## 1. Introduction

Turbulent wall-bounded shear flows are a class of flows of great practical importance, and one for which a relatively clear picture has emerged for the mean flow and the production–dissipation process of turbulent kinetic energy (TKE). In the last decades, since the prediction by Tchen (1953) [as noted in Kader and Yaglom (1989)] and the pioneering work of Perry et al. (1986), evidence has emerged for a distinct scaling in a range of the energy spectrum at lower wavenumbers *k* than those of the inertial subrange. In this range some common characteristics for several types of unstratified shear flow emerge, among which is a distinctive *z*, and “inactive turbulence,” which scales with the boundary layer thickness *δ*, Perry et al. (1986) arrived at scalings for the velocity spectra. Their theory, which is based on dimensional analysis and matched asymptotics, yields expressions for the spectra that upon integration produce dimensionless velocity variances in line with Townsend’s (1976) law of active and inactive turbulence.

Shortly thereafter, the shape of the longitudinal velocity spectrum in the unstable atmospheric surface layer (ASL) was investigated apparently for the first time by Kader (1988), using what was termed “directional dimensional analysis” by Kader and Yaglom (1990); the results about the spectrum appeared in English in Kader and Yaglom (1989). Their prediction for the dynamic sublayer (the neutral part of the surface layer) was the appearance of the same *L* is the Obukhov length scale) the longitudinal spectrum displays a

On the observational front, the existence of the

Davidson et al. (2006) chose a slightly different approach and showed that the *u* is the streamwise velocity component and *δ* of the boundary layer).

*z*is a main scaling parameter for the “active” (flux carrying) turbulence. Recently, it has been proposed by Davidson and Krogstad (2014) that, instead of

*z*, the length scale formed from the friction velocity

*ε*is the correct scale for the longitudinal structure function

*z*with an

*ε*-based length scale implies a link between the production range of the longitudinal structure function and turbulent motions in the universal equilibrium range. With the hypothesis of an

*ε*-based scaling that was not formally justified, Davidson and Krogstad (2014) obtained for unstratified flowswhere the dissipation-based length scale is defined to beand

*δ*is an “outer length scale” (e.g., the boundary layer height).

The experimental results presented by Davidson and Krogstad (2014) supporting the superiority of *z* as a length scale were for somewhat moderate-friction Reynolds numbers *z*, and the two scalings are equivalent.

Pan and Chamecki (2016) recast the discussion in terms of the integral scale for the active motions

In the atmospheric surface layer, balance between shear production and dissipation is impacted by the buoyancy production/destruction term in the TKE equation. The integral length scale is modified by buoyancy, and it is no longer clear that it should be proportional to *z*. This calls into question the validity of *z* scaling for the longitudinal structure function. Yet most theoretical and observational works utilize *z* in the scaling of *r* or *k* and attempt to include the effects of stability via explicit dependence on stability parameters. In this work we take a different route and generalize the dimensional analysis of Pan and Chamecki (2016) to include buoyancy effects on the scaling of

This work is organized as follows. The scaling theory is modified to include the effects of buoyancy in section 2, and the field data employed here are described in section 3. Results from data analysis are presented in section 4, followed by a short discussion and conclusions in section 5.

## 2. Dimensional analysis including stability

*η*is the Kolmogorov length scale, and

*ε*. Note also that the three scaling ranges have overlapping regions, where two scalings must be valid and the asymptotic matching takes place.

*ε*, leading toOnly two dimensionless groups can be formed from the four physical variables in (7). Here we choose to write

*κ*is the von Kármán constant). It is perhaps interesting to point out that a third option would be to write

The next step in the analysis is to match the production range with the inactive range. However, contrary to the unstratified case, the choice of velocity scale is no longer unique. The following analysis is split into two cases to separate conditions under which the energy content in

### a. Case I: Shear-dominated regime

### b. Case II: Buoyancy-dominated regime

*r*, as long as the function

*z*as the length scale for

*r*, then (27) becomesindicating that the local free-convection velocity

### c. Applicability to the atmospheric surface layer

A classification of turbulent flows in the presence of shear and buoyancy is typically based on three independent time or length scales. One common choice is to use a turbulence time scale

*P*and buoyancy production/destruction

For the stable surface layer, analysis is restricted to the regimes in which shear production of TKE is strong enough to sustain continuous turbulence (i.e., cases of intermittent turbulence are excluded). The Kansas data suggest a perfect balance between shear production and dissipation for neutral and stable conditions (Wyngaard and Coté 1971), which is manifested in the ratio

Conversely, in the unstable ASL, the importance of buoyancy effects as measured by

### d. Other useful considerations

*r*, the derivatives with respect to

*r*will be plotted, which gives a better view of how the production/destruction is distributed over the length scales.

## 3. Dataset

The data used in this study were collected as part of the AHATS (UCAR/NCAR–Earth Observing Laboratory 1990), which took place near Kettleman City, California, during the period from 25 July to 16 August 2008. The field site was surrounded by short grass stubble and was predominantly horizontally homogeneous and level, with an estimated roughness length *z* = 1.51, 3.30, 4.24, 5.53, 7.08, and 8.05 m during the period from 25 June to 17 July were used. The CSAT-3 anemometers sampled the three components of the velocity vector and virtual temperature at 60 Hz. Mean temperature data were collected at 1 Hz at all heights using calibrated SHT 75 transducers (Sensirion AG).

Data were divided into blocks of 36.4 min (equivalent to 2^{17} points). Only data with wind directions of *α* is the angle of incidence with respect to the sonic’s main horizontal axis, were included in the data analysis. The coordinate system was aligned with the mean wind direction so that *τ* is the time lag].

The values of *L* were calculated using data from the top sonic (these values did not change appreciably with height for the top five sonics). Mean velocity gradients were estimated by fitting a second-order polynomial in ln(*z*) to the measurements following Högström (1988). Note that the values of *κ* enters only into the definition of *L*, and the more traditional value of *ε* were estimated from the inertial subrange in the second- and third-order structure functions following the approach described by Chamecki and Dias (2004), which consists of averaging the dissipation obtained at each *r* within the inertial subrange (here assumed to be

Before proceeding to results, we note that the discussion on self-correlation that has become frequent in analysis of ASL data does not apply to the two main points being investigated in this work: the functional form of the structure function and the collapse of data from different heights or blocks. This is because all the points in the structure function for a given block are normalized by the same velocity and length scales so that the normalization does not change the relative position of the points within the structure function and it does not affect its functional form [i.e., ln(*r*) or

## 4. Results and discussion

Ratios of shear and buoyancy production to TKE dissipation rate obtained from the AHATS data for all measurement heights are shown in Fig. 1, together with ensemble-averaged values based on the stability groups listed in Table 1. Results from AHATS are in good agreement with the two sets of MOS functions shown in the figure. For near-neutral conditions, the ratio of shear production to dissipation is

Stability groups used for ensemble averages. The coefficients in the log-law [(1)] fitted to the data are represented by

Before proceeding to assess the *z* scalings for

Townsend (1958) argued that for neutral flows the integral length scale is proportional to the dissipation-based length scale (i.e.,

### a. Analysis for individual flow realizations

The initial analysis presented here follows the approach of Davidson and Krogstad (2014): for each data block, here interpreted as one realization of the flow field, the collapse of *z* and *r*. Three blocks, characterizing typical behavior for near-neutral (Nb1), unstable (Ub1), and stable (Sb1) temperature stratification are chosen to illustrate the results. An additional unstable block (Ub2) with large imbalance between shear production and dissipation of TKE is also analyzed. Statistical characterization of these blocks is presented in Table 2. In addition, vertical profiles of the dimensionless mean velocity gradient, the dissipation length scale, and the ratio between TKE shear/buoyancy production and dissipation are shown in Fig. 3.

Statistical characterization of selected blocks (all quantities evaluated at the top sonic height

Despite the near-neutral stability parameter

The second-order structure function analysis for the near-neutral run Nb1 is presented in Fig. 4. The difference between *z* scaling and *z* is the appropriate scale for shear production under near-neutral conditions. In addition, at least a decade of the log-scaling behavior predicted by Davidson and Krogstad (2014) is observed in Fig. 4d (approximately *ε* and issues associated with the statistical convergence of *r*).

Structure functions involved in shear and buoyancy production are presented in Figs. 4e and 4f. Note that we plot *r* using *δ*. As expected for near-neutral stratification, buoyancy production/destruction is negligible.

The analysis for the stable block Sb1 (for this run, *z* in collapsing the measurements at different heights and that the entire *z* is no longer an appropriate estimate of the shear production length scale and cannot collapse

The analysis for the unstable block Ub1 (for this run, *z* in collapsing

Some unstable blocks behave differently from Ub1. This is true, in our data, of some blocks with larger values of

Despite the general superiority of *z* in the scalings of *τ* in the form

Analysis of these four individual representative blocks of data lends support to the theory presented in section 2. More specifically,

### b. Analysis for ensemble averages based on atmospheric stability

Herein, the analysis is divided in two parts. In the first part, all the 1296 samples (216 blocks × 6 heights) are split into four stability groups according to the value of *L* is always estimated with data from the highest sonic). These stability groups are near neutral

The initial analysis using four stability groups is shown in Fig. 8; the ensemble average for each stability group is displayed on top of lines corresponding to 24 selected blocks of data at each height, for a total of 144 samples spanning the entire stability range of the dataset. For all stability groups, the collapse between the different curves is far superior when *z*. In addition, the spread between different runs seems quite small up to

The increase of error bars with increasing *r*, as the number of equivalent independent samples in the evaluation of the structure function decreases as

To further investigate the effect of stability on the dependence of *z* was used as the relevant length scale), the point of departure from inertial range behavior displayed strong dependence on stability via *ε* must be used to collapse the inertial subrange, and Kaimal et al. (1972) opted to multiply the energy spectrum by

Figure 9 shows that the near-neutral curve transitions smoothly from the inertial subrange behavior to the logarithmic scaling in (20), in excellent agreement with the empirical fit of de Silva et al. (2015). Once again, unlike the results from Kaimal et al. (1972), the unstable groups are well organized according to *r*) scaling in near-neutral conditions to an approximate power law in strongly unstable conditions. Note that using *z* as the representative length scale, Kaimal et al. (1972) were unable to identify a clear trend for the low-wavenumber region of the unstable runs and ended up lumping all unstable runs in a “shaded region” in their figures. Using the present normalization, not only is there a clear trend with instability, but also the most unstable group approaches the

As pointed out in the theory section, a prediction for the range of stabilities in which buoyancy and shear are equally important is not available at the moment. Nevertheless, the gradual transition from the ln(*r*) to the

A clearer confirmation of the validity of the *y* axis together with a line corresponding to the fitted values of

Because the logarithmic scaling is expected to hold for shear-dominated regimes, it is interesting to investigate its applicability to the weakly unstable groups as well. Results for groups U1 and U2 are displayed in Fig. 11. The

The coefficients resulting from fitting the

## 5. Conclusions

In this paper, we extended the theoretical results obtained by Davidson and Krogstad (2014) for the scaling of the second-order longitudinal structure function in the production range to include the buoyancy effects that are ubiquitous in the atmospheric surface layer. The theoretical development presented here has two components. The first is the dimensional analysis following Perry et al. (1986) and Pan and Chamecki (2016), but including the effects of buoyancy via a buoyancy flux *z* in the dimensional analysis is that

Theoretical predictions for the scaling of the longitudinal structure function in the production range are possible for the limiting cases in which either shear or buoyancy dominates the energy content in

Several conclusions can be drawn from the analysis of turbulence data measured by sonic anemometers as part of the AHATS experiment:

