## 1. Introduction

*F*

_{Δ}is the (homogeneous) filter function for a scale Δ. Subgrid scales (smaller than Δ) are parameterized using subgrid-scale (SGS) models. The SGS model is crucial for an LES to generate realistic turbulent fields in the ABL, especially in regions where the local integral scale is smaller than the filter scale Δ. SGS models parameterize the SGS stress

*τ*

_{ij}, whose divergence enters the filtered Navier–Stokes equations. The SGS stress,

*S̃*

_{ij}is the strain-rate tensor, |

*S̃*| =

*S̃*

_{ij}

*S̃*

_{ij}

*ν*

_{T}is the eddy viscosity. When this closure is used in a traditional LES,

*c*

^{(Δ)}

_{s}

It has often been remarked (Pope 2000; Meneveau 1994) that a single value of *c*^{(Δ)}_{s}

Once the eddy-viscosity closure is accepted, the Smagorinsky coefficient *c*^{(Δ)}_{s}_{Δ} is defined according to Π_{Δ} = −〈*τ*_{ij}*S̃*_{ij}〉. The dissipation predicted by the Smagorinsky model, ^{Smag}_{Δ}*τ*^{Smag}_{ij}*S̃*_{ij}〉 depends upon *c*^{(Δ)}_{s}*c*^{(Δ)}_{s}*c*^{(Δ)}_{s}

*c*

^{(Δ)}

_{s}

*c*

^{(Δ)}

_{s}

*c*

^{(Δ)}

_{s}

*L*); typically, the characteristic length scale

*c*

^{(Δ)}

_{s}

*c*

^{(Δ)}

_{s}

*c*

^{(Δ)}

_{s}

*z,*filter scale Δ, and atmospheric stability. In particular,

*c*

^{(Δ)}

_{s}

*z*/Δ. In a more detailed, recent study of the Smagorinsky coefficient, Kleissl et al. (2003, hereafter KMP) analyzed nearly 160 h of data from the Horizontal Array Turbulence Study (HATS), a field experiment in the ABL with 14 sonic anemometers in two arrays. KMP quantified the decrease of

*c*

^{(Δ)}

_{s}

*c*

^{(Δ)}

_{s}

*z*and Δ/

*L.*Sullivan et al. (2003) find that the measured values of

*c*

^{(Δ)}

_{s}

_{w}, the length scale at the peak of the spectrum of vertical velocity, is used to scale the filter size. In LES, parameterization of

*c*

^{(Δ)}

_{s}

_{w}/Δ is feasible if Λ

_{w}is known a priori or it can be determined from energy spectra that are computed in space (such as in LES with homogeneous boundary conditions). Still, such parameterizations require knowing empirical formulas for the coefficient as function of Λ

_{w}/Δ. In other words, even if one can obtain Λ

_{w}from the simulation's spectra during an LES, a functional form for

*c*

^{(Δ)}

_{s}

_{w}must be prescribed, with an associated need for empirical coefficients. Along a fundamentally different line of thinking, Germano et al. (1991) proposed the “dynamic model.” Instead of prescribing

*c*

^{(Δ)}

_{s}

*L*

_{ij}is the resolved stress tensor and

*α*Δ [an overline

*α*Δ]. If one applies this dynamic procedure by replacing

*T*

_{ij}and

*τ*

_{ij}by their prediction from the basic Smagorinsky model the result is

*c*

^{(Δ)}

_{s}

*c*

^{(αΔ)}

_{s}

*c*

^{(Δ)}

_{s}

*c*

^{(Δ)}

_{s}

*c*

^{(Δ)}

_{s}

*α*Δ is smaller than the local integral scale of turbulence.

*c*

^{(Δ)}

_{s}

*c*

^{(αΔ)}

_{s}

*α*Δ, a test filter at

*α*

^{2}Δ (denoted by a hat below) delivers another equation similar to Eq. (5):

*β*is defined according to

*β*does not depend on scale, Eqs. (5) and (9) can be solved for the two unknowns,

*c*

^{(Δ)}

_{s}

*β*(Porté-Agel et al. 2000a). Note that this assumption is equivalent to assuming a power-law behavior (

*c*

^{(Δ)}

_{s}

^{2}∼ Δ

^{Φ}, or dimensionally appropriate,

*c*

^{(αΔ)}

_{s}

^{2}

*c*

^{(Δ)}

_{s}

^{2}

*α*

^{Φ}

*β*see the appendix. It needs to be emphasized that

*β*is determined dynamically from the resolved scales; the power-law behavior in Eq. (11) is the only empirical assumption. Porté-Agel et al. (2000a) applied the scale-dependent dynamic SGS model to an LES of a neutral boundary layer and obtained good agreement with observations for mean velocity gradients and streamwise energy spectra.

