissipation rate.

These results of our numerical study are discussed in the context of experimental results (Warhaft 2000) and kinematic simulations (Holzer & Siggia 1994). The theoretical arguments developed here follow from earlier work of Batchelor & Townsend (1956), Betchov (1956) and Dresselhaus & Tabor (1991).