Probability Density Functions of Velocity Increments in the Atmospheric Boundary Layer

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1 Boundary-Layer Meteorol (2010) 134: DOI /s z ARTICLE Probability Density Functions of Velocity Increments in the Atmospheric Boundary Layer Lei Liu Fei Hu Xue-Ling Cheng Li-Li Song Received: 17 November 2008 / Accepted: 2 November 2009 / Published online: 27 November 2009 Springer Science+Business Media B.V Abstract The probability density functions (pdf s) of the wind increments are measured under different weather conditions in the atmospheric boundary layer, including the extreme weather of a typhoon and sand storm. It is found that in each case the measured pdf s with respect to different time lags coincide by suitable scaling transformation. This property is similar to that of the stable distributions. However, fitting results show that the tails of the stable distributions are generally heavier than that of the measured ones. Beside, the stable distributions (except for the Gaussian distribution) have infinite variance, which implies infinite average kinetic energy. In fact, it can be proved that if the tails of the pdf s are heavy enough, the variance will be infinite. Therefore, the tail-truncated stable distributions with finite variances are introduced to fit the data and the fitting results are excellent. Keywords Atmospheric turbulence Probability density functions Stable distribution Truncated stable distribution Velocity increments 1 Introduction Wind variations are commonly divided into two parts according to the time scale. Each part gives rise to different problems in the recently growing use of wind energy (Burton et al. 2001; Peinke et al. 2004). On the scale of hours and longer, the wind variation is associated with the passage of weather systems, and with the diurnal variation (Stull 1988). The prediction of this variation is important for integrating large amounts of wind power into the L. Liu (B) F. Hu X.-L. Cheng LAPC, Institute of Atmospheric Physics, Chinese Academy of Sciences, Beijing, China liulei@mail.iap.ac.cn F. Hu hufei@mail.iap.ac.cn L.-L. Song Institute of Tropical and Marine Meteorology, China Meteorological Administration, Guangzhou, China

2 244 L. Liu et al. electricity network, to allow the other generating plant supplying the network to be organized appropriately. On the shorter scale below one hour down to several seconds, the wind variations known as turbulence have a very significant effect on the design and performance of the individual wind turbine, as well as on the quality of power delivered to the network and its effect on the customer. Thus, a better understanding of the nature of wind variations is the basis for a more efficient use of the wind energy. The study of wind increments is helpful for understanding wind variations: firstly, wind variations may be viewed as a special stochastic process (Peinke et al. 2004; Gontier et al. 2007). Since the famous work of Einstein (1956), it has been realised that the stochastically meaningful quantity is the increment of the stochastic process but not the stochastic process itself. Many concepts and methods have been developed to understand the stochastic behaviour of increments (Mandelbrot 1999). Therefore, since we choose a stochastic process view, we choose to study the increments. Secondly, large wind variations during short times correspond to wind gusts. A precise statistical description of the occurrence of gusts is important for many applications, such as the estimates of extreme mechanical loads for wind turbines. Some authors have used the wind increments as a measure of gusts (Peinke et al. 2004; Boettcher et al. 2003). Thus, an accurate stochastic analysis of wind increments may be helpful for a better understanding of gusts. Thirdly, the probability density functions (pdf s) of velocity increments are commonly used for the intermittency analysis of small-scale turbulence. Thus, in laboratory turbulence experimental and numerical studies have paid much attention to them (e.g. Van Atta and Park 1972; Gagne et al. 1990; Vincent and Meneguzzi 1991; Noullez et al. 1997; Tabeling et al. 1996). It was generally found that the pdf s are stretched exponential at small scales and almost Gaussian at large scales in laboratory turbulence (Frisch 1995). Various theoretical interpretations have been proposed. Based on a physical picture of fully developed turbulence as a collection of weakly correlated random eddies and strongly correlated structured eddies, She (1991) (seealso,she and Orszag 1991) proposed a model to describe the evolution of the pdf s of transverse velocity gradients in three-dimensional isotropic turbulence. From an analytical closure for Burgers turbulence, Kraichnan (1990) also proposed a model for the evolution of the pdf s of velocity increments. Castaing et al. (1990) derived a relation based on the Kolmogorov s cascade picture to describe these pdf s. Using multifractal methods, Benzi et al. (1991) derived a relation for pdf s of velocity increments and of the velocity gradient. Trying to explain the origin of non-gaussian statistics of velocity increments, Li and Meneveau (2005) derived a simple nonlinear dynamical system for Lagrangian evolution of longitudinal and transverse velocity increments from the Navier Stokes equation. All the methods and concepts developed for laboratory turbulence can be borrowed in the study of atmospheric turbulence and the corresponding results of the two types of turbulence can also be compared. Unlike laboratory turbulence, there is no agreement as to which pdf s of the velocity increments should be in the atmospheric boundary layer. Chu et al. (1996) found that the velocity gradient pdf s exhibit long exponential tails and are largely independent of atmospheric stratification. Ragwitz and Kantz (2001) found that the pdf s of velocity increments, measured in the atmospheric boundary layer, behave like those from laboratory measurements. Their results deviated from Boettcher et al. (2003), where it was claimed that the velocity increments in the direction of the mean wind do show similar statistics to the laboratory data, but only if they are conditioned on an averaged wind speed value. In this paper, we measure the pdf s of velocity increments in the atmospheric boundary layer under different weather conditions. It is found that in each case the measured pdf s