The objective of the present study is to examine field data at various length scales and determine whether the dynamic model yields realistic predictions of the coefficient *c*^{(Δ)}_{s}

The present paper is organized as follows. In section 2, we describe the field experiment and the data processing techniques. Section 2 also contains a brief review of the results of KMP: measured distributions of *c*^{(Δ)}_{s}*c*^{(Δ)}_{s}

## 2. Dataset and processing

### a. The HATS dataset

*c*

^{(Δ)}

_{s}

*z*, filter scale Δ, and the Obukhov length

*L.*Parameter

*L*is defined as

*u*∗ = (−〈

*u*′

*w*′〉)

^{1/2}is the friction velocity,

*θ*

_{0}is the mean air temperature,

*g*is the gravitational acceleration, and

*κ*= 0.4 is the von Kármán constant. Data from two sensor setups were used to dynamically determine the Smagorinsky coefficients. These setups are presented in Table 1.

Figure 1 shows a schematic of the instrument setup for arrays 1 and 2. To compute SGS quantities, the velocity fields have to be spatially filtered in two dimensions at a scale Δ. Since the velocities will also be filtered at two larger scales, *α*Δ and *α*^{2}Δ, Δ is chosen to be smaller than the values used in KMP. Here we use twice the lateral instrument spacing as the basic filter length, that is, Δ = 2*δ*_{y}, where *δ*_{y} is the lateral spacing of the sonic anemometers. As in Sullivan et al. (2003) and Horst et al. (2004), convolutions with the top-hat filter of width Δ are evaluated using the trapezoidal integration rule. This is equivalent to using discrete weights (0.5, 1, … , 1, 0.5)/(*n* −1) for data from *n* sensors spaced in the lateral (*y*) direction. Note that other options exist to define weights for discretely sampled data. For instance, Vasilyev et al. (1998) use the second moment of the filter's discrete Fourier transfer function to relate filter width with the weights. The difference among these methods is not large when considering the compounded two-dimensional filtering: in the streamwise direction, the filtering occurs on the much finer time-sampling grid (see later), and thus there is negligible ambiguity how to relate filter width and weight factors. This streamwise filter is responsible for removing most of the SGS variance, rendering the effects of the less accurate cross-stream filters less important. A smooth Gaussian filter is used in the streamwise (*x*) direction, where 20-Hz sampling results in a higher resolution with a spacing of 0.05 s 〈*u*〉 m s^{−1} ≈ 0.13 m, using Taylor's hypothesis. The Fourier transform of the Gaussian filter function *Ĝ*_{Δ} = exp[−(*k*^{2}_{1}^{2}/ 24)] is multiplied with the Fourier transform of each 8192 data points segment (∼6.8 min) of the velocity time series. Before the convolution, the mean of the velocity time series is subtracted and a Bartlett window is applied. Derivatives are computed from the filtered time series. Due to edge effects of the filter and the streamwise derivative, a segment of duration (*α*^{2}Δ/2 + *δ*_{x})/〈*u*_{d}〉 is discarded from beginning and end of the time series of all filtered variables. Then averages for various time scales *T*_{c} are computed. For an analysis of filter accuracy, see Horst et al. (2004) and Cerutti and Meneveau (2000).