- The recent suggestion by Davidson and Krogstad (2014) that
is a better scaling than *z*(not only better, but actually the correct one) for the longitudinal structure functionholds for the atmospheric surface layer. This is true independent of stability. In the neutral case, the equivalence of the two scalings pointed out by de Silva et al. (2015) is not observed because of imbalance between shear production and dissipation despite the very large Reynolds number. - The use of
in scaling *r*(or*k*) also provides a better assessment of the effects of buoyancy on the large scales, aspresents a systematic dependence on in unstable conditions that is not observed when *z*is used (Kaimal et al. 1972). In addition,is a good estimate of the end of the inertial subrange for all stability cases. - Scaling predictions from dimensional analysis are supported by the AHATS data. Results presented here show for the first time clear evidence that the energy density in the large scales displays a
scaling (or, equivalently, in the spectrum) for all stably stratified cases. Therefore, the effect of buoyancy on the energy distribution in the large scales of the stable ASL is mostly captured by the large decrease in with increasing stability. Residual effects may be present in the dependence of the coefficients on stability, but no clear trend has emerged from the AHATS data. - Results suggest that
is superior to for characterizing the effects of stability on . This is seen in the analysis of individual blocks as well as in the trends and when the ln( *r*) is fitted to data in shear-dominated blocks. In addition,seems to be the range of validity for the shear-dominated regime. - The existence of a
scaling in the longitudinal structure function is independent of the attached-eddy hypothesis of Townsend (1976), as suggested by theoretical predictions and confirmed by observations under stable conditions. This finding confirms the argument raised for the first time by Davidson and Krogstad (2014). Therefore, the existence of a scaling in the longitudinal spectra should not be interpreted as a confirmation of the attached-eddy hypothesis.

A clear picture emerges from the combination of the present results with those reported by Davidson and Krogstad (2014), de Silva et al. (2015), and Pan and Chamecki (2016). For unstratified flows, when there is an approximate local balance between shear production and dissipation *z* scaling. If *z* (*z* provides good scaling for

MC and NLD are grateful for support from the Brazilian National Council for Scientific and Technological Development (CNPq) under Research Grant 401146/2014-6. The AHATS data were collected by NCAR’s Integrated Surface Flux Facility. We would like to acknowledge operational, technical and scientific support provided by NCAR’s Earth Observing Laboratory, sponsored by the National Science Foundation.

# APPENDIX

## Estimation of the TKE Dissipation Rate from Sonic Data

*r*within the inertial subrange and

A summary of the analysis of the estimated TKE dissipation rate is presented in Fig. A1. There is practically no difference between the two estimates obtained by using different averaging intervals in the second-order structure functions (Fig. A1b), suggesting that setting the upper limit to *ε* as using the variable *r* than

## REFERENCES

Banerjee, T., G. G. Katul, S. Salesky, and M. Chamecki, 2015: Revisiting the formulations for the longitudinal velocity variance in the unstable atmospheric surface layer.

,*Quart. J. Roy. Meteor. Soc.***141**, 1699–1711, doi:10.1002/qj.2472.Banerjee, T., D. Li, J.-Y. Juang, and G. Katul, 2016: A spectral budget model for the longitudinal turbulent velocity in the stable atmospheric surface layer.

,*J. Atmos. Sci.***73**, 145–166, doi:10.1175/JAS-D-15-0066.1.Bolgiano, R., 1959: Turbulent spectra in a stably stratified atmosphere.

,*J. Geophys. Res.***64**, 2226–2229, doi:10.1029/JZ064i012p02226.Chamecki, M., and N. L. Dias, 2004: The local isotropy assumption and the turbulent kinetic energy dissipation rate in the atmospheric surface layer.

,*Quart. J. Roy. Meteor. Soc.***130**, 2733–2752, doi:10.1256/qj.03.155.Davidson, P. A., and P.-Å. Krogstad, 2014: A universal scaling for low-order structure functions in the log-law region of smooth- and rough-wall boundary layers.

,*J. Fluid Mech.***752**, 140–156, doi:10.1017/jfm.2014.286.Davidson, P. A., T. B. Nickels, and P.-Å. Krogstad, 2006: The logarithmic structure function law in wall-layer turbulence.

,*J. Fluid Mech.***550**, 51–60, doi:10.1017/S0022112005008001.Deardorff, J. W., 1970: Convective velocity and temperature scales for the unstable planetary boundary layer.

