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  • View in gallery

    Ratio between local shear production P (solid lines and circles) or buoyancy production/destruction (dashed lines and squares) and local rate of TKE dissipation ε as a function of the stability parameter . Orange and black lines represent the two ratios from MOS functions corresponding to Kansas and the fits by Högström (1990), respectively. Individual runs are shown with open gray symbols, and ensemble averages using the stability intervals in Table 1 are shown with filled symbols. Error bars represent one standard deviation from the ensemble mean.

  • View in gallery

    Ratio between the dissipation-based length scale and the usual surface-layer length scale as a function of the stability parameter . Individual runs are shown with open gray symbols, and ensemble averages using the stability intervals in Table 1 are shown with filled symbols. Error bars represent one standard deviation from the ensemble mean. Orange and black lines represent the two ratios from MOS functions corresponding to Kansas (Kaimal and Finnigan 1994) and the fits by Högström (1990), respectively.

  • View in gallery

    Vertical profiles of (a) dimensionless mean velocity gradient, (b) dissipation length scale, (c) ratio between shear production and dissipation of TKE, and (d) ratio between buoyancy production/destruction and dissipation for the three sample runs listed in Table 2. The gray line in (b) represents the log-layer production length scale .

  • View in gallery

    Scaling of second-order structure functions for near-neutral case Nb1. Longitudinal structure functions scaled by (a),(c) z and (b),(d) , shown in (a),(b) a log–log scale to emphasize the power-law behavior in the inertial subrange and (c),(d) a log–linear scale to emphasize the ln(r) scaling. (e) Shear production density and (f) buoyancy production/destruction density are also shown. The gray line in (b) and (d) is the fit from de Silva et al. (2015) with and . Dashed vertical gray lines indicate the estimated end of the inertial subrange at .

  • View in gallery

    As in Fig. 4, but for the stable block Sb1.

  • View in gallery

    As in Fig. 4, but for the unstable block Ub1. In (b), the 2/3 slope predicted for the buoyancy-dominated case is also shown.

  • View in gallery

    As in Fig. 4, but for the unstable block Ub2.

  • View in gallery

    Second-order longitudinal structure functions for 24 selected blocks (a),(c),(e),(g) scaled by z and (b),(d),(f),(h) scaled by for four stability groups: (a),(b) near-neutral, (c),(d) unstable, (e),(f) strongly unstable, and (g),(h) stable. Filled circles in (b),(d),(f), and (h) indicate the ensemble average for each stability group, and error bars indicate one standard deviation.

  • View in gallery

    Ensemble-average second-order longitudinal structure functions for each of the 11 stability groups described in Table 1. Gray lines represent inertial range scaling and the logarithmic scaling in (1) with the fit from de Silva et al. (2015).

  • View in gallery

    Ensemble-average second-order longitudinal structure functions for stability groups (a) N, (b) S1, (c) S2, (d) S3, (e) S4, and (f) S5 described in Table 1. Green lines represent the log-law [(1)] with coefficients fitted to the data, and gray lines indicate the fits from de Silva et al. (2015) for comparison. Magenta arrows indicate regions where the log-law scaling is observed. Error bars indicate one standard deviation about the ensemble average.

  • View in gallery

    As in Fig. 10, but for stability group U1. Red crosses represent data for stability group U2.

  • View in gallery

    (a) TKE dissipation rate estimated from using a fixed interval for the inertial subrange normalized by surface-layer scales displayed as a function of . Orange and black lines represent the MOS functions corresponding to Kansas and the fits by Högström (1990), respectively. (b) Comparison between TKE dissipation rates obtained from using fixed and variable (i.e., based on ) intervals. The linear fit (magenta dashed line) yields a slope of with . (c) Comparison between TKE dissipation rates obtained from and using fixed intervals. The linear fit (magenta dashed line) yields a slope of with . (d) Power-law exponents obtained for the inertial subrange of and using the fixed interval. Solid horizontal black lines indicate the values expected from Kolmogorov’s theory, and the gray areas indicate a variation of ±10% around the theoretical values.

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Scaling Laws for the Longitudinal Structure Function in the Atmospheric Surface Layer

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  • 1 Department of Atmospheric and Oceanic Sciences, University of California, Los Angeles, Los Angeles, California
  • 2 Department of Environmental Engineering, Federal University of Paraná, Curitiba, Brazil
  • 3 Department of Civil Engineering, University of British Columbia, Vancouver, British Columbia, Canada
  • 4 Department of Meteorology, The Pennsylvania State University, University Park, Pennsylvania
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Abstract

Scaling laws for the longitudinal structure function in the atmospheric surface layer (ASL) are studied using dimensional analysis and matched asymptotics. Theoretical predictions show that the logarithmic scaling for the scales larger than those of the inertial subrange recently proposed for neutral wall-bounded flows also holds for the shear-dominated ASL composed of weakly unstable, neutral, and all stable conditions (as long as continuous turbulence exists). A 2/3 power law is obtained for buoyancy-dominated ASLs. Data from the Advection Horizontal Array Turbulence Study (AHATS) field experiment confirm these scalings, and they also show that the length scale formed by the friction velocity and the turbulent kinetic energy dissipation rate consistently outperforms the distance from the ground z as the relevant scale in all cases regardless of stability. With this new length scale, the production range of the longitudinal structure function collapses for all measurement heights and stability conditions. A new variable to characterize atmospheric stability emerges from the theory: namely, the ratio between the buoyancy flux and the TKE dissipation rate.

Current affiliation: National Center for Atmospheric Research, Boulder, Colorado.