Gradients are calculated with finite differences (FDs). In the vertical direction (*x*_{3} = *z*), the setup necessitates a first-order one-sided FD ∂*ũ*/∂*z*|_{zd}*z*_{s} − *z*_{d})^{−1}[*ũ*(*z*_{s}) − *ũ*(*z*_{d})]. In the horizontal directions, a second-order centered FD scheme is used, for example, for the *y* direction: ∂*ũ*_{i}/∂*y*|_{y0}*δ*_{y})^{−1}[*ũ*_{i}(*y*_{0} + *δ*_{y}) − *ũ*_{i}(*y*_{0} − *δ*_{y})]. Assuming Taylor's hypothesis, the same formula with *δ*_{x} = *δ*_{y} is used in the streamwise direction to compute ∂*ũ*_{i}/∂*x.*

In order to depict the available data as a function of stability and array, KMP divided the data into segments of length 6.8 min. These segments were classified according to stability, characterized in terms of Obukhov length *L,* defined as in Eq. (12) and yielding a dimensionless parameter Δ/*L.* The distribution of data by stability can be seen in KMP's Fig. 2 for various heights (characterized in terms of Δ/*z*). In the present paper we use the same procedure and data classification. In the following the procedures to compute the model coefficient will be described in more detail.

### b. Empirically determined Smagorinsky coefficient: Procedures and results

_{Δ}(Clark et al. 1979; KMP),

*T*

_{c}. KMP analyzed the behavior of

*c*

^{(Δ,emp)}

_{s}

*z*and Δ/

*L.*It was found that the data can be described by a function of the form:

*R*is the ramp function. The parameters of Eq. (14) were determined as

*n*= 3 and

*α*=

*c*

_{0}= 0.135.

In the present paper, the filter size is half of that in KMP. Figure 1a provides a sketch of the filtering procedures in the transverse (*y* or *x*_{2}) direction. A three-point top-hat filter with trapezoidal weights [0.25, 0.5, 0.25] is used in the lower array and a two-point filter with weights [0.5, 0.5] is used in the upper array. In the streamwise direction, the Gaussian filter is used as described in the preceding section. Thus filtered velocities *ũ*_{i}, and SGS stresses *τ*_{ij}, at a scale Δ = 2*δ*_{y} are available at locations 7–13 and between locations 1 and 5 (Fig. 1b). As a result, the filtered strain-rate tensors can be obtained at locations 9 and 11, using a second-order centered FD in the horizontal and first-order one-sided FD in the vertical directions, respectively. Since *τ*_{ij} is available at these locations as well, the Smagorinsky coefficients *c*^{(Δ,emp)}_{s}

A first question to address is whether the data analyzed at scale Δ = 2*δ*_{y} provide results that are consistent with those of KMP that were obtained at a larger scale, using more sensors from each array. To compare our results with KMP, data from array 2 (Δ/*z* ∼ 1.1) are divided into stability bins from Δ/*L* = −1 to Δ/*L* = 5 and further divided into subsegments of length *T*_{c} = 3.2 s. This corresponds roughly to a length scale *T*_{c}〈*u*〉 ∼ 8.7 m, which is on the order of twice the filter scale Δ ∼ 4.3 m. The empirically determined Smagorinsky model coefficient *c*^{(Δ,emp)}_{s}*T*_{c}. In order to isolate the dependence on Δ/*L,* we compute the conditional PDF of (*c*^{Δ,emp}_{s}^{2}, *P*(*c*^{2}_{s}*L*) = *P*(*c*^{2}_{s}*L*)/*P*(Δ/*L*), where *P*(Δ/*L*) is the fraction of data contained in each Δ/*L* bin. The (*c*^{Δ}_{s}^{2} range [−0.03 < (*c*^{Δ}_{s}^{2} < 0.1] is divided into 260 bins. Figure 2 shows the conditional PDF of (*c*^{Δ,emp}_{s}^{2} using color contours. The figure confirms the results of KMP: *c*^{Δ,emp}_{s}*c*^{(Δ,emp)}_{s}

The comparison with KMP is repeated using a larger averaging time scale *T*_{c}. Figure 3a shows a direct comparison of data from array 1 (Δ/*z* ∼ 2.1) of the present paper with data from a better-resolved filter but same Δ/*z* from array 2 of KMP for an averaging time scale of *T*_{c} = 6.8 min. The results agree very well, even though they are obtained from two different arrays. The agreement confirms that the curves collapse for a given Δ/*z,* independent of the dimensional values of Δ or *z.* Finally, in Fig. 3b we perform a comparison based on the global time averages of SGS dissipations. Here we average the terms in Eq. (13) over all data available in each Δ/*L* bin, obtaining a single measured value of *c*^{(Δ,emp)}_{s}*c*^{(Δ)}_{s}

To provide a systematic description of the effects of averaging time *T*_{c} upon the statistics of *c*^{(Δ,emp)}_{s}*c*^{(Δ,emp)}_{s}*T*_{c}. Figure 4 displays the median of *c*^{(Δ,emp)}_{s}*T*_{c} for different stabilities. As reported in KMP, the median of *c*^{(Δ,emp)}_{s}*T*_{c}, in neutral and unstable conditions. As reported in KMP, the decrease is weaker in stable conditions, which can be attributed to larger intermittency in stable conditions.

### c. Scale-invariant dynamic model: Procedures

In order to obtain the dynamic model coefficient from Eq. (7), filtered strain-rate tensors and velocity vectors at a scale Δ have to be filtered at *α*Δ to evaluate *L*_{ij} and *M*_{ij}. Usually *α* = 2, but the limited maximum filter width in the lateral direction requires us to use *α* = 1.75 in the present study. As shown in Germano et al. (1991), the sensitivity of the dynamic coefficient to *α* is not expected to be important. Figure 1b shows that *S̃*_{ij} at a scale Δ can be obtained at locations 7, 9, 11, and 13. At locations 9 and 11 *S̃*_{ij} is computed from centered horizontal FD and one-sided vertical FD. At locations 7 and 13, the horizontal and the vertical FDs are one-sided. A filter of size 1.75Δ is applied to *ũ*_{i}, *S̃*_{ij}, |*S̃*|, and |*S̃*|*S̃*_{ij}. The filter weight *w*_{i} associated with a variable (already filtered at scale Δ) at location *y*_{i}, used to compute a test-filtered variable at location *y*_{αΔ}, is evaluated as follows: *w*^{*}_{i}*y*_{i} − Δ/2, *y*_{i} + Δ/2] ∩ [*y*_{αΔ} − *α*Δ/2, *y*_{αΔ} + *α*Δ/2]|, where [*y*_{i} − Δ/2, *y*_{i} + Δ/2] is the segment of length Δ surrounding the point *y*_{i}, and [*y*_{αΔ} − *α*Δ/2, *y*_{αΔ} + *α*Δ/2] is the segment of length *α*Δ surrounding the point *y*_{αΔ}. Variables *y*_{i} and *y*_{αΔ} are the *y* coordinates of the instrument at location *i* and the test-filtered variable, respectively. Weights *w*^{*}_{i}*w*_{i} = *w*^{*}_{i}_{i} *w*^{*}_{i}*w*_{i} = [0.214, 0.571, 0.214] for locations *i* = [7, 9, 11] and *i* = [9, 11, 13]. Using the test-filtered variables, the time series of *L*_{ij}*M*_{ij} and *M*_{ij}*M*_{ij} are computed at locations 9 and 11, averaged over a time scale *T*_{c}, and divided to obtain *c*^{(Δ,dyn)}_{s}*α,* we follow the prevalent usage in practical implementations of the dynamic model of not taking into account the effects of compound filtering (see, however, Najjar and Tafti 1996 for a discussion of effects of compound filters and a quantification of its effects on LES using the dynamic model).

### d. Scale-dependent dynamic model: Procedures

The scale-dependent dynamic coefficient is obtained similarly to procedures described in section 2c. The filtered strain-rate tensors and filtered velocity vectors of Fig. 1b are now, however, filtered at *α*^{2}Δ = 1.75^{2}Δ. The same weighting scheme as in section 2c produces weights of *w*_{i} = [0.18, 0.32, 0.32, 0.18] for strain-rate tensors at locations [7, 9, 11, 13]. The resulting *N*_{ij}, while *Q*_{ij}.

It is important to note that *N*_{ij} is a function of *β.* Parameter *β* is computed using procedures identical to those in Porté-Agel et al. (2000a, hereafter POR). Six coefficients of a fifth-order polynomial in *β* are obtained from averaging products of strain rates and resolved stresses over *T*_{c}, as described in the appendix [Eqs. (A2)–(A10)]. Then the roots of the polynomial in *β* are determined by the “roots” function in MATLAB (The Mathworks Inc.). As argued in POR, only the largest real root is physically meaningful. A time series of *Q*_{ij} and *N*_{ij} is obtained from Eq. (9) using the *β* value derived from quantities averaged over *T*_{c}. Finally, the scale-dependent dynamic procedure yields the coefficient at a scale Δ as (*c*^{(Δ,sd–dyn)}_{s}^{2} = 〈*Q*_{ij}*N*_{ij}〉/〈*N*_{ij}*N*_{ij}〉.