,*J. Atmos. Sci.***27**, 1211–1213, doi:10.1175/1520-0469(1970)027<1211:CVATSF>2.0.CO;2.Deissler, R. G., 1961: Effects of inhomogeneity and of shear flow in weak turbulent fields.

,*Phys. Fluids***4**, 1187–1198, doi:10.1063/1.1706194.Deissler, R. G., 1962: Turbulence in the presence of a vertical body force and temperature gradient.

,*J. Geophys. Res.***67**, 3049–3062, doi:10.1029/JZ067i008p03049.Deissler, R. G., 1998:

*Turbulent Fluid Motion.*Taylor & Francis, 478 pp.De Silva, C. M., I. Marusic, J. D. Woodcock, and C. Meneveau, 2015: Scaling of second- and higher-order structure functions in turbulent boundary layers.

,*J. Fluid Mech.***769**, 654–686, doi:10.1017/jfm.2015.122.Drobinski, P., R. Newsom, R. Banta, P. Carlotti, R. Foster, P. Naveau, and J. Redelsperger, 2002: Turbulence in a shear driven nocturnal surface layer during the CASES’99 experiment.

*Extended Abstracts, 15th Conf. on Boundary Layer and Turbulence*, Wageningen, Netherlands, Amer. Meteor. Soc., P3.8. [Available oline at https://ams.confex.com/ams/BLT/techprogram/paper_43842.htm.]Hinze, J. O., 1975:

*Turbulence*. McGraw-Hill Publishing Company, pp.Högström, U., 1988: Non-dimensional wind and temperature profiles in the atmospheric surface layer: A re-evaluation.

,*Bound.-Layer Meteor.***42**, 55–78, doi:10.1007/BF00119875.Högström, U., 1990: Analysis of turbulence structure in the surface layer with a modified similarity formulation for near-neutral conditions.

,*J. Atmos. Sci.***47**, 1949–1972, doi:10.1175/1520-0469(1990)047<1949:AOTSIT>2.0.CO;2.Högström, U., J. C. R. Hunt, and A. Smedman, 2002: Theory and measurements for turbulence spectra and variances in the atmospheric neutral surface layer.

,*Bound.-Layer Meteor.***103**, 101–124, doi:10.1023/A:1014579828712.Kader, B. A., 1988: Three-layer structures of an unstably stratified atmospheric surface layer.

,*Izv. Atmos. Oceanic Phys.***24**, 1235–1250.Kader, B. A., and A. M. Yaglom, 1989: Spectra and correlation functions of surface layer atmospheric turbulence in unstable thermal stratification.

*Turbulence and Coherent Structures*, O. Metais, and M. Lesieur, Eds., Kluwer Academic Press, 387–412, doi:10.1007/978-94-015-7904-9_24.Kader, B. A., and A. M. Yaglom, 1990: Mean fields and fluctuation moments in unstably stratified turbulent boundary layers.

,*J. Fluid Mech.***212**, 637–662, doi:10.1017/S0022112090002129.Kaimal, J. C., and J. Finnigan, 1994:

*Atmospheric Boundary Layer Flows.*Oxford University Press, 289 pp.Kaimal, J. C., J. C. Wyngaard, Y. Izumi, and O. R. Cot, 1972: Spectral characteristics of surface-layer turbulence.

,*Quart. J. Roy. Meteor. Soc.***98**, 563–589, doi:10.1002/qj.49709841707.Katul, G., and C. Chu, 1998: A theoretical and experimental investigation of energy-containing scales in the dynamic sublayer of boundary-layer flows.

,*Bound.-Layer Meteor.***86**, 279–312, doi:10.1023/A:1000657014845.Kolmogorov, A. N., 1941: The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers (in Russian).

,*Dokl. Akad. Nauk SSSR***30**, 299–303.Lauren, M. K., M. Menabde, A. W. Seed, and G. L. Austin, 1999: Characterisation and simulation of the multiscaling properties of the energy-containing scales of horizontal surface-layer winds.