© 2017 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author e-mail: Marcelo Chamecki, chamecki@ucla.edu

Abstract

Scaling laws for the longitudinal structure function in the atmospheric surface layer (ASL) are studied using dimensional analysis and matched asymptotics. Theoretical predictions show that the logarithmic scaling for the scales larger than those of the inertial subrange recently proposed for neutral wall-bounded flows also holds for the shear-dominated ASL composed of weakly unstable, neutral, and all stable conditions (as long as continuous turbulence exists). A 2/3 power law is obtained for buoyancy-dominated ASLs. Data from the Advection Horizontal Array Turbulence Study (AHATS) field experiment confirm these scalings, and they also show that the length scale formed by the friction velocity and the turbulent kinetic energy dissipation rate consistently outperforms the distance from the ground z as the relevant scale in all cases regardless of stability. With this new length scale, the production range of the longitudinal structure function collapses for all measurement heights and stability conditions. A new variable to characterize atmospheric stability emerges from the theory: namely, the ratio between the buoyancy flux and the TKE dissipation rate.

Current affiliation: National Center for Atmospheric Research, Boulder, Colorado.

© 2017 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author e-mail: Marcelo Chamecki, chamecki@ucla.edu
Keywords: Buoyancy; Turbulence

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 behavior. On the basis of the concept that, near the wall, shear flows are affected by “active turbulence,” which scales with the distance to the wall 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 range previously predicted by Tchen (1953) and Perry et al. (1986) for wall turbulence; for the “dynamic–convective sublayer” (the region where buoyancy becomes important for the production of turbulent kinetic energy), they predicted a range; finally, for the “free-convection sublayer,” the prediction is a range that is distinct from the inertial subrange scaling. A few years later, Yaglom (1994) also arrived at the scaling using an approach akin to the one introduced by Perry et al. (1986). More recently, Tong and Nguyen (2015) applied dimensional analysis to the multipoint velocity difference probability density function and arrived at the conclusion that, for weak unstable stratification (characterized by , where L is the Obukhov length scale) the longitudinal spectrum displays a slope in the low-wavenumber range, which is connected to the inertial subrange by a slope. Banerjee et al. (2015) used a spectral budget approach to obtain the scaling in near-neutral conditions and the region in strongly unstable conditions . For stable ASLs, recent work by Banerjee et al. (2016) also based on spectral budgets suggests the validity of scaling for weak stratification (characterized by ) and for stronger stratification . These two studies suggested a certain degree of symmetry in the spectral response to buoyancy, with both stable and unstable flows displaying scaling when buoyancy is weak and scaling when buoyancy is very strong. For the case of turbulence under strong stable stratification (characterized by the assumption that dissipation is much larger than shear production), the prediction from Bolgiano (1959) results in a steeper decay in the spectral density given by . Bolgiano’s study was followed by the classic scaling proposed by Lumley (1964) in the so-called buoyancy range of the spectrum. A large body of literature has developed since these pioneering papers by Bolgiano (1959) and Lumley (1964), and a review of the topic is outside the scope of the present paper.

On the observational front, the existence of the scaling in the near-neutral surface layer is supported by a number of datasets (e.g., Kader and Yaglom 1989; Katul and Chu 1998; Högström et al. 2002). Similarly, several independent datasets show slopes close to at scales much larger than those in the inertial subrange as free convection is approached (Kader and Yaglom 1989; Banerjee et al. 2015). The region for the dynamic–convective sublayer does not seem to have received support after the work of Kader and Yaglom (1989). Instead, data presented by Banerjee et al. (2015) suggest a smooth transition from the regime to the scaling as instability increases. Very few observational studies exist for stable conditions. Drobinski et al. (2002) showed one single run displaying a region, but results from Lauren et al. (1999) suggest that the negative power-law exponent increases in magnitude with increasing stability and deviates from the even for weak stratification.

Davidson et al. (2006) chose a slightly different approach and showed that the region of the spectrum corresponds to a range in the longitudinal structure function , where u is the streamwise velocity component and is a unit vector in the streamwise direction. The authors showed that the scaling in is more clearly identified than the corresponding scaling in the spectrum, effectively bringing the discussion of energy density in the large scales to physical space (Kolmogorov’s choice for the inertial subrange 60 yr before). Although the nomenclature varies somewhat depending on the authors, it is now widely recognized [after Townsend (1976), Perry et al. (1986), Kader and Yaglom (1989), and Davidson et al. (2006)] that in shear flows three regions can be identified, either in the longitudinal spectrum or in the longitudinal structure function, before the dissipation range. They are, in order of increasing spatial scales, (i) the inertial subrange proper with or behavior; (ii) a production range with a or behavior (in the case of unstratified flows), and (iii) an inactive range that is influenced by the “outer scales” (e.g., the thickness δ of the boundary layer).

In all of the aforementioned works, it was explicitly assumed that the distance from the lower boundary (the wall) 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 and the TKE dissipation rate ε is the correct scale for the longitudinal structure function for both smooth and rough wall boundary layer flows. This approach, which also predicts a range beyond the inertial subrange, relied on a matching of the energy density similar to that used by Perry et al. (1986). Note that replacing 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 flows
e1
where the dissipation-based length scale is defined to be
e2
and 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 over z as a length scale were for somewhat moderate-friction Reynolds numbers , on the order of 4000 for smooth and 12 000 for rough turbulent flow (estimated on the basis of their published data). De Silva et al. (2015) showed that, as increases and an equilibrium between shear production and viscous dissipation of TKE is attained, becomes proportional to z, and the two scalings are equivalent.