3 Pdf s of Velocity Increments 245 coincide by a suitable scaling transformation (see Sect. 2.2). In mathematics, there is a kind of distribution with a similar property. These distributions are called stable distributions that have been used in many different branches of scientific research (e.g. Mandelbrot 1963; Liu and Meng 2004). Here, the stable distributions are used to fit the measured pdf s. It is found that the measured pdf s cannot be well approximated by the stable distributions in the extreme tails. Beside, the stable distributions except for the Gaussian one have infinite variance that implies infinite average kinetic energy. This seems unphysical. As a result, the Koponen s truncated stable distributions with finite variance are introduced to fit the data (see Sect. 2.3). 2 Probability Density Functions of Velocity Increments 2.1 Datasets In our following analysis, three datasets measured under different weather conditions will be used. Dataset1. The data were recorded on the 325-m meteorological tower in the city of Beijing during 1 4 May 2005 (Hu 1995; Hu and Peng 2006). Wind velocity was measured with a sampling rate of 10 Hz by an ultrasonic anemometer (Campbell CSAT3) at 47 m height. After the investigation of the quality of the data we choose a representative 97-h period (see the top of Fig. 1). During this period the wind velocity was recorded continuously without breaks. Dataset2. The data were also recorded on the 325-m meteorological tower in the city of Beijing during April Wind velocity was measured by the same ultrasonic anemometer used for dataset1. During this period, a sand storm struck this city (Quan and Hu 2008). We select a representative 86 h that was recorded continuously without breaks (see the middle of Fig. 1). velocity components along mean wind(m s 1 ) time(h) Fig. 1 Datasets before spikes are detected and replaced by the linear interpolation. Top: Dataset1. Middle: Dataset2, where there is a dust storm. Bottom: Dataset3, where a typhoon was recorded. The details of each dataset are referred to Sect. 2.1

4 246 L. Liu et al. Dataset3. The data shown in the bottom of Fig. 1 are supplied by Guangdong Meteorological Bureau. They were recorded in the eastern of Guangdong Province near the southern coastline of China during May Wind velocity was measured with a sampling rate of 10 Hz by an ultrasonic anemometer (Campbell CSAT3) at 5-m height. During the observation, the typhoon Pearl swept across this area. The observation lasted for about 22 h and the data were recorded continuously without breaks. We focus on the velocity component in the direction of the mean wind. Wind velocity time series V m (t) are commonly decomposed into a mean speed value V and the turbulent fluctuations V (t) around V : V m = V + V. (1) We measure the probability density functions (pdf s) of the increments of the turbulent fluctuations V τ,definedby V τ = V (t + τ) V (t), (2) where τ is the time lag. The following describes steps or algorithms to produce V τ from original datasets. First, the velocity component along the direction of mean wind, which is denoted by V m, can be computed by transforming the instrument s reference frame to the streamline reference frame (Kaimal and Finnigan 1994). In this algorithm, average wind components are computed over successive 10- min blocks. Second, post-field ultrasonic data always contain spikes that can be caused by random electronics noise or pollutants (such as water or dust) collecting on the transducers of the ultrasonic anemometer. Thus, an algorithm is applied to detect spikes in the values of V m and replace the detected spikes by linear interpolation (Vickers and Mahrt 1997). All the parameters used in this algorithm are set as those in Vickers and Mahrt (1997). As a result, the percentage of detected spikes is 0.89, 0.57 and 0.65% in dataset1, dataset2 and dataset3, respectively. The small values suggest that spikes in our datasets will not affect the statistics of velocity increments. Third, block averaging is used to compute the mean wind velocity V. A 10- min block width is chosen as that in the first step. We have also performed the statistical analysis with different block widths ranging from 1 min to 1 h as in Boettcher et al. (2003), and no significant changes in our results are observed. Indeed, except for a few boundary points of the average periods, the increments of V m are not significantly different from these of V. The exceptions do not affect the results of the measured pdf s as long as the averaging time is not too short. In this paper, only the statistical analysis of V τ basedona 10- min block width is presented (see Sect. 2.2). At last, V τ can be computed from Eqs. 1 2 using the despiked values of V m. 2.2 Evidence of Stability In this section, the measured probability density functions (pdf s) are tested for the property of stability, which is similar to that of stable distributions (see Eq. 8 in Appendix A). If so, the measured pdf s are then fitted with the stable distributions. We measured the pdf s of velocity increments V τ with τ = 0.1s, 0.2s and τ = 0.3s. Then we find that by a suitable scaling transformation, the pdf s of V 0.2s and V 0.3s will coincide with V 0.1s,thatis, Cτ 1 f τ (x) = f 0.1s (C τ x), (3) where C τ (τ = 0.2 s, 0.3 s) are scaling constants. The results of dataset1 are presented in Fig. 2a, where the scaling constants C 0.2s = 0.85 and C 0.3s = The results of data-

5 Pdf s of Velocity Increments 247 C 1 P(C V τ σ 1 ) (a) τ=0.1s C=1 τ=0.2s C=0.85 τ=0.3s C=0.78 τ=0.4s C=0.72 τ=0.6s C= C V σ 1 τ C 1 P(C V τ σ 1 ) (b) τ=0.1s C=1 τ=1s C=0.6 τ=1min C=0.2 τ=5min C=0.2 τ=30min C= