Field Experimental Study of Dynamic Smagorinsky Models in the Atmospheric Surface Layer

a. The HATS dataset

In HATS two vertically displaced horizontal arrays of 14 Campbell Scientific three-component sonic anemometer–thermometers (CSAT3) were installed in the atmospheric surface layer in the San Joaquin Valley close to Kettlemen City, California, from 31 August until 1 October 2000. The HATS experiment was described in detail in KMP (Kleissl et al. 2003) and Horst et al. (2004). The focus of KMP was to determine relationships between c^(Δ)_s and different relevant length scales: height above ground z, filter scale Δ, and the Obukhov length L. Parameter L is defined as

View Expanded

where u∗ = (−〈u′w′〉)^1/2 is the friction velocity, θ₀ 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, αΔ and α²Δ, 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 = 2δ_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[−(k²₁Δ²/ 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 (α²Δ/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 (x₃ = 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 = (2δ_y)⁻¹[ũ_i(y₀ + δ_y) − ũ_i(y₀ − 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

The Smagorinsky coefficient is measured from the field data by matching mean measured and modeled SGS dissipations Π_Δ (Clark et al. 1979; KMP),

View Expanded

where 〈 〉 denotes Eulerian time averaging of HATS data over a time scale T_c. KMP analyzed the behavior of c^(Δ,emp)_s from HATS data as function of parameters Δ/z and Δ/L. It was found that the data can be described by a function of the form:

View Expanded

Here, R is the ramp function. The parameters of Eq. (14) were determined as n = 3 and α = c₀ = 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 x₂) 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 = 2δ_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 are evaluated at locations 9 and 11. The results from these two locations are essentially identical and only results from location 9 are presented.

A first question to address is whether the data analyzed at scale Δ = 2δ_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 is obtained by evaluating the averages in Eq. (13) over time T_c. In order to isolate the dependence on Δ/L, we compute the conditional PDF of (c^Δ,emp_s)², P(c²_s|Δ/L) = P(c²_s, Δ/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 decreases in stable conditions and its PDF shows a large spread in unstable conditions with a considerable number of negative values. The most likely value of c^(Δ,emp)_s corresponds well to the empirical fit of KMP. Liu et al. (1995) obtained the eddy-viscosity field without averaging and also found a highly variable eddy-viscosity field with negative values, which causes numerical instabilities in LES.

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 in each bin. The coefficients are very close to the lines that are the predictions from the fit of KMP. Only in unstable conditions are the predictions about ∼10% too small. Besides confirming the collapse of the data, this comparison shows that the coarser filter resolution in the lateral direction (using only two or three sensors) does not affect the prediction of c^(Δ)_s.

To provide a systematic description of the effects of averaging time T_c upon the statistics of c^(Δ,emp)_s, the main aspects of the PDF of c^(Δ,emp)_s are documented as functions of T_c. Figure 4 displays the median of c^(Δ,emp)_s as a function of T_c for different stabilities. As reported in KMP, the median of c^(Δ,emp)_s is constant with averaging time and much smaller in stable conditions than in unstable and neutral conditions. In unstable conditions the median increases slightly with averaging time. A measure of the spread of the PDF is documented in terms of the difference between third and first quartile normalized by the second quartile. As expected, this measure decreases with increasing 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 αΔ 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 − αΔ/2, y_αΔ + αΔ/2]|, where [y_i − Δ/2, y_i + Δ/2] is the segment of length Δ surrounding the point y_i, and [y_αΔ − αΔ/2, y_αΔ + αΔ/2] is the segment of length αΔ 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 are normalized so that they sum up to 1: w_i = w^*_i/Σ_i w^*_i. This procedure gives weights of 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_ijM_ij and M_ijM_ij are computed at locations 9 and 11, averaged over a time scale T_c, and divided to obtain c^(Δ,dyn)_s using Eq. (7). Note that in defining 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 α²Δ = 1.75²Δ. 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 and are used to compute N_ij, while and are used to compute 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_ijN_ij〉/〈N_ijN_ij〉.

a. Scale-invariant dynamic model: Results

To begin, the scale-invariant, dynamically determined Smagorinsky model coefficient c^(Δ,dyn)_s is obtained according to section 2c by evaluating the averages over time 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 in Fig. 2, it can be seen that the dynamically determined coefficients are too small, especially in conditions of stable stratification (Δ/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. Finally, the same results are obtained when performing the averages over all available data as shown in Fig. 7, where one value of c^(Δ,dyn)_s is plotted for each Δ/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 is examined in Fig. 8 by plotting the quartiles of the (c^Δ,dyn_s)² distribution for different averaging times T_c. The median of c^(Δ,dyn)_s is very similar for 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 consistently show that the dynamic procedure underpredicts the Smagorinsky coefficient when Δ is close to, or exceeds, 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 [Eq. (14)], for any fixed value of 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

Analysis for the scale-dependent dynamic model first requires computation of the parameter describing scale dependence of the Smagorinsky coefficient, 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

View Expanded

&〈&|&〉 are evaluated over a time scale 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).

Parameter 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 for small L: c^(Δ)_s ∝ (Δ/L)⁻¹. Consequently,

View Expanded

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, predicted from the scale-dependent dynamic model, is obtained by replacing the measured 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 for the case Δ/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 in Fig. 5, but the spread in the PDF is considerably larger. Results from the intermediate time scale 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 quite well in unstable and neutral conditions. In stable conditions, the prediction is still improved compared to the dynamic model (Fig. 6), but significant scatter persists. Finally, we present results using the longest 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 obtained from long time averaging predicts c^(Δ,emp)_s and its dependence on stability and height quite accurately.

The variability of c^{(Δ,sd–dyn)}_s is examined in Fig. 13b. The variability is larger than for c^(Δ,emp)_s and it reduces subsequently for 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 to differ less than 10% from the median of c^(Δ,emp)_s, then Fig. 13b suggests that the Eulerian averaging time scale 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, and c^{(Δ,sd–dyn)}_s feature symmetric PDFs with a strong peak at (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.