It is argued that the mixing efficiency of naturally occurring stratified shear flows, γ=Rf/(1−Rf), where Rf is the flux Richardson number, is dependent on at least two governing parameters: the gradient Richardson number Ri and the buoyancy Reynolds number Reb=ε/vN2. It is found that, in the range approximately 0.03<Ri<0.4, which spans 104<Reb<106, the mixing efficiency obtained via direct measurements of fluxes and property gradients in the stable atmospheric boundary layer and homogeneous/stationary balance equations of turbulent kinetic energy (TKE) is nominally similar to that evaluated using the scalar balance equations. Outside these Ri and Reb ranges, the commonly used flux-estimation methodology based on homogeneity and stationarity of TKE equations breaks down (e.g. buoyancy effects are unimportant, energy flux divergence is significant or flow is non-stationary). In a wide range, 0.002<Ri<1, the mixing efficiency increases with Ri, but decreases with Reb. When Ri is in the proximity of Ricr∼0.1–0.25, γ can be considered a constant γ≈0.16–0.2. The results shed light on the wide variability of γ noted in previous studies.
The specification of eddy viscosity and diffusivities remains a central problem in developing predictive models for oceans, lakes and the atmosphere. For stably stratified flows, the vertical diffusivities can be defined in terms of vertical momentum (, ) and buoyancy ( or temperature () fluxes as 1.1 and 1.2 where KM≡KU=KV and KT are the eddy viscosity and diffusivity, respectively, and their ratio Prtr=KM/KT is the turbulent Prandtl number . When temperature T is the major contributor to density ρ, as in dry atmosphere, freshwater lakes and some marine environs, the buoyancy fluctuations become b′=−g(ρ−ρo)/ρo=−gρ′/ρo≈αgT′, and KT can be interpreted as the mass diffusivity. Here, g is the gravity, ρo the reference density and α the thermal expansion coefficient. The instantaneous velocities (u,v,w) in the (x,y,z) directions are written in the usual forms u=U+u′, v=V +v′, w=W+w′, the temperature and the density where U,V,W, and represent the mean and the primes are the fluctuations. In the atmosphere, sonic anemometers and gradient measurements allow direct evaluation of KM and KT from (1.1) and (1.2) by turbulent fluxes (using covariance calculation) and the gradients of temperature and velocity components at a suitable vertical separation. Here, the linear overbar stands for averaging over specific time segments (assuming Taylor frozen turbulence hypothesis). In oceans and other water bodies, however, the measurement of fluxes still remains a technical challenge, and eddy diffusivities are obtained either indirectly via local microstructure measurements or directly by tracer dispersion observations over large space–time domains. In the former, the simplified equations for the budget of turbulent kinetic energy (TKE) and turbulent thermal variance , 1.3 and 1.4 are used with the assumption of homogeneity (neglecting advection and diffusion of q2 and Θ2), where is the shear production of TKE and is the buoyancy flux . Here, mixing is assumed to be internal; that is, TKE is produced and approximately balanced locally vis-à-vis external mixing where the energy flux divergence and advective terms transport turbulence, which is generated elsewhere, to the mixing location . Here, 1.5 are the dissipation rates of q2and Θ2 in isotropic approximation, and ν and D are the molecular viscosity and diffusivity, respectively. As mentioned previously, ε and χ in natural waters are estimated using microstructure profiling measurements [3–8] as well as acoustic Doppler current profiler and acoustic Doppler velocimeter records [9–12].
Defining traditional flux Richardson number as  1.6 or using a generalized definition of Rf for non-stationary and inhomogeneous turbulence , 1.6a where the denominator accounts for all local and non-local sources of the TKE production, and introducing the mixing efficiency  1.7 it is possible to write the stationary balance of TKE using equations (1.3) and (1.2) as 1.8 or in non-dimensional form, 1.9 where 1.10 is the mixing Reynolds number. Note that (1.6a) is equivalent to (1.6) if pure internal mixing is occurring, that is shear-produced turbulence is dissipating locally and consumed for buoyancy flux. For ocean mixing, Osborn  suggested Rf≤0.17, which gives an upper bound for KT in (1.8) with γ≈0.2.
For the case where the temperature flux and gradients are measured directly Kf≡KT, the mixing Reynolds number (1.10) becomes 1.10a and the buoyancy Reynolds number is given by 1.11 where is the squared buoyancy frequency.
Conversely, following (1.2) and (1.4), the temperature diffusivity KT for stationary turbulence can be estimated as 1.12 where can be directly measured, and χ can be estimated from temperature microstructure data. The corresponding mixing Reynolds number is (cf. equation (1.10a)) 1.10b and the mixing efficiency γχ≡γ can be estimated using (1.8) and (1.12) , provided that the assumptions of stationarity and homogeneity are satisfied simultaneously . Another approach of estimating γ≡γf is the flux-based measurement of KT≡Kf using (1.2) and (1.8) or γ≡γo using (1.7) with (1.6) or (1.6a), where the temperature flux is measured directly. For convenience, these are listed in table 1.
The objectives of this paper are (i) to compare the mixing efficiencies γχ and γf calculated by the two methods and (ii) to discuss the dependence of mixing efficiency on Ri and Reb. The canonical flow type of interest here is the stratified shear flow that is common in oceans, lakes and the atmosphere. We will use data taken in the atmospheric nocturnal boundary layer at very high Reynolds numbers.
The present work is particularly helpful in delving into the variability of γ and therefore Rf observed in different studies that have employed a variety of techniques by analysing mixing mechanisms [15–17]. In the laboratory, Linden , Rohr et al. , Rohr & Van Atta  and Monti et al. , among others, have used non-stationary turbulence, whereas Strang & Fernando  and McEwan  used stationary or quasi-stationary flows, yielding somewhat different results. Numerical studies with evolving [17,24,25] and stationary flows  show disparate Rf and γ behaviour. In all, past results illustrate the dependence of Rf and γ on the mixing mechanism , the state of evolution of mixing [21,24,25,28], the nature of forcing (internal versus external [2,29]) and even the effect of topography . Results of several experimental and theoretical studies that showed a clear dependence of γ on Ri were summarized in fig. 10 of Lozovatsky et al. , and the repercussions of variable mixing efficiency are discussed in Balmforth et al. . The notion of Phillips  that γ grows with Ri, reaches a maximum at some Richardson number (0<Ri<∼1), and then decreases has been supported by laboratory experiments [18,34], measurements in a stably stratified atmospheric boundary layer  and used in the Goddard Institute for Space Studies (GISS) oceanographic numerical model of Canuto et al. . Initial attempts to estimate γ from oceanic flux measurements and balance equations of q2 and Θ2 yielded a γχ that is significantly smaller than γf (a single depth towing measurements in a tidal front by Gargett & Moum ), but the mixing efficiency was confined to a narrow range (0.13–0.17) on the basis of vertical profiling through turbulent patches . The authors noted a large uncertainty for γ estimates owing to technical difficulties of flux measurements in the ocean interior and a limited length of observational records.
It was our hope that the evaluations of γ from a comprehensive dataset that allows direct measurement (using γo and γf) and indirect evaluation (via γχ) of γ will help clarify the wide variability of mixing efficiencies reported in the literature. Furthermore, our dataset allows evaluation of γ as a function of governing variables of natural stratified sheared flows.
2. Governing parameters
Consider a stratified shear flow away from the boundaries, which is characterized by the buoyancy frequency N and mean shear , with their length scales of variation LN and LSh, respectively. The turbulence in the flow is specified by a characteristic velocity q, length scale ltr and time scale τ. The molecular parameters involved are v and D. Using ε≈q3/ltr, the governing dimensional variables become N, Sh, ε, q, v, D, LN, LSh and τ, yielding the governing non-dimensional variables Ri=N2/Sh2, Reb=ε/vN2, SN=Sh×(q2/ε), τ×Sh, ltr/LSh, ltr/LN and Pr=v/D, where ltr∼q3/ε has been reintroduced. For slowly spatially varying flows or turbulent regions that are much smaller than LN and LSh, the governing parameters become the gradient Richardson number Ri, buoyancy Reynolds number Reb and Prandtl number Pr. The additional constraint is the evolution of turbulence in equilibrium with shear (whence the shear number SN becomes a constant ). At high Reynolds numbers, or at geophysical scales, it is possible to invoke the Reynolds number similarity, and assume that the flows are Pr independent.
If local shear production of turbulence is the dominant source of (internal) mixing with ltr=(LSh,LN), the simplest stationary TKE balance KMSh2≈ε gives the following relationship between Reb, Ri and Rm: 2.1 where the turbulent Prandtl number Prtr=KM/KT itself can be a function of Ri whence Ri exceeds a critical Richardson number Ricr≈0.1–0.25 [40,41]. Conversely, Prtr tends to be close to unity for Ri≪Ricr [1,31,40,42]. Shih et al.  obtained an inverse dependence between Reb and Ri similar to equation (2.1) (their eqn (5.1)), and also showed a relatively weak decrease in Prtr (from approx. 1 to 0.85) when Reb increases from approximately 10 to 100. For Reb>∼100, Prtr remained approximately constant. Although Ri and Reb are formally independent governing parameters, but for shear-generated turbulence, they are related to each other in a specific parameter range. Nevertheless, it is important to note the external (the mean shear for Ri) and internal (the dissipation rate for Reb) nature of these parameters in turbulent stratified flows.
3. Observations and data processing
The data (wind velocity and sonic temperature) used in this study were obtained during the vertical transport and mixing experiment (VTMX) in Salt Lake City, Utah conducted from 30 September to 7 October 2000 (see Monti et al.  and Princevac et al.  for a general description of the experiment). A meteorological mast at a gentle (approx. 4°) mountain slope was equipped with two three-cup anemometers at heights h=2.0 (CA-1) and h=7.3 (CA-2) m above ground level (agl) and with two thermistor sensors at h=1.8 (T-1) and h=6.9 (T-2) m agl. The threshold speed of the anemometers was 0.5 m s−1, with an accuracy of 1.5 per cent. The 5 min averaged data were collected to characterize the mean wind speed and air temperature. The high-frequency (10 Hz sampling rate) records of velocity components u (downslope), v (cross-slope), and w (upward) and temperature T were obtained at h=4.5 m agl using a sonic anemometer/thermometer (Applied Technologies, Inc. and Metek GmbH). The resolution and accuracy of data were 0.01 and 0.05 m s−1, and 0.01°C and 0.05°C, respectively. A sketch showing positions of the instruments at the mast is given in figure 1c. Clear skies and light synoptic winds characterized the weather conditions from 30 September until 6 October. During the last night, wind increased up to 12 m s−1 owing to synoptic influence. Because of the very dry atmosphere (relative humidity during the experiment did not exceed 5–8%), we did not apply a moisture correction to the sonic temperature.
The 5 min averaged records of temperature, along-slope wind velocity and vertical components of temperature and momentum fluxes at h=4.5 m agl are shown in figure 1a,b. The night-time were predominantly negative as a result of the stable stratification near the ground (note 30 September 2000 is day 274 of 2006; this differs by 1 day from Monti et al. , where the leap year adjustment was not made). To obtain accurate estimates of γ, specific 20 min segments of data with almost constant fluxes and winds were selected (see a number of such segments for each night in figure 1a). Note that the homogeneity and stationarity of flow at specified segments were confirmed by comparing (1.6) and (1.6a). An example of , |Sh| and Ri records during a typical night (3–4 October) is given in figure 2, with seven segments chosen for the calculation of γ. The flow was usually weakly stable at the beginning of the night (Ri<0.1 in this case). At about midnight (figure 2), a local maximum of Ri≈2 was observed owing to a sharp drop of shear and a continuous increase in wherein the flow was dominated by quasi-periodic internal-wave oscillations . Oscillations with longer periods of approximately 2–3 h were also observed  that could be attributed to global intermittency associated with periods of intense turbulence production and enhance stability. Continuous cooling of the surface and general reduction of mean shear towards the end of the night led to an increasing Richardson number. The natural meteorological variability during the observational period of seven nights allowed estimates of γ to be obtained over a wide range of Ri with high statistical confidence.