Pan and Chamecki (2016) recast the discussion in terms of the integral scale for the active motions . The authors showed that the logarithmic scaling (1) can be obtained for unstratified flows even if the integral scale is explicitly included in the dimensional analysis. In fact, they showed that is a requirement for the dimensional analysis by Perry et al. (1986) to be consistent with Kolmogorov’s (1941, hereafter K41) theory, providing a more formal basis to (1). Therefore, the logarithmic scaling is independent of any assumptions about the integral scale, and the only requirements for it to hold are (i) existence of one single velocity scale and (ii) the existence of “inactive eddies” that are much larger than the local integral scale . Pan and Chamecki (2016) used large-eddy simulation to show that the logarithmic scaling holds in the roughness sublayer above a model canopy, where shear production is much larger than the local dissipation rate.

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 . No assumptions are made about the integral length scale. The focus is on the scales larger than those for the inertial subrange, and the analysis is divided into two limiting cases: shear-dominated and buoyancy-dominated regimes. The applicability of these regimes to ASL flows is discussed. Measurements in the surface layer obtained during the Advection Horizontal Array Turbulence Study (AHATS) field campaign are then used to verify theoretical predictions.

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

Following Perry et al. (1986) and Pan and Chamecki (2016), we start from three separate scalings for the universal, local production (or active), and inactive scales, and include in the analysis the rate of production/destruction of TKE by buoyancy (or buoyancy flux) , where is the buoyancy parameter, is virtual temperature, and is the buoyancy flux at the ground (hereinafter an overbar indicates an ensemble-averaged variable, and a prime indicates turbulent fluctuations around the ensemble average). The proposed scalings are, respectively,
e3
e4
e5
Here, is the local integral length scale, η is the Kolmogorov length scale, and is an outer scale such as the boundary layer height . The integral scale is included in the scaling of the active motions as done by Perry et al. (1986), except that is not assumed here. Buoyancy effects are included explicitly in the inactive and production ranges, but possible buoyancy effects in the universal range can only enter via the rate of energy transfer across scales in the inertial subrange ε. Note also that the three scaling ranges have overlapping regions, where two scalings must be valid and the asymptotic matching takes place.
In the methodology proposed by Perry et al. (1986), the overlap region of universal and local production scales gives rise to the inertial subrange. Therefore, both and should be consistent with the K41 theory for the inertial subrange
e6
In practice, the requirement that be consistent with K41 requires that the effects of , , and enter the scaling via ε, leading to
e7
Only two dimensionless groups can be formed from the four physical variables in (7). Here we choose to write and , yielding
e8
The choice of dimensionless groups is not unique. This specific combination is useful because, in the neutral case for which , (8) reduces to , as hypothesized by Townsend (1958) and demonstrated by Pan and Chamecki (2016) to be a requirement for consistency between the analysis of Perry et al. (1986) and K41. Nevertheless, it is also useful to point out that one could have defined and , where is the Obukhov length scale. This alternative choice yields
e9
where . Note that this definition is similar to that of the Monin–Obukhov similarity variable (except for use of the length scale instead of , where κ is the von Kármán constant). It is perhaps interesting to point out that a third option would be to write , which becomes the traditional Monin–Obukhov similarity function if the integral length scale is replaced by .

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 is dominated by shear from those under which it is dominated by buoyancy. The applicability of these two cases to ASL turbulence is discussed in section 2c.

a. Case I: Shear-dominated regime

If shear is the dominant mechanism in determining the energy content of , then is the natural choice of velocity scale for both local production and inactive scales. The matching in the overlap region between (4) and (5) is done for the energy density , so we write
e10
e11
Similar to , we have defined .
In the overlap region of local production and inactive scales, both (10) and (11) must be valid, and we have
e12
Expanding around the neutral state using Taylor’s series yields
e13
where represents “on the same order of magnitude,” and the fact that the correction due to buoyancy cannot be larger than the neutral term itself in a shear-dominated regime was used. Therefore, we can write
e14
with the term inside square brackets being of order 1. Similarly, one obtain for
e15
Because the two terms in brackets are of order 1, the neutral state alone can explain the ratio in (12) being of order , and we can write
e16
Therefore, we have
e17
which can be substituted into (10) to yield an ordinary differential equation for :
e18
Before integration, it is more convenient to rewrite (18) in terms of :
e19
Note that is possible using and (8).
Integration of (19) yields
e20
Thus, in the shear-dominated regime, the longitudinal second-order structure function displays logarithmic dependence on similar to the one found for neutral conditions. However, stability corrections enter via the constants of integration that now become functions of the stability parameter (i.e., the ratio between rates of buoyancy production/destruction and dissipation of TKE).

b. Case II: Buoyancy-dominated regime

If buoyancy is the dominant mechanism in determining the energy content of , then the shear velocity is not as important, and and are the characteristic velocity scales for local production and inactive scales, respectively. This leads to the following scalings for the production and inactive ranges:
e21
and
e22
Note that is the velocity scale for free convection (Deardorff 1970), and is the local free-convection velocity (Wyngaard et al. 1971) evaluated at .
In the overlap region of local production and inactive scales, both (21) and (22) must be valid, and we have
e23
We follow the same procedure of using Taylor’s series expansion to express and , this time around the free-convection limit . Because shear effects are small compared with buoyancy effects [now is a small parameter with and ], we keep only the first-order term in the expansions. Following the same approach of (13)(16) yields
e24
Plugging (24) into (21) yields
e25
We can write , invoking (9) and integrating (25) to obtain
e26
Note that appears on both sides of the equation, and its presence is only needed for the solution to be in dimensionless form. Alternatively, (26) can be written in dimensional form as
e27
Any length scale could be used to normalize r, as long as the function and the velocity scale required to normalize are adjusted accordingly. Note that in (27) contains the local integral length scale . In the present analysis, is not specified, and the only relationship available to eliminate it from the final result is (8). Therefore, using as a length scale is the only way available at the moment to obtain an equation with all parameters depending only on . The corresponding velocity scale becomes , and the solution can be recast in the form
e28
Here we have defined and , the former being possible once again by invoking (9). This choice does not imply that is the characteristic velocity scale for local production scales in the buoyancy-dominated regime [as already pointed out, is the characteristic velocity scale]. The advantage of using stems from the fact that it brings in as the length scale. Thus, the selection of is the only way to collapse data measured at various heights for a given stability condition characterized by . Thus, (28) is a convenient form of expressing the solution for the buoyancy-dominated regime in (27) in the presence of finite shear (note that this is not a free-convection solution).
If one decides to use z as the length scale for r, then (27) becomes
e29
indicating that the local free-convection velocity must be the velocity scale for . This result is in agreement with the spectral scaling for the strongly unstable surface layer obtained by Kader and Yaglom (1989) and Yaglom (1994). However, here this prediction is made for the buoyancy-dominated regime, which does not necessarily require vanishing shear. Note that when is used in (29), even if (8) is employed to eliminate , the final result is given by , requiring two dimensionless parameters ( to describe and to fully characterize the solution.

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 or (here is the TKE) and the shear and buoyancy time scales and , respectively. From these three time scales, two independent dimensionless numbers can be formed, and a number of combinations is possible (e.g., Mater et al. 2013). In the previous subsections, the effects of buoyancy and shear on the energy content of have been used to determine the appropriate velocity scale for . For consistency, a classification based on length scales associated with the main terms of the TKE budget is employed here. These length scales are the TKE dissipation-based scale , the shear production length scale , and the buoyancy production/destruction length scale . From these three length scales, we form two dimensionless numbers and to classify the flow regime (only in the special case in which the TKE budget is in local balance the two dimensionless numbers are not independent and are related by ).

Within the framework of Monin–Obukhov similarity (MOS) theory, which synthesizes most of the current knowledge of surface-layer turbulence, the ratios of shear production P and buoyancy production/destruction to dissipation can be expressed as
e30
e31
Here, and are the similarity functions for the mean shear and the dissipation rate of TKE, respectively. These two ratios are shown as a function of in Fig. 1 using the empirical similarity functions from the Kansas experiment (Kaimal and Finnigan 1994) and the ones proposed by Högström (1988) and Högström (1990). A discussion of the data presented in Fig. 1 is deferred to section 4.
Fig. 1.
Fig. 1.

Ratio between local shear production P (solid lines and circles) or buoyancy production/destruction (dashed lines and squares) and local rate of TKE dissipation ε as a function of the stability parameter . Orange and black lines represent the two ratios from MOS functions corresponding to Kansas and the fits by Högström (1990), respectively. Individual runs are shown with open gray symbols, and ensemble averages using the stability intervals in Table 1 are shown with filled symbols. Error bars represent one standard deviation from the ensemble mean.

Citation: Journal of the Atmospheric Sciences 74, 4; 10.1175/JAS-D-16-0228.1

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 in Fig. 1. This is not the case for the functions proposed by Högström, which yield for neutral conditions and values slightly larger than 1 for most of the stable range. Nevertheless, both sets of functions suggest an approximate balance between shear production and dissipation, with a smaller buoyancy destruction. This view is also supported by the DNS studies of Shah and Bou-Zeid (2014) showing that the main effect of buoyancy in stable Ekman boundary layers is to reduce production and that the direct destruction of TKE is secondary. Therefore, there is enough evidence to suggest, that for the stable ASL, the scaling of is dominated by shear with only a secondary influence of buoyancy, and the logarithmic scaling in (20) is applicable.

Conversely, in the unstable ASL, the importance of buoyancy effects as measured by grows quickly with increasing instability. Thus, the logarithmic scaling in (20) is expected to be valid only for weakly unstable conditions. On the other hand, for strongly unstable conditions, we expect the scaling to follow (28), but a theoretical prediction for the transition between the two scalings is still not at hand. At any rate, on the basis of the experimental spectra presented by Banerjee et al. (2015), a gradual transition between the shear-dominated scaling in (20) and the buoyancy-dominated scaling in (28) is expected for unstable conditions.

d. Other useful considerations

Two other structure functions that are useful in the interpretation of the observational results for are presented next. In unstratified shear flows, energy is injected into the flow along the longitudinal direction and redistributed to the other directions by pressure. For flows with unstable (stable) density stratification, energy is injected (removed) directly into (from) the vertical direction by the buoyancy term. A full analysis of budgets of spectra or structure functions is beyond the scope of the present work, but a fair amount of insight into the scales involved can be obtained by looking at the corresponding mechanical and buoyant production terms in structure function form. From the budgets of the correlation functions found, for example, in Hinze (1975) and Deissler (1961, 1962, 1998) and from the relationship between the correlation and structure functions,
e32
the structure functions involved in mechanical and buoyancy production (or destruction) are found to be, respectively, and . Because structure functions show cumulative covariances up to a distance 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 (Salesky and Chamecki 2012). Data from the AHATS profile tower, consisting of six CSAT-3 sonic anemometers (Campbell Scientific Inc.) at heights of 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 217 points). Only data with wind directions of , where 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 was followed for each block of time. Small biases in the mean horizontal velocity of at and at (with respect to the other four sonics) were removed from the raw data for all runs during preprocessing (Salesky and Chamecki 2012). No other preprocessing was applied to the dataset. Blocks of data were rejected if the normalized vertical velocity standard deviation exhibited more than a 30% deviation from the value predicted by MOS, a common quality control criterion in micrometeorology (Lee et al. 2004). Nonstationary ratios (Vickers and Mahrt 1997) for the along-wind (RNu), crosswind (RNv), and vector-wind (RNS) velocity components were computed, and runs where RNu, RNv, or RNS ≥ 0.5 were excluded. A total of 216 blocks passed all quality control criteria and were used in the present analysis. These 216 blocks of data times 6 separate sonics totals 1296 samples of structure functions under various stability conditions. Taylor’s hypothesis was used to interpret temporal lags as spatial lags in [i.e., , where τ is the time lag].

The values of and 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 determined from the AHATS data follow traditional MOS closely (Salesky and Chamecki 2012), except that the data suggested a von Kármán constant . For the purpose of the present analysis, the value of κ enters only into the definition of L, and the more traditional value of is employed. TKE dissipation rates 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 ). The values obtained from the second-order structure function were considered more reliable and were used in all the analyses. More details of the method used to estimate the dissipation rate and a comparison between estimates obtained from second- and third-order structure functions are presented in the appendix. Shear and buoyancy production were calculated from their definitions and .

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 ]. The normalization with is designed to collapse the inertial subrange. However, the collapse of the production range is not guaranteed by the collapse of the inertial subrange.

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 , which falls between the perfect balance found from the Kansas data and the larger imbalance suggested by Högström (1990). A study of the causes for such imbalance is beyond the scope of the present investigation, but its consequences for scaling the structure function are important and will be discussed below. At this point, Fig. 1 suggests that the applicability of the shear- and buoyancy-dominated regimes discussed in section 2c should hold for the AHATS data.

Table 1.

Stability groups used for ensemble averages. The coefficients in the log-law [(1)] fitted to the data are represented by and (see Figs. 10 and 11).

Table 1.

Before proceeding to assess the and the z scalings for , it is useful to compare the stability dependence of the two length scales. For that purpose, is shown as a function of in Fig. 2. Note that if one starts from the assumption that under neutral conditions shear production is given by , then . Therefore, as for the near-neutral data (Fig. 1), one would expect as well. This is confirmed by Fig. 2. This interpretation is only valid for neutral conditions. However, it is always true that , and a comparison using the expressions for from Kansas and Högström (1990) is also provided in Fig. 2 (the latter being a better representation of the AHATS data). The possible existence of self-correlation in Fig. 2 is not relevant for the present analysis, since the goal of the figure is to emphasize the difference in behavior of the two length scales ( and ) with stability and not to establish a relationship between and .

Fig. 2.
Fig. 2.

Ratio between the dissipation-based length scale and the usual surface-layer length scale as a function of the stability parameter . Individual runs are shown with open gray symbols, and ensemble averages using the stability intervals in Table 1 are shown with filled symbols. Error bars represent one standard deviation from the ensemble mean. Orange and black lines represent the two ratios from MOS functions corresponding to Kansas (Kaimal and Finnigan 1994) and the fits by Högström (1990), respectively.

Citation: Journal of the Atmospheric Sciences 74, 4; 10.1175/JAS-D-16-0228.1

Townsend (1958) argued that for neutral flows the integral length scale is proportional to the dissipation-based length scale (i.e., ). Pan and Chamecki (2016) have shown that this is indeed a requirement for the dimensional analysis of Perry et al. (1986) to be consistent with K41. For the stratified case, the same requirement only guarantees that is a function of the stability parameter , as indicated by (8). Therefore, the same interpretation is not possible in the presence of stratification. Nevertheless, the strong reduction in with increasing stability will play an important role in the next sections.

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 obtained from the six measurement heights is assessed using z and to scale the separation distance 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.

Table 2.

Statistical characterization of selected blocks (all quantities evaluated at the top sonic height ). The symbol is a lower bound on the true value, estimated with a very low .

Table 2.
Fig. 3.
Fig. 3.

Vertical profiles of (a) dimensionless mean velocity gradient, (b) dissipation length scale, (c) ratio between shear production and dissipation of TKE, and (d) ratio between buoyancy production/destruction and dissipation for the three sample runs listed in Table 2. The gray line in (b) represents the log-layer production length scale .

Citation: Journal of the Atmospheric Sciences 74, 4; 10.1175/JAS-D-16-0228.1

Despite the near-neutral stability parameter , block Nb1 actually displays a slightly stable mean velocity gradient profile. As expected, unstable (stable) stratification leads to reduced (increased) mean shear (Fig. 3a). The ratio between shear production and dissipation is fairly similar for the neutral and stable cases (Fig. 3c), except for the two intermediate points that yield fairly low values of for the stable case (these small values are not typical of the stable data shown in Fig. 1). However, because of the strong effects of buoyant production of TKE (, Fig. 3d), the unstable cases present much larger imbalance between shear production and dissipation (this is particularly accentuated for Ub2).

The second-order structure function analysis for the near-neutral run Nb1 is presented in Fig. 4. The difference between z scaling and scaling for is small, with the collapse of the curves being only slightly better when is employed. This result is expected, given the small imbalance between shear production and dissipation ( for this run; see Fig. 3) and the fact that 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 ). The agreement with the empirical fit of de Silva et al. (2015) shown as a gray line is also very good (small differences may be due to uncertainties in the estimation of ε and issues associated with the statistical convergence of for large r).

Fig. 4.
Fig. 4.

Scaling of second-order structure functions for near-neutral case Nb1. Longitudinal structure functions scaled by (a),(c) z and (b),(d) , shown in (a),(b) a log–log scale to emphasize the power-law behavior in the inertial subrange and (c),(d) a log–linear scale to emphasize the ln(r) scaling. (e) Shear production density and (f) buoyancy production/destruction density are also shown. The gray line in (b) and (d) is the fit from de Silva et al. (2015) with and . Dashed vertical gray lines indicate the estimated end of the inertial subrange at .

Citation: Journal of the Atmospheric Sciences 74, 4; 10.1175/JAS-D-16-0228.1

Structure functions involved in shear and buoyancy production are presented in Figs. 4e and 4f. Note that we plot and so that the area under the curve is preserved and the contributions of each scale to the total production can be inferred from the figure. The normalization of r using also collapses the shear production density at all levels, at least within the range where the estimates are reliable . At all levels most of the shear production occurs around and becomes negligible before and after . This result indicates that (i) the logarithmic scaling in occurs in scales corresponding to the shear production range, (ii) based on shear production the inertial subrange extends up to , and (iii) the large-scale end of the log-scaling region is more or less coincident with the scales at which shear production shuts off. This may be an indication that the logarithmic scaling holds for scales in which shear production is active and that the large-scale end may not be directly related to the boundary layer depth δ. As expected for near-neutral stratification, buoyancy production/destruction is negligible.

The analysis for the stable block Sb1 (for this run, ; ) shown in Fig. 5 is very revealing. Note that is far superior to z in collapsing the measurements at different heights and that the entire collapses onto one single curve that seems to follow the logarithmic scaling in (20) as predicted for shear-dominated cases in section 2. The scales contributing the most to shear production are located around , which is similar to the neutral case. Despite the approximate balance between local shear production and dissipation in stable blocks, z is no longer an appropriate estimate of the shear production length scale and cannot collapse . This result suggests that provides an adequate description of the reduction in length scales with increasing stability, as anticipated from Fig. 2. Buoyancy destruction is very small, and TKE is dominated by a near balance between shear production and dissipation. As in the neutral block, a very good collapse of the shear production density is obtained.

Fig. 5.
Fig. 5.

As in Fig. 4, but for the stable block Sb1.

Citation: Journal of the Atmospheric Sciences 74, 4; 10.1175/JAS-D-16-0228.1

The analysis for the unstable block Ub1 (for this run, ; ) is shown in Fig. 6. Shear production is the dominant production mechanism only in the first level, and it shuts off around . Buoyancy is the dominant mechanism of TKE production for this block, and it is active up to or so. Consequently, the curves in Fig. 6 follow the neutral log-scaling fit for a very small range of scales around and quickly diverge from it. However, the collapse of obtained at different heights against remains good far beyond . Specifically, at the lowest level separates around , while the curves for the other levels remain together almost up to . In the shear-dominated regime, the good collapse of against suggests that is capable of capturing much of the modification of the energy distribution in introduced by buoyancy production. In the buoyancy-dominated regime, although the definition of the integral length scale is arbitrary (discussed in section 2b), is better than z in collapsing when is used as the velocity scale for normalization.

Fig. 6.
Fig. 6.

As in Fig. 4, but for the unstable block Ub1. In (b), the 2/3 slope predicted for the buoyancy-dominated case is also shown.

Citation: Journal of the Atmospheric Sciences 74, 4; 10.1175/JAS-D-16-0228.1

Some unstable blocks behave differently from Ub1. This is true, in our data, of some blocks with larger values of and smaller values of (these blocks also tend to have larger turbulence intensity). As an example, results for Ub2 ( and ) are shown in Fig. 7. Despite the similar values of for Ub1 and Ub2 (see Table 2), Ub2 has much larger values of (see Fig. 3d). Using as the velocity scale for normalization, the collapse of obtained at different heights is highly superior when is used. This is particularly clear for the four highest sonics, for which buoyancy dominates the production of TKE ( and ). As shown in the log–log plot in Fig. 7b, the structure functions follow the prediction for the buoyancy-dominated regime in (28) very closely. Note also in the inset that the lowest level deviates the most from the scaling, as expected from its lower value of (and consequently higher value of ).

Fig. 7.
Fig. 7.

As in Fig. 4, but for the unstable block Ub2.

Citation: Journal of the Atmospheric Sciences 74, 4; 10.1175/JAS-D-16-0228.1

Despite the general superiority of over z in the scalings of , some caution must be used in the interpretation of this result. Pan and Chamecki (2016) found that, under conditions of high turbulent intensity, Taylor’s hypothesis breaks down in the sense that as a function of is different from as a function of . In view of that, the results in Fig. 7 may, in all fairness, be strictly representative of scaling with time lags τ in the form and .

Analysis of these four individual representative blocks of data lends support to the theory presented in section 2. More specifically, is a better length scale for collapsing in the production range of ASL data not only for near-neutral conditions but also for stable conditions. The same is true for unstable conditions if is chosen as the velocity scale used to normalize , as indicated by the theory. In addition, the shear-dominated scaling in (20) is applicable to the neutral and stable runs as suggested by the values of and . Increasing instability breaks down the validity of (20), and the buoyancy-dominated scaling in (28) seems to be a good approximation for runs with small values of and large values of .

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 , unstable , strongly unstable , and stable . For the second part of the analysis, the division is further refined into 11 stability groups, as indicated in Table 1 (the few blocks with that satisfied the data quality control are not used).

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 is used in place of z. In addition, the spread between different runs seems quite small up to (which corresponds approximately to the position of the peak in local production, well outside the inertial subrange; see Figs. 47). This confirms the superiority of the scaling in the inertial and part of the production ranges for neutral, stable, and unstable runs. For the unstable runs, it shows that is an appropriate length scale if is used to normalize , as indicated by the theory presented in section 2.

Fig. 8.
Fig. 8.

Second-order longitudinal structure functions for 24 selected blocks (a),(c),(e),(g) scaled by z and (b),(d),(f),(h) scaled by for four stability groups: (a),(b) near-neutral, (c),(d) unstable, (e),(f) strongly unstable, and (g),(h) stable. Filled circles in (b),(d),(f), and (h) indicate the ensemble average for each stability group, and error bars indicate one standard deviation.

Citation: Journal of the Atmospheric Sciences 74, 4; 10.1175/JAS-D-16-0228.1

The increase of error bars with increasing in Fig. 8 has two possible contributions. First, the theoretical results suggest that is the most appropriate stability parameter determining the behavior of . Even though there is a clear relation between and , there is also considerable scatter in the data (this is very pronounced for the unstable blocks; see Fig. 1). Second, the importance of random errors increases with increasing r, as the number of equivalent independent samples in the evaluation of the structure function decreases as . Therefore, the total number of equivalent independent samples at is one-hundredth of the number of samples at .

To further investigate the effect of stability on the dependence of on , conditional averages are calculated using the 11 stability groups described in Table 1. Results are shown in Fig. 9 together with the prediction from Kolmogorov (1941) for the inertial subrange and the prediction from Davidson and Krogstad (2014) using the empirical coefficients fit by de Silva et al. (2015). As expected, follows the inertial range behavior predicted by Kolmogorov independently of atmospheric stability. All curves seem to deviate from inertial range scaling around , suggesting that marks the end of the inertial subrange for all stabilities (note that, at , shear and buoyancy production densities are not zero and can be as large as about , but the slope of the second-order structure functions are already very close to the inertial range prediction). This result is not trivial; in the classic spectral analysis of Kaimal et al. (1972) (where z was used as the relevant length scale), the point of departure from inertial range behavior displayed strong dependence on stability via . Note that ε must be used to collapse the inertial subrange, and Kaimal et al. (1972) opted to multiply the energy spectrum by . Results presented here suggest that, if they had divided the wavenumber by instead, that would have resulted in . This would not only collapse the inertial subrange but also ensure that the end of the inertial subrange would be independent of stability.

Fig. 9.
Fig. 9.

Ensemble-average second-order longitudinal structure functions for each of the 11 stability groups described in Table 1. Gray lines represent inertial range scaling and the logarithmic scaling in (1) with the fit from de Silva et al. (2015).

Citation: Journal of the Atmospheric Sciences 74, 4; 10.1175/JAS-D-16-0228.1

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 , transitioning smoothly from the ln(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 power-law behavior in (28). The slope never reaches the buoyancy-dominated prediction, even though it is clearly very close for some individual blocks (e.g., see Fig. 7b). Once again, this is likely because the conditional average based on is not ideal, given that it mixes a fairly large range of values of . Normalized structure functions for the stable groups display a clear scaling, confirming the validity of (20).

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 scaling with increasing instability suggests a gradual steepening of the power-law exponent. It seems possible that a simple interpolation of the exponents of the energy density between −1 for shear-dominated and −1/3 for buoyancy-dominated regimes would yield reasonable agreement with the observed trends (as an example, note an intermediate power law clearly seen in the production range of the unstable block UB1 shown in Fig. 6b).

A clearer confirmation of the validity of the scaling in (20) is presented in Fig. 10, where each stability case for neutral and stable conditions is shown in an individual panel using a linear scale in the y axis together with a line corresponding to the fitted values of and for each stability group. More than one decade of scaling can be observed in neutral conditions and about a decade in all stable groups .

Fig. 10.
Fig. 10.

Ensemble-average second-order longitudinal structure functions for stability groups (a) N, (b) S1, (c) S2, (d) S3, (e) S4, and (f) S5 described in Table 1. Green lines represent the log-law [(1)] with coefficients fitted to the data, and gray lines indicate the fits from de Silva et al. (2015) for comparison. Magenta arrows indicate regions where the log-law scaling is observed. Error bars indicate one standard deviation about the ensemble average.

Citation: Journal of the Atmospheric Sciences 74, 4; 10.1175/JAS-D-16-0228.1

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 scaling also works for the weakly unstable case U1, but here the effects of buoyancy are already a bit stronger than in the stable groups (reducing the range of scaling to less than one decade). Finally, for the group U2, there is no scaling, and buoyancy production is already strongly impacting the energy content of . This analysis, together with the results presented in Fig. 1 and Table 1, suggests that the shear-dominated regime holds for .

Fig. 11.
Fig. 11.

As in Fig. 10, but for stability group U1. Red crosses represent data for stability group U2.

Citation: Journal of the Atmospheric Sciences 74, 4; 10.1175/JAS-D-16-0228.1

The coefficients resulting from fitting the prediction (1) to the data (Figs. 10, 11) are presented in Table 1. For neutral conditions, the values of and are in excellent agreement with those presented by de Silva et al. (2015), with being slightly closer to their theoretical prediction than their own experimental fits. For stable conditions, the fitted coefficients change significantly in the range (increasing from 2.48 to 2.98) but remain approximately constant for (ranging between 2.90 and 3.01). This coincides more or less with the dependence of on , which shows significant changes only in the interval , lending additional support to the conclusion that is the most appropriate stability parameter for the scaling of .

