NOTES AND CORRESPONDENCE. Relationship between the Vertical Velocity Skewness and Kurtosis Observed during Sea-Breeze Convection

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1 885 NOTES AND CORRESPONDENCE Relationship between the Vertical Velocity Skewness and Kurtosis Observed during Sea-Breeze Convection S. ALBERGHI, A.MAURIZI, AND F. TAMPIERI ISAC-CNR, Bologna, Italy 25 February 2001 and 4 March 2002 ABSTRACT Fourth-order moments of vertical velocity in the convective boundary layer (CBL) are seldom available because the unsteady nature of atmospheric flows makes time statistics unreliable for the evaluation of high-order moments. In this paper, sound detection and ranging (sodar) measurements in well-developed, almost steady convective conditions are selected and analyzed to investigate the relationship between skewness and kurtosis. Moments of vertical velocity up to the fourth order are computed with special attention to the assessment of the errors connected to the number of independent data and to the inherent spatial and temporal filtering of the measuring system. Results show that a significant correlation between skewness and kurtosis exists in the lower half of the CBL, and a quadratic relationship is proposed that is similar to the one valid for alongwind velocity in shear-dominated boundary layers. 1. Introduction Turbulence in the atmospheric convective boundary layer (CBL) has specific signatures, important for both meteorological modeling and dispersion studies. Although most of the available experimental information and frequently used models handle the first two moments of the velocity probability density functions (pdf), the importance of skewness S of the vertical component of the velocity in the CBL has long been recognized, especially in relation to dispersion modeling (e.g., Baerentsen and Berkowicz 1984). In fact, the main motivation of our work stems from the application of Lagrangian dispersion models to the CBL. From Thomson s (1987) formulation, it turns out to be necessary to assume a given pdf shape starting from the knowledge of few moments. This closure problem has been addressed by some authors [see Maurizi and Tampieri (1999) for a review], and it appears that an optimal choice (in the sense of the maximum missing information pdf; Jaynes 1957; Du et al. 1994) can be made if the moments are known up to a given even order. Gaussian pdf is the minimal optimal choice; if non-gaussianity is important, as occurs in the CBL, a knowledge of the moments up to at least the fourth order is necessary. Corresponding author address: A. Maurizi, ISAC-CNR, via Gobetti 101, I Bologna, Italy. a.maurizi@isac.cnr.it Because of the difficulty in obtaining reliable determinations of the high-order moments in atmospheric flows (Lenschow et al. 1994), which are often unsteady, only sparse values of fourth-order moments are found in literature. Thus, to allow model applications even in the absence of detailed measurements, suitable parameterizations are appropriate. Relationships between S and kurtosis K implicitly result from the definition of stochastic processes such as those used in Lagrangian stochastic modeling (e.g., Maurizi and Tampieri 1999, their Fig. 2) or, as another example, the one that results from the assumptions made by Lenschow et al. (1994, their Fig. 1) for the estimation of statistical errors on higher-order moments. These relationships, resulting from the a priori definition of a stochastic process, cannot be used to parameterize K as a function of S in absence of a proper experimental verification. An alternative approach would be to look for a direct relationship based on data to be used for the definition of a realistic stochastic process. Measurements of high-order moments in the sheardominated boundary layers (in the absence of stratification effects), performed mainly in wind tunnel experiments, show that the fluctuating component of the velocity aligned with the mean flow displays an increasing negative skewness as the distance from the wall increases and a correspondingly increasing kurtosis (Durst et al. 1987). The existence of this relationship between S and K, for which a quadratic formula was 2002 American Meteorological Society

2 886 JOURNAL OF APPLIED METEOROLOGY VOLUME 41 TABLE 1. Selected stationary periods in convective conditions. Reference wind is computed as a time average over the whole period and vertical average over the selected z n. Site Date Local time 30 Jul Aug Aug Aug Sep Jul Aug Jul Aug Aug Aug Reference wind (m s 1 ) FIG. 1. Map of the area in which the measurements sites are located. Urban areas are represented by dashed regions. proposed (Shaw and Seginer 1987; Tampieri et al. 2000), stimulated further investigations in different stability conditions. A similar formula for CBL vertical velocity was in fact proposed by Alberghi et al. (2000) based on the statistical limit on S and K for the existence of a pdf. The purpose of this paper is to give a better assessment of this relationship based on new data obtained from sound detection and ranging measurements (sodar) and existing data from literature. 2. Dataset description and stationarity analysis SODAR measurements were performed at two sites near Rome, Italy (Practica di Mare, hereinafter ; and Agenzia Municipale per l Ambiente, hereinafter ), during the summers of 1995, 1996, and 1997 by researchers of the Istituto di Fisica dell Atmosfera of the Consiglio Nazionale delle Ricerche (IFA-CNR, Rome). The two sites are at different distances from the coast with respect to the prevailing wind direction, which is mainly driven by a south-southwest sea land breeze during cases under study. is located about 6 km from the shoreline, and is about 15 km inland (Fig. 1). Instantaneous vertical velocity w was measured in sampling volumes centered at heights z n 39 (n 1)27 m with n 1,..., 32, with a 6 1 Hz sampling frequency. The radius R of the horizontal section of the sampling volume varies with height z as R z tan, where assumes values in the range 8 10, depending on ambient conditions. The actual maximum height reached depends on ambient noise and is typically m. Quality control on data based on the signal-tonoise ratio was performed according to Mastrantonio and Fiocco (1982) and Greenhut and Mastrantonio (1989). An analysis of steadiness of the time series was performed so as to ensure statistically homogeneous data. In particular, the focus is on the central part of the day, when the thickness z i of the CBL reaches its maximum and the breeze cell is well developed. Periods with a clearly observable (from facsimile) transient in z i were excluded, as were those having significant variations in horizontal wind intensity U and direction. It is assumed that a convective equilibrium exists at least in the lower part of the boundary layer where the majority of data was collected. In these conditions, no observable thermal discontinuity is detected from facsimile. From this analysis, 10 convective periods were identified as fulfilling the above prerequisites, and then more-quantitative analysis of steadiness was performed on them. The selected periods were subdivided into 0.5-h intervals on which mean w and variance w 2 of vertical velocity were calculated for each time series, that is, for each height. Subperiods with sufficiently steady firstand second-order statistics (mostly within a 20% band of variability) for each z n were identified. Table 1 reports results of the selection process. Note that in three cases (30 July, 4 August, and 6 August 1995) all referring to the site, vertical mean velocity is significantly less than 0 at all levels z n. This aspect was investigated by Mastrantonio et al. (1994), who attributed it to a displacement of the breeze-cell center. In the following, we will consider only centered moments and assume that vertical advection has negligible influence on the CBL structure. In fact, considering the three periods with significant departure of w from 0, advection term w z w 2 and turbulent transport term z w 3 were estimated using profiles for w 2 and w 3 from Stull (1988). It results that, using w * 1.24 m s 1 and z i 500 m (estimated fitting the w 2 profile on data) and for a typical value w 0.3ms 1 at an elevation