The mixing efficiency, as mentioned in table 1, was calculated directly using equations (1.6) or (1.6a) and (1.7) and equations (1.9) and (1.2) to obtain γo and γf, respectively, as well as indirectly to yield γ≡γχ (equations (1.9) and (1.12)). In order to estimate the mixing efficiency directly (i.e. γo in table 1), we first compare the flux Richardson numbers Rf and RfII defined differently for stationary (equation (1.6)) and non-stationary (equation (1.6a)) balances of TKE. Strong linear correlation between the two variables can be seen in figure 3 with r2=0.97 for the best least-square linear regression RfII=1.1Rf. This result suggests that statistically, RfII and Rf are almost identical, supporting the assumption of homogeneity and stationarity for the dataset considered. Thus, we can confidently use equation (1.6) paired with equation (1.7) to calculate mixing efficiency γo≡γf.
The covariance and Reynolds stresses were computed for selected 20 min segments using the direct covariance method and integrating the corresponding co-spectra (see several examples of in figure 4a). A characteristic relative difference between the two temperature flux estimates was 11 per cent by amplitude, with 21 per cent standard deviation. The gradients of the mean temperature were taken as the finite differences between the 20 min averaged temperatures and located Δh=5.1 m apart. The TKE and temperature dissipation rates, ε and χ, were obtained from the inertial and inertial-convective subranges of the velocity and temperature spectra 3.1 at each i segment, and the 20 min averaged wind velocities were used to convert the frequency spectra of w′ and T′ to the corresponding wavenumber spectra Ew(κ) and ET(κ), where κ is the horizontal wavenumber. To ensure high quality of the dissipation estimates, only those spectra that exhibited a near-perfect −5/3 subrange were used with the canonical values of spectral constants ; for the longitudinal flow component (which is u in our case), cku=0.52 and for transversal components (w or v), ckw=ckv=4/3cku=0.67 (an example is given in figure 4b). A concern was the significant variation of the Obukhov–Corrsin constant cT reported in previous studies. Sreenivasan  suggested 0.3<cT<0.5 with a tendency for lower cT at higher Reynolds numbers; for that reason, cT=0.3 was used in this study. The spectra for u′,v′ and w′ exhibited clear inertial subranges, which, as expected, were wider for Eu(κ) and Ev(κ) compared with Ew(κ). At many segments, the flow was not horizontally unidirectional, thus posing problems of selecting the longitudinal and transverse directions. Therefore, the estimates of ε were made using Ew(κ) with ckw=0.67 because w is the unequivocal transversal component. Two methods were used for the Rm (equation (1.10)) calculation, based on KT≡Kf using the temperature flux and gradient measurement in (1.2), i.e. Rmf, and on KT≡Kχ obtained via (1.12) by estimating χ through ET(κ), i.e. Rmχ.
4. Dependence of mixing Reynolds numbers Rmχ and Rmf on Ri
On the basis of arguments of §2, the diffusivities must be dependent on the gradient Richardson number Ri as well as the buoyancy Reynolds number Reb. To explore the Rm(Ri) dependence, a combined plot of Rmf and Rmχ versus Ri is shown in figure 5. Both diffusivities therein are almost identically affected by Ri following approximately Rm∼Ri−3/2 . This simple parametrization has been previously used in several numerical models . The Ri−3/2 fit is, however, applicable for a relatively narrow range of Ri.
In figure 5, Rmf flattens at a value of Rmf≈2.2×103 in the range of Ri between Ricr=0.25 and Ri=1, and may even show a slightly rising tendency at larger Ri, although the latter cannot be substantiated owing to availability of few experimental points. It is, however, a possibility that the Rmf samples at Ri>1 represent intermittent turbulent patches that were advected to the observational site rather than generated locally by weak vertical shear. In this case, for Ri>1, the local Ri does not represent the state of turbulence in the same way as that for continuous turbulence.