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 . An important difference between our approach and previous work in the atmospheric surface layer is that a generic integral length scale is included in the dimensional analysis instead of the distance from the ground. A consequence of the absence of z in the dimensional analysis is that replaces as the similarity variable. More interestingly, dimensional analysis suggests that is a function of an alternative similarity variable, the ratio between buoyancy flux and the rate of dissipation of TKE . Note that and are the two dimensionless parameters that characterize the importance of shear and buoyancy in the ASL and are used to classify the ASL into shear- and buoyancy-dominated regimes (Fig. 1).

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 . The second component of the theory is the identification of the applicability of these two regimes to ASL turbulence, which is based on the ratio of shear and buoyancy production to the dissipation rate ( and , respectively). In summary, the theory predicts for neutral and stable conditions (both shear dominated), as long as shear production is strong enough to sustain continuous turbulence. For unstable conditions, the theory predicts the same scaling for weak instability (shear dominated) and a power law for strong instabilities (buoyancy dominated). The latter can be conveniently expressed as if is chosen as the velocity scale for buoyancy-dominated conditions in the presence of finite shear. A prediction for intermediate instabilities is not available at the moment. Note that these results suggest a strong asymmetry in the scaling of the second-order longitudinal structure function between stable and unstable stratification. This asymmetry is a consequence of buoyancy effects not being symmetric: while buoyancy production plays a critical role in the TKE budget for unstable conditions, buoyancy destruction plays only a minor role for stable conditions.

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 function holds 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, as presents 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 , then and both are equally capable of scaling the second-order structure function. For the specific case of a high-Reynolds number log layer in which , this leads to the success of the z scaling. If , as in the canopy shear layer, then the length scales and are distinct, and the former provides the correct scaling (suggesting that, in the absence of a local balance, the energy flux into the inertial subrange is more capable of scaling the production range than local production itself). In the stable surface layer, for the range of conditions studied here, (shear-dominated regime), implying , and both scales are expected to provide good scaling for the production range. However, it is clear that is no longer proportional to z ( requires to be approximately constant: the increase with height of for stable cases corresponds to an increase of the production length scale with height slower than the neutral linear relationship), and it is no longer reasonable to expect that z provides good scaling for . Finally, in strongly unstable conditions (buoyancy-dominated regime), no clear length scale emerges from the analysis. As discussed here, the advantage of choosing is that it constrains the dependence of the empirical coefficients to one single dimensionless parameter, . The results obtained here for indicate that further analysis of ASL data on the basis of (or extending) scaling and the use of the alternative variables to characterize stability ( and ) may prove fruitful in better understanding turbulent atmospheric flows.

Acknowledgments

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

Following Chamecki and Dias (2004), the TKE dissipation rate was estimated from the inertial subrange in the second- and third-order longitudinal structure functions, and , respectively. Following Kolmogorov’s theory, the dissipation can be estimated as
ea1
ea2
where represents an average in r within the inertial subrange and is the Kolmogorov constant for the second-order structure function, is the corresponding constant for the longitudinal spectrum, and is Kolmogorov’s constant for the TKE spectrum (Pope 2000). Note that these relationships between constants yield , in agreement with observational data (Sreenivasan 1995) and . As discussed in detail by Chamecki and Dias (2004), the path averaging performed by sonic anemometers introduces errors in the smallest spatial scales sampled by the sensor. For the longitudinal component, the pathlength of the CSAT3 is . Visual inspection of many samples’ structure functions suggest that the energy density attenuation becomes significant for , so this threshold is used as the cutoff for calculation of structure functions, and it sets the lower limit of the inertial subrange used here. For the large-scale limit, Chamecki and Dias (2004) used for all stabilities. Here, this is used as a first approximation so that and are determined by averaging (A1) and (A2) in the interval . Because our analysis suggests that the end of the inertial subrange is located at , which displays strong dependence on measurement height and atmospheric stability, we refined by calculating a corrected estimate using the averaging interval , where is based on .

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 provides as good estimates of ε as using the variable . We prefer to use the constant limit because there are some blocks for which the . The values of are, on average, 31% smaller than the corresponding , in agreement with the results obtained by Chamecki and Dias (2004) for a different dataset. To investigate this issue further, we fitted power-law exponents to the inertial subrange region of and . Results clearly show that, while is already in its inertial subrange region in the interval adopted here, is not (Fig. A1d). This is also in agreement with the conclusions of Chamecki and Dias (2004), who noted that attains its inertial subrange scaling for larger values of r than and that the inertial subrange of may not be properly resolved by sonic anemometers deployed close to the ground (i.e., for or so). Thus, we use obtained from within the averaging interval as our best estimate of the TKE dissipation rate. Note that the values obtained spread around the usually accepted Monin–Obukhov similarity functions for the TKE dissipation rate (Fig. 5a), building confidence in our estimates.

Fig. A1.
Fig. A1.

(a) TKE dissipation rate estimated from using a fixed interval for the inertial subrange normalized by surface-layer scales displayed as a function of . Orange and black lines represent the MOS functions corresponding to Kansas and the fits by Högström (1990), respectively. (b) Comparison between TKE dissipation rates obtained from using fixed and variable (i.e., based on ) intervals. The linear fit (magenta dashed line) yields a slope of with . (c) Comparison between TKE dissipation rates obtained from and using fixed intervals. The linear fit (magenta dashed line) yields a slope of with . (d) Power-law exponents obtained for the inertial subrange of and using the fixed interval. Solid horizontal black lines indicate the values expected from Kolmogorov’s theory, and the gray areas indicate a variation of ±10% around the theoretical values.

Citation: Journal of the Atmospheric Sciences 74, 4; 10.1175/JAS-D-16-0228.1

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