## 3. Smagorinsky coefficients determined from dynamic SGS models

### a. Scale-invariant dynamic model: Results

To begin, the scale-invariant, dynamically determined Smagorinsky model coefficient *c*^{(Δ,dyn)}_{s}*T*_{c} = 3.2 s for array 2. Figure 5 shows the PDF of (*c*^{(Δ,dyn)}_{s}^{2} conditioned on Δ/*L* using color contours. It is apparent that the most likely value of (*c*^{(Δ,dyn)}_{s}^{2} depends on stability. It is very close to zero for Δ/*L* > 1 and increases strongly in near-neutral conditions (Δ/*L* ∼ 0). In neutral and unstable conditions, the spread in the PDF is large with a considerable number of negative values. These trends are consistent with those of the empirical coefficient reported in section 2b. However, comparing the color contours with the line from the fit in Eq. (14) and with the conditional PDF of *c*^{(Δ,emp)}_{s}*L* > 0).

Figure 6 shows the empirically and dynamically determined coefficient for a longer averaging time *T*_{c} = 6.8 min and for arrays 1 and 2. At this averaging scale too, the results confirm that the dynamic model predicts a coefficient that is significantly smaller than *c*^{(Δ,emp)}_{s}*c*^{(Δ,dyn)}_{s}*L* bin.

The dynamic procedure predicts the correct basic trends of the coefficient with stability (Δ/*L*) and height (Δ/*z*), but the magnitudes of the coefficients are too small by significant factors. In unstable and neutral conditions, factors range from 2 to 5. In very stable conditions this factor is as large as an order of magnitude or more. Thus, the energy transfer (Π_{Δ}) from resolved scales to SGS is too small, and in an LES using such a model one would expect a high-wavenumber pileup of energy in the spectra near the wall. This weakness of the dynamic model was already observed in an LES of the ABL (POR) in neutral conditions, and present results suggest that this weakness would be exacerbated in conditions of stable stratification.

The variability of *c*^{(Δ,dyn)}_{s}*c*^{Δ,dyn}_{s}^{2} distribution for different averaging times *T*_{c}. The median of *c*^{(Δ,dyn)}_{s}*T*_{c} ranging from 0.05 s (no averaging) to hours. The relative spread of the PDF decreases with averaging time, which agrees with results from KMP and Fig. 4 for *c*^{(Δ,emp)}_{s}

In summary, the results for *c*^{(Δ,dyn)}_{s}*L* or *z*, or both. This deficiency is not surprising. As suggested by the same empirical fit through the available data for *c*^{(Δ,emp)}_{s}*z* or *L* the coefficient is dependent upon Δ unless Δ ≪ *L* and Δ ≪ *z.* Thus, the expected behavior of the coefficient contradicts the basic assumption of scale invariance underlying the dynamic model. This was already noted in POR for the neutral case but Δ > *z.* The scale-dependent dynamic model described in section 2d addresses this problem. In the following section we analyze the data to study whether the scale-dependent model yields more realistic predictions of the coefficient compared to the standard dynamic model.

### b. Scale-dependent dynamic model: Results

*β*= (

*c*

^{αΔ}

_{s}

^{2}/ (

*c*

^{Δ}

_{s}

^{2}. Again, data from array 2 (Δ/

*z*∼ 1.1) are divided into bins of different stabilities ranging from Δ/

*L*= −1 to Δ/

*L*= 5, and divided into subsegments of length

*T*

_{c}. Parameter

*β*is obtained according to section 2d. Specifically, we use Eqs. (A2)–(A10). Averages such as

*T*

_{c}. Figure 9 shows a few representative polynomials

*P*(

*β*) for the case

*T*

_{c}= 6.8 min for three values of Δ/

*L.*The largest root is the value of

*β*that solves the condition of Eqs. (A1) and (A2) (POR).