On the contrary, Rmχ continuously decreases beyond Ricr, but much slower than in the Ri0−Ricr range, where Ri0≈0.025, being about an order of magnitude smaller than Ricr. For Ri>1, Rmχ tends to a background value Rmb=600. In the original stably stratified layer, shear-induced mixing onsets at Ri=Ricr, which rapidly grows until the Richardson number decreases to Ri=Ri0, and then it reaches a saturation level typical of non-stratified shear flows. At Ri<Ri0, Rmf flattens, deviating from the −3/2 power law towards Rmf∼3×105, but Rmχ continuously increases at low Ri, reaching approximately 107 at Ri≈10−3.
The dependences of Rmf and Rmχ on Ri evident from figure 5 can be approximated by a scaling formula 4.1 with s=2. For Rmχ, the fit is shown by the dashed line (b), with Rmn=2×107 and R0=2.5×10−3, whereas the heavy line (c) is drawn for Rmf with Rmn=3×105 and R0=2.5×10−2. The formula (4.1) belongs to the family of negative Ri power-law parametrizations of eddy viscosity and diffusivity in stably stratified flows [49–55]. Different s values have been proposed, ranging between 1 and 2.5. The value of Ri0 in (4.1) is usually taken from 0.1 to 0.3; however, Ri0 as low as 0.02–0.05 has been used sometimes to satisfy experimental data . The fitted Ri0=2.5×10−3 is much smaller than the previously suggested values, and the corresponding Rmn=2×107 is also unusually large. Perhaps, the presence of wall-induced turbulence in the present case may explain the anomalies; previous comparisons of (4.1) have been conducted with data taken from the thermocline or free shear flows.
This can be checked by applying the law-of-the-wall  test for vertical diffusivity Kz=κu*z, where u* and κ=0.4 are the friction velocity and von Karman constant, respectively. All the highest values of Rmf and Rmχ in figure 5 that correspond to the lowest Ri<10−2 represent seven segments of data obtained on 7 October under strong katabatic winds (the along-slope wind component varied between −10 and −12 m s−1 (figure 1). A characteristic estimate of u* for these data is approximately 1 m s−1 (the quadratic law formula). Thus, at h=4.5 m agl, , and hence Rm≈1.5×105. This value matches well with the normalized diffusivity Rmf in figure 5 at the lower end of the Ri axis, but it is more than an order of magnitude smaller than the corresponding Rmχ for the same values of Ri. The test implies that the balance-based calculation of Rmχ produces unreliable estimates of the normalized diffusivity for very low Richardson numbers (Ri<0.03). Probably, we can make the same conclusion about the flux-based estimates Rmf for high Ri>1. Hence, the dependence of mixing Reynolds number Rm on Ri is best represented by equation (4.1) with Rmn=3×105 and R0=2.5×10−2, which is shown by line (c) in figure 5.
5. Mixing efficiency
(a) Calculations of γ and its dependence on Reb
The estimates of γχ and γf can be obtained from the regression plots Rmχ(Reb) and Rmf(Reb) shown in figure 6a,b. According to (1.8), γ must be constant when Rm and Reb are linearly dependent, which is satisfied for the regression Rmχ(Reb) shown in figure 6a as 5.1 For Rmf(Reb), however, the least-squared fit yields 5.2 (figure 6b), where ξ≈33 is a non-dimensional regression coefficient and the exponent p=1/2. The above leads to the Reb dependence of 5.3 In addition, γo, which is equivalent to γmf, was evaluated (table 1) using independent measurements of B and P, and the results are approximated in figure 7 as 5.4a where co≈1.5. Note that the approximations (5.1), (5.2) and (5.4a) are obtained with high statistical confidence, with coefficients of determination r2=0.96, 0.92 and 0.93, respectively.
At first glance, γχ≈0.16 obtained in (5.1) for the nocturnal stably stratified atmospheric boundary layer is in good agreement with many previous estimates of γ for natural waters. In ocean mixing studies, γ=0.2 is frequently used [4,14,54,56–63]; however, higher (up to 0.4) and lower (approx. 0.1) values of γ have also been suggested (e.g.  and [38,65,66]). Limnologists [13,67,68] usually prefer γ=0.15–0.17, or in some cases, even smaller values, γ=0.04–0.06 , at very low stabilities. Recent analysis of DNS data [15,25] showed an approximately linear regression between Rmχ and Reb, also supporting γ=0.17, but only in a relatively narrow range of Reb=7–102. The authors identified this range as a stationary transition turbulent regime sandwiched between decaying and developing turbulence.