,*Bound.-Layer Meteor.***90**, 21–46, doi:10.1023/A:1001749126625.Lee, X., W. Massman, and B. Law, 2004:

*Handbook of Micrometeorology: A Guide for Surface Flux Measurement and Analysis*. Atmospheric and Oceanographic Sciences Library, Vol. 29, Kluwer Academic Publishers, 250 pp.Lumley, J. L., 1964: The spectrum of nearly inertial turbulence in a stably stratified fluid.

,*J. Atmos. Sci.***21**, 99–102, doi:10.1175/1520-0469(1964)021<0099:TSONIT>2.0.CO;2.Mater, B. D., S. M. Schaad, and S. K. Venayagamoorthy, 2013: Relevance of the Thorpe length scale in stably stratified turbulence.

,*Phys. Fluids***25**, 076604, doi:10.1063/1.4813809.Pan, Y., and M. Chamecki, 2016: A scaling law for the shear-production range of second-order structure functions.

,*J. Fluid Mech.***801**, 459–474, doi:10.1017/jfm.2016.427.Perry, A. E., S. Henbest, and M. S. Chong, 1986: A theoretical and experimental study of wall turbulence.

,*J. Fluid Mech.***165**, 163–199, doi:10.1017/S002211208600304X.Pope, S. B., 2000:

*Turbulent Flows.*Cambridge University Press, 802 pp.Salesky, S. T., and M. Chamecki, 2012: Random errors in turbulence measurements in the atmospheric surface layer: Implications for Monin–Obukhov similarity theory.

,*J. Atmos. Sci.***69**, 3700–3714, doi:10.1175/JAS-D-12-096.1.Shah, S. K., and E. Bou-Zeid, 2014: Direct numerical simulations of turbulent Ekman layers with increasing static stability: Modifications to the bulk structure and second-order statistics.

,*J. Fluid Mech.***760**, 494–539, doi:10.1017/jfm.2014.597.Sreenivasan, K. R., 1995: On the universality of the Kolmogorov constant.

,*Phys. Fluids***7**, 2778–2784, doi:10.1063/1.868656.Tchen, C. M., 1953: On the spectrum of energy in turbulent shear flow.

,*J. Res. Natl. Bur. Stand. (U.S.)***50**, 51–62, doi:10.6028/jres.050.009.Tong, C., and K. X. Nguyen, 2015: Multipoint Monin–Obukhov similarity and its application to turbulence spectra in the convective atmospheric surface layer.

,*J. Atmos. Sci.***72**, 4337–4348, doi:10.1175/JAS-D-15-0134.1.Townsend, A. A., 1958: Turbulent flow in a stably stratified atmosphere.

,*J. Fluid Mech.***3**, 361–372, doi:10.1017/S0022112058000045.Townsend, A. A., 1976:

*The Structure of Turbulent Shear Flow.*2nd ed. Cambridge University Press, 442 pp.UCAR/NCAR—Earth Observing Laboratory, 1990: NCAR Integrated Surface Flux System (ISFS). NCAR/Earth Observing Laboratory, doi:10.5065/D6ZC80XJ.

Vickers, D., and L. Mahrt, 1997: Quality control and flux sampling problems for tower and aircraft data.

,*J. Atmos. Oceanic Technol.***14**, 512–526, doi:10.1175/1520-0426(1997)014<0512:QCAFSP>2.0.CO;2.Wyngaard, J. C., and O. R. Coté, 1971: The budgets of turbulent kinetic energy and temperature variance in the atmospheric surface layer.

,*J. Atmos. Sci.***28**, 190–201, doi:10.1175/1520-0469(1971)028<0190:TBOTKE>2.0.CO;2.Wyngaard, J. C., O. R. Coté, and Y. Izumi, 1971: Local free convection, similarity, and the budgets of shear stress and heat flux.

,*J. Atmos. Sci.***28**, 1171–1182, doi:10.1175/1520-0469(1971)028<1171:LFCSAT>2.0.CO;2.Yaglom, A. M., 1994: Fluctuation spectra and variances in convective turbulent boundary layers: A reevaluation of old models.

,*Phys. Fluids***6**, 962–972, doi:10.1063/1.868328.