*β*is computed for the short-duration averaging time of

*T*

_{c}= 3.2, and

*β*is obtained in each segment. The conditional PDF of

*β*is presented in Fig. 10a, where the

*β*range (0 <

*β*< 1.5) is divided into 150 bins. Note that

*β*also depends on stability. In very stable conditions most

*β*values are close to 0.3. The lower bound of

*β*can be explained by considering the limit of

*c*

^{(Δ)}

_{s}

*L*:

*c*

^{(Δ)}

_{s}

*L*)

^{−1}. Consequently,

For Δ/*L* < 0.5, *β* increases and reaches a most likely value of *β* ∼ 0.5. Recall that for scale invariance one would expect a limiting behavior of *β* ∼ 1. Here we obtain *β* < 1 since even in the neutral case Δ > *z* and thus *β* < 1 for the reasons explored in POR. The data analysis is repeated by increasing the averaging time *T*_{c} to cover segments of length *T*_{c} = 6.8 min, as well as over very long averaging covering all data segments in each stability bin. Results are shown in Figs. 11a and 12a, respectively. The observations from results for *T*_{c} = 3.2 s (Fig. 10a) are confirmed since *β* is close to its lower bound 0.327 for Δ/*L* > 1 and increases to values between 0.5 and 0.7 in neutral and unstable conditions. The parameter *β* is very similar for Δ/*z* ∼ 2.1 and for Δ/*z* ∼ 1.1. The magnitude of *β* in the present analysis compares well with Fig. 10 in POR. They obtain a significant increase from *β* ∼ 0.5 at Δ/*z* = 2 to *β* ∼ 0.65 at Δ/*z* = 1.1 in neutral conditions (Δ/*L* = 0), quite consistent with present field measurement results. The limit of large *z*/Δ (Δ ≪ *z*), where the turbulence is better resolved, cannot be verified with the HATS data for which Δ is comparable or larger than *z.* Figure 13 shows that the median of *β* is constant with averaging time and the variability decreases with *T*_{c}.

The model coefficient, *c*^{(Δ,sd–dyn)}_{s}*β* value in the expression for *N*_{ij} (see section 2d). The analysis is performed again using several averaging times *T*_{c} = 3.2 s, *T*_{c} = 6.8 min, as well as a large *T*_{c} encompassing all available data in each bin. As before, results for *T*_{c} = 3.2 s are presented in terms of a conditional PDF for *c*^{(Δ,sd–dyn)}_{s}*z* ∼ 1.1 and −1 < Δ/*L* < 5 in Fig. 10b. The general trend in the relationship with stability is similar to that observed for *c*^{(Δ,dyn)}_{s}*T*_{c} = 6.8 min, in which *β* computed at that time scale is used, are shown in Fig. 11b. Results clearly show that the scale-dependent dynamic model predicts *c*^{(Δ,emp)}_{s}*T*_{c}, by averaging over the entire dataset in each stability bin. Results are shown in Fig. 12b. As can be seen *c*^{(Δ,sd–dyn)}_{s}*c*^{(Δ,emp)}_{s}

The variability of *c*^{(Δ,sd–dyn)}_{s}*c*^{(Δ,emp)}_{s}*T*_{c} > 3.2 s. Also, in unstable conditions the median increases significantly with averaging time for *T*_{c} > 3.2 s. If a reasonable criterion is introduced that requires the median of *c*^{(Δ,sd–dyn)}_{s}*c*^{(Δ,emp)}_{s}*T*_{c} should correspond to at least 12.8 s, or about eight filter scales (8 ≈ 12.8〈*u*〉/Δ).

To confirm that we have obtained results that are unique to turbulence signals under the present physical conditions and do not occur for any time series of random numbers, the procedure to compute dynamic and scale-dependent dynamic coefficients is tested with a time series of random velocity vectors. We generate random velocity fluctuations by distributing 3D vectors whose length is sampled from a uniform distribution in [0, 1] m s^{−1}, and whose direction is uniformly distributed over a sphere. Both white-noise and colored-noise signals (with a −5/3 energy spectrum for each velocity component) are used. The resulting *c*^{(Δ,emp)}_{s}*c*^{(Δ,dyn)}_{s}*c*^{(Δ,sd–dyn)}_{s}*c*^{(Δ)}_{s}^{2} = 0, that is, as expected random signals do not have the correlations between *L*_{ij} and *M*_{ij} associated with net energy flux to smaller scales and a nonzero value of the coefficient. The resulting PDF for *β* is positively skewed, increasing for *β* > 0.327 and but reaching a peak at *β* ∼ 0.45. This is significantly different from the results of the present paper, where for example the peak in *P*(*β*|Δ/*L*) for stable conditions in Fig. 10a is narrow and much closer to 0.327.