Further, the direct numerical simulation (DNS) data  produced Rmf that appeared to be proportional to 2PrRe1/2b for Reb=102–103. This is identical to the functional dependence (5.3) shown in figure 6b based on our atmospheric dataset for much larger buoyancy Reynolds numbers, Reb>103, and therefore the dimensionless constant ξ is different. Results of several laboratory experiments [70,71,72] compiled in Shih et al. , together with DNS data, suggest that Rm∼Re1/3b (in our notation) when Reb increases from 102 to 105. Note that the diffusivity in the laboratory experiments of Barry et al.  with grid-generated turbulence was calculated using the change of system's background potential energy before and after mixing events, and hence is an integral measure of different turbulent regimes. A slightly weaker than Re1/2b dependence can be suggested for the eight most energetic () samples shown in figure 6b that can be approximated by a power function Rmf∼Re0.4b. It is, however, reasonable to conclude that for the atmospheric nocturnal boundary layer Rmf∼Re1/2b for Reb>≈104. The difference points to the sensitivity of γ to the calculation methods used as well as to the nature of the flow.
(b) Interdependence between Rmf and Rmχ
In order to examine the disparity between the constant (equation (5.1)) and Reb-dependent (equation (5.3)) mixing efficiencies evaluated using different methods, we analysed the relationship between mixing Reynolds numbers Rmχ and Rmf (figure 8). It appeared that 68 per cent of data in the plot occupy a relatively narrow range of intermediate values of Rmχ and Rmf (within the rectangle), wherein the linear regression Rmf=cRRmχ is valid with a relatively high coefficient of determination r2=0.64 and a regression coefficient cR=0.65, having 95 per cent confidence bounds from 0.53 to 0.79. The deviation of cR from the ‘perfect agreement’ case of cR=1 can be attributed to the uncertainties of evaluating ε and χ using Kolmogorov and Obukhov–Corrsin spectra, especially those associated with spectral constants ckw and cT in equation (3.1) and flux measurements.
The difference between Rmf versus Rmχ is substantial at high and low ends of the Rm diagram, where approximately 106<Rmχ<2×107 and Rmf<(2–3)×105. This is equivalent to high and low values of Reb (figure 6a,b). Note, however, that large and small values of turbulent variables are usually subjected to highest uncertainties.
The striking loss of the parity between Rmf and Rmχ (or Kf and Kχ) at small and large Reb invites explanation, which has a bearing on the estimation of fluxes in natural waters. Because Reb∼(LO/LK)4/3, where LO=(ε/N3)1/2 and LK=(ε/ν3)1/4 are the buoyancy (or Ozmidov) and Kolmogorov scales, respectively, Reb≫1 implies very weakly stratified turbulence, where the TKE production essentially balances the dissipation, and fluxes are determined by r.m.s. velocity and temperature fluctuations (, cθ being a correlation coefficient ). This flux saturation, a reflection of a weak gradient or entraining fluid from non-turbulent regions, also implies non-stationarity and hence unsuitability of indirect methods of flux evaluation. On the other hand, small Reb implies lack of the inertial subrange, which is essential for indirect flux estimation. A stationary balance between production of TKE, buoyancy flux and the rate of dissipation can be established only in a specific (figure 8) intermediate range of Reb.
In other words, in fully developed turbulence at high Reb=107–108, a constant rate of mixing sustains until the density/temperature gradient almost completely erodes to a level that cannot uphold continuous growth of buoyancy (temperature) flux, whence the normalized diffusivity Rmf tends to saturation (compare the highest Rmf values in figures 5 and 6b). Less energetic turbulence (Reb<104) confounded by stratification produces weak mixing, which is characterized by the smallest Rmf≈(2–3)×103. The results indicate that highly energetic (high Reb) as well as underdeveloped (low Reb) turbulence does not support stationary, non-diffusive and non-advective balance of Θ2 in stratified flows. These findings are consistent with the interpretation of DNS data by Shih et al. . The difference is the actual range of Reb (a narrow one in both studies) where stationary turbulence prevails. The results can also be cast in terms of Ri, which is discussed below.