3 887 FIG. 2. Power spectra at different z n for the 6 Aug 1995 case: plus sign is z 2 66 m, times sign is z m, asterisk is z m, open square is z m, and filled square is z m. Continuous thick line represents the inertial subrange decay k 5/3. Vertical dotted line represents the k vol value; the dot dashed vertical line estimates the L 1 wavenumber. of z 150 m, the two terms are, respectively, 10 4 m 2 s 3 and m 2 s 3, justifying the above assumption. Estimates of w * and z i using other profiles reported in literature are found to be within a range that does not affect this result. 3. Error estimates Before computing statistics for the above-selected periods, an analysis of error sources is in order. A preliminary stage for the reduction of statistical error was performed that rejected those series with more than 15% of missing data. This criterion was used by Greenhut and Mastrantonio (1989), but with a less severe threshold (25%), and is necessary for reliable estimates of high-order moments. Because the signal-tonoise ratio generally decreases with height, it implies that if the time series for z n is rejected, time series for z k z n are also rejected. Sodar measurements are inherently filtered both in volume and in time. This means that part of the energy associated with high wavenumbers is not accounted for. To evaluate the influence of spatial (increasing with height) and temporal filtering, power spectra of the considered time series were computed. The Lomb (1976) correlogram algorithm for irregularly spaced data was used to account for missing data without the need of any interpolation process. Figure 2 reports power spectra at different levels for the worst case (4 August 1995), that is, the one for which accepted measurements reach the maximum height (z m). The cutoff wavenumber k vol (2R) m 1 introduced by volume filtering at z 16, with R evaluated under the more severe conditions ( 10 ), can be estimated to be at least 20 times as large as the wavenumber relative to integral scale L (Fig. 2). According to Kristensen and Gaynor (1986), because of spatial filtering, the correction to variance is 1.2(2R/L) 2/3, which in this case is on the order of 15%. This value is the upper limit for the error because L is almost uniform with height in CBL, whereas R decreases as height decreases. Furthermore, in the current dataset usually n 16. For time filtering, note that it has a small influence, being Nyq U 1 k L 1, where Nyq 12 1 Hz is the Nyquist frequency. It should be observed that both cutoffs fall within an energy cascade range that is supposed to be nearly isotropic and thus has a velocity distribution that does not greatly differ from Gaussian. For this and the above reasons, it is assumed that errors on skewness S w 3 / w 2 3/2 and kurtosis K w 4 / w 2 2, on which the main attention is focused in this study, are negligible as well. When computing statistics, reliable estimates require that the number of statistically independent data must be large enough. This number is proportional to the ratio T/, where T is the total length of the time series and is the Eulerian integral timescale. Errors on parameter estimations can be evaluated following Lenschow et al. (1994). In accord, for a process with typical values w m s 2, S 0.59, and K 3.47, the error variance in the estimate of w n 2 is n k n /T, where k 2 6, k 3 34, and k , and, for the evaluation of S and K, it holds that k S 5 and k K 68 (Mann et al. 1995). Thus, for a typical calculated from data ( 40 s), a total length T 2.5 h is required in order to have , 3 0.4, 4 2.3, S 0.15, and K 0.6. This could be taken as the maximum error estimates for the whole dataset. Note that the number of independent data needed for a good estimate increases with the order of the moment desired. Furthermore, note that relative errors on estimates of nondimensionalized parameters S and K are smaller than those on third- and fourthorder velocity moments, respectively. 4. Skewness kurtosis relationship Kurtosis is reported in Fig. 3 as a function of S, together with some sparse values obtained from published material (Hanna 1982; Caughey et al. 1983; Deardorff and Willis 1985; Lenschow and Stankov 1986; Lenschow et al. 1994, 2000; Luhar et al. 1996). The first evidence is that the majority of points displays an ordered trend. Moreover, they have nearly the same behavior as that observed in shear-produced turbulence (Durst et al. 1987) and can also be fairly represented by a parabolic relationship. It can be noted, however, that data referring to observations above the heights of the skewness maximum tend to show reduced values of S with increasing height. Because they refer to conditions in which the ground heating effect is diminishing while entrainment and other more complex effects at

4 888 JOURNAL OF APPLIED METEOROLOGY VOLUME 41 FIG. 3. Skewness kurtosis relationship (thick continuous line), statistical limit (thin continuous line), current dataset ( ), and data from literature ( ). the CBL top become important (waves, capping clouds, etc.; e.g., LeMone 1990), they are not considered in establishing the S K relationship. Thus, using the expression K S 2 1 of the statistical limit (Kendall and Stuart 1977) as a reference, the formula 2 K (S 1), (1) with one free parameter, is proposed as a model for the skewness kurtosis relationship in the part of the CBL strongly dominated by convection. Fitting (1) on data referring to this part of the CBL gives , which is similar to that obtained for streamwise velocity in shear-produced turbulence: 2.3 (Tampieri et al. 2000). This allows an estimation of K from S with a fair approximation, as shown by the dashed line in Fig. 3. In pure shear-dominated turbulence (although based only on laboratory data), using (1) to fit vertical velocity data, Tampieri et al. (2000) obtained 3.3. Thus, horizontal wind shear can be expected to act as a parameter leading from a pure convective case with 2.5 to a pure mechanical case with 3.3. A further remark is in order. Although the data do not cover an interval around S 0 in this case, when S is small, K 3. The proposed relationship gives K 2.5 for S 0, and therefore it is very unlikely that Gaussian turbulence will result in convective conditions. 5. Conclusions Vertical velocity in the CBL from sodar data has been analyzed, with special emphasis on error assessment. For the selected dataset, skewness and kurtosis turn out to be correlated. A quadratic model with one parameter is proposed for this relationship, which, though not explaining the underlying process, allows for estimation of K from S with a fair degree of confidence. Furthermore, this parameterization is comparable to that obtained for the horizontal velocity component in shearproduced turbulence. Because the vertical velocity component in shear-dominated turbulence displays a different value of, it is expected that the coefficient will depend on stability. As a final remark concerning pdf modeling, it should be noted that the implicit relationships between thirdand fourth-order moments defined by most of the bi- Gaussian models [in particular, the Baerentsen and Berkowicz (1984) model] are reflected in a relationship between S and K similar to (1). This result appears to be fortuitous in that those relationships are a consequence of different, more or less physically founded assumptions. Nevertheless, this fact could explain the fair accordance of Lagrangian dispersion models based on bi-gaussian pdfs with data. However, it results that some non-gaussian processes are associated with S K relationships that qualitatively resemble the one presented here. Examples based on very different assumptions can be found in Lenschow et al. (1994) and Maurizi and Lorenzani (2001). Acknowledgments. The authors are indebted to G. Mastrantonio (IFA-CNR, Rome, Italy) for making sodar data available and for valuable support during data treatment. REFERENCES Alberghi, S., F. Tampieri, S. Argentini, G. Mastrantonio, and A. Viola, 2000: Analysis of the pdf of the vertical velocity in the buoyancy-driven atmospheric boundary layer. Advances in Turbulence VIII. Proc. Eighth European Turbulence Conference, C. Dopazo, Ed., Barcelona, Spain, CIMNE, [Available from International Center for Numerical Methods in Engineering, Gran Capitán s/n, Barcelona, Spain.] Baerentsen, J. H., and R. Berkowicz, 1984: Monte-Carlo simulation of plume diffusion in the convective boundary layer. Atmos. Environ., 18, Caughey, S. J., M. Kitchen, and J. R. Leighton, 1983: Turbulence structure in convective boundary layers and implications for diffusion. Bound.-Layer Meteor., 25, Deardorff, J. W., and G. E. Willis, 1985: Further results from a laboratory model of the convective planetary boundary layer. Bound.-Layer Meteor., 32, Du, S., J. D. Wilson, and E. Yee, 1994: Probability density functions for velocity in the convective boundary layer, and implied trajectory models. Atmos. Environ., 28, Durst, F., J. Jovanovic, and L. Kanevce, 1987: Probability density distributions in turbulent wall boundary-layer flow. Turbulent Shear Flow 5, F. Durst et al., Eds., Springer, Greenhut, G. K., and G. Mastrantonio, 1989: Turbulence kinetic energy budget profiles derived from Doppler sodar measurements. J. Appl. Meteor., 28, Hanna, S. R., 1982: Applications in air pollution modeling. Atmospheric Turbulence and Air Pollution Modeling, F. T. M. Nieuwstadt and H. van Dop, Eds., Reidel,

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