## 4. Conclusions

Predictions of the scale-invariant dynamic SGS model (Germano et al. 1991) and the scale-dependent dynamic SGS model (Porté-Agel et al. 2000a) for the Smagorinsky coefficient *c*^{(Δ)}_{s}*c*^{(Δ,emp)}_{s}*z* considered. Clearly, the scale-invariance assumption of the dynamic model breaks down when the filter size is large (Δ > *z* or Δ > *L*), resulting in coefficients that are too small. In an LES of the ABL this is expected to lead to unrealistic velocity profiles near the surface and a pileup of energy reflected in flat velocity spectra.

The scale-dependent dynamic model accounts for scale dependence of the coefficient. As a result the predicted coefficient is close to the value measured by the dissipation balance. It needs to be stressed that the additional parameter introduced by the scale-dependent dynamic model *β* is not empirically tuned, but rather determined dynamically from the large scales. Despite the resulting improvement in predicting the coefficient that produces the correct SGS dissipation compared to the scale-invariant dynamic model, it is reiterated that even “perfect” prediction of the coefficient does not increase the correlation between measured and modeled SGS stresses. This deficiency of the eddy-viscosity closure is related to misalignment of the eigenvectors of SGS stress and strain-rate tensors.

The results for the scale-dependent dynamic model show that short time averaging yields predicted coefficients that fluctuate greatly. This can be problematic in implementations where the extent of averaging is limited (e.g., flows in complex geometries). The data also suggest that the scatter in the prediction is reduced when the Eulerian averaging time scale is greater than ∼8 times the time scale associated with the filter scale. Such a time scale is somewhat larger than averaging time scales usually employed in the Lagrangian dynamic model (Meneveau et al. 1996). However, due to the fundamental differences between Lagrangian and Eulerian averaging the applicability of the result to Lagrangian averaging is uncertain and remains to be explored in simulations.

## Acknowledgments

HATS measurements were made by the NCAR Integrated Surface Flux Facility. The authors wish to thank Tom Horst, Donald Lenschow, Chin-Hoh Moeng, Peter Sullivan, and Jeffrey Weil from the NCAR-ATD and MMM divisions for the fruitful collaboration during the field experiment. Thanks also to Profs. W. Eichinger, F. Porté-Agel, S. Richardson, and J. Wyngaard for the loan of sonic anemometers. We thank the anonymous reviewers of this paper for their constructive feedback. The authors gratefully acknowledge funding from the National Science Foundation, Grant NSF-ATM 0130766.

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## APPENDIX

### Evaluation of β

*c*

^{(Δ)}

_{s}

^{2}= 〈

*L*

_{ij}

*M*

_{ij}〉/〈

*M*

_{ij}

*M*

_{ij}〉 = 〈

*Q*

_{ij}

*N*

_{ij}〉/〈

*N*

_{ij}

*N*

_{ij}〉. This equality can be rewritten as

*L*

_{ij}

*M*

_{ij}

*N*

_{ij}

*N*

_{ij}

*Q*

_{ij}

*N*

_{ij}

*M*

_{ij}

*M*

_{ij}

*β*= (

*c*

^{(αΔ)}

_{s}

^{2}/(

*c*

^{(Δ)}

_{s}

^{2}and

*θ*= (

*c*

^{(α2Δ)}

_{s}

^{2}/(

*c*

^{(Δ)}

_{s}

^{2}. As shown in POR, one unknown can be eliminated by assuming a basic functional form of the scale dependence of the coefficient. A power-law assumption (

*c*

^{(αΔ)}

_{s}

^{2}= (

*c*

^{(Δ)}

_{s}

^{2}

*α*

^{ϕ}yields

*θ*=

*β*

^{2}. After substituting, Eq. (A1) can be written as a fifth-order polynomial in

*β*:

Array properties for the HATS experiment. Here, *d* = double-filtered array; *s* = single-filtered array; *d*_{0} = displacement height; δ_{y} = lateral instrument spacing; Δ = filter size