(c) Dependence of γ on Ri
The parametrization of mixing Reynolds number Rmf as a function of Ri given in §4 is a good representation for Richardson numbers below approximately 1. We have directly evaluated γf≡γo (equation (1.6a) paired with equation (1.7)) and plotted it in figure 9 as a function of Ri. The growing trend of γo(Ri) up to Ri≈1 seen here has been reported in several laboratory experiments, numerical and field studies [20,32,69,74–76]. Lozovatsky et al.  proposed an approximation, 5.5 to the GISS modelling results of Canuto et al. , in which the eddy coefficients are parametrized using specific damping functions. If we discard in figure 9 the two largest samples of γo corresponding to Ri≫1 as outliers (per discussion on two largest Rmf samples shown in figure 5 for Ri>4), then the data trend broadly mimics equation (5.4), although the scatter is high. A better fit to this specific dataset is 5.6 which nicely captures the main trend shown in figure 9 by a bold line. The squared box in figure 9 encompasses the same range of Ri as that in figure 5 (0.03<Ri<0.4) and the corresponding range of Rm boxed in figures 6 and 8. The box-averaged mixing efficiency appeared to be equal 〈γo〉=0.2, with the box-median value [γo]=0.165. This is close to γ=0.16 obtained using equations (1.8) and (1.12) in the range 3.7–5.7 (figure 6a). The box-averaged Richardson number 〈Ri〉=0.1 is smaller than the conventional critical value Ricr=0.25, but still is in the range where shear-induced turbulence and internal mixing are dominant.
The above result can explain why a majority of microstructures in stratified oceans and lakes, where mixing is generated mainly by shear instabilities with Ri slightly below critical, leads to γ=0.16–0.2 when evaluated using indirect methods (table 1). Note that atmospheric and laboratory measurements have spanned a wide range of Ri and Reb, and hence exhibited significant variations of γ. A monotonic increase in γo with Ri can be seen from about 0.01 at Ri≈(2–3)×10−3 to 0.4–0.5 when Ri is approaching unity. However, in a limited range around Ri approximately 0.1, the mixing efficiency can be considered approximately constant close to 0.2. Note that in a weakly stratified upper oceanic layer, the median of Ri can be as low as 0.1 . The cumulative distribution of the Richardson number is often well approximated by a lognormal probability law, showing that the probability of Ri<0.25 can be above 60 per cent and for Ri<1, can approach 80 per cent.
Considering that two variables Ri and Reb are involved in determining fluxes (§2), we plot contours of γo(Ri,Reb) in figure 10, which show that γo in the boxed area (Ri and Reb range associated with shear-generated turbulence) is between 0.1 and approximately 0.3. On the basis of our data, the upper cut-off Reupb was identified as approximately 5×105 (Ri<0.4), beyond which buoyancy effects are insignificant, whereas the lower limit was Relwb≈104 (Ri>0.03), signifying the absence of a clear inertial subrange due to suppression of local production of TKE. The variability of γo outside this Ri−Reb range is possibly due to other types of TKE sources unrelated to mean shear and/or to non-stationary turbulence (developing at Ri<0.03, Reb>∼106 or decaying at Ri>0.3, Reb<104).
6. Discussion and conclusions
Vertical heat, mass and momentum fluxes have not been directly measured in oceans and lakes until recently, and thus the eddy viscosity and diffusivities that are central to predictive modelling are inferred indirectly. One method relies on the assumption that the mixing efficiency γ is a constant in the expression KTN2=γε. Alternatively, a simplified scalar balance equation is used. The possible variability of γ was investigated in this paper based on sonic anemometer data obtained from the stable atmospheric boundary layer. The diffusivity KT≡Kf and the corresponding mixing Reynolds number Rmf (equation (1.10a) and table 1) were evaluated using temperature flux and gradient measurements and ε via the kinetic energy spectra; γ so obtained was designated as γ=γf. The results were compared with γ=γx, which was evaluated using KT≡Kχ and the corresponding Rmχ (equation (10b)) based on the commonly used (in physical oceanography) scalar dissipation technique (equation (1.12)). Dimensional arguments suggest that, at high buoyancy Reynolds numbers Reb,γ is a function of Ri and Reb, but the dependence on molecular parameters such as Pr is negligibly weak .
It was found that γ≈0.16 is nominally a constant in the range 104<Reb<106, and for our data, this corresponds to approximately 0.03<Ri<0.4. Both normalized diffusivities Rmf and Rmχ coincide in this regime (figures 5 and 8), leading to Rm∼Ri−3/2, indicating an approximate equivalence between γf and γx. Outside the above parameter ranges, Rm can be parametrized as a function of Ri using equation (4.1). For very low and high Reb and Ri, the basic assumptions underlying KTN2=γε were untenable, and should be used with circumspection. Furthermore, the accuracy of measurements, calibration of sensors as well as the usage of Obukhov–Corrsin spectral forms all affect the numerical value of γ.
The mixing efficiency was found to be a growing function of Ri (equation (5.5)) in a wide range, 0.002<Ri<1. It was a decreasing function of Reb, according to γ∼Re−1/2b (equation (5.3)), in the range 3×104<Reb<3×107, which is in agreement with the DNS results of Shih et al.  for 102<Reb<103, but at odds with Gargett's  suggestion of γ∼Re−1b for approximately the same range of Reb (based on laboratory data of Itsweire et al. ). If turbulence is anisotropic at smaller Reynolds numbers, as argued by Gargett , then it impacts the dissipation estimates and hence mixing efficiency calculations. Detailed studies on the dependence of γ with Ri and Reb were not possible, given the difficulty of obtaining γ versus Ri and Reb in nature when one or the other parameter is constant (equation (2.1)). The mixing efficiency may vary from approximately 0.01 at Ri≈(2–3)×10−3 to approximately 0.5–0.6 at Ri∼1. Phillips  suggested an increase of Ri up to a critical value Ri0, following a γ decrease at higher Ri. This trend was implemented by Canuto et al.  in modelling, and was experimentally observed by Strang & Fernando , Guyez et al.  and others. A decrease in γ at high Ri was evident (figure 9 and equation (5.5)), but could not be confirmed using the present dataset, as in the case of stably stratified Arctic boundary-layer data presented by Grachev et al. [80,81]. In our case, the number of high Ri data points are only a handful, thus precluding any inferences.
At Richardson numbers close to or below a critical value Ricr∼0.1–0.25 (viz. approx. 0.03<Ri<0.4), direct measurements of γ=γo via fluxes and gradients could be approximately treated as a constant with a characteristic value between 0.16 and 0.2. This Ri range is most pertinent to shear-generated stratified turbulent layers of oceans and lakes, which may explain why γ is often observed to be close to 0.2 and treated as such . In this range, the microstructure-based estimates γx and γf are consistent with γo, showing γ≈0.16. Comparison of different observations, nonetheless, is stymied by the dependence of Ri on measurement resolution, in particular, the separation with which the gradients are estimated. DeSilva et al.  showed that when this separation is greater than the buoyancy scale, the measurement may not be representative of local Ri, and our data were on the verge of this limit.
Our results help shed light on the differences between mixing efficiencies often encountered in oceanographic and limnological studies. Both constant and variable values of mixing efficiency are used, which forms the basis of closure in numerical models . The present results show that a constant mixing efficiency can be used only in a limited range of governing parameters (Ri, Reb), and many oceanographic measurements appear to be in this range ; therein turbulence is internally generated and approximately satisfies conditions of stationarity and homogeneity.
We are thankful to the participants of the VTMX experiment who conducted the field measurements and to Andrea Dato (a former student of University of Rome ‘La Sapienza’, Italy) who actively participated in the data processing during his visit to the USA. This work was supported by the National Science Foundation (CMG) and ONR grants nos N00014-10-1-0738 and N00014-11-1-0709 (MATERHORN Programme).
One contribution of 13 to a Theme Issue ‘Turbulent mixing and beyond: non-equilibrium processes from atomistic to astrophysical scales I’.
- © 2012 The Author(s) Published by the Royal Society. All rights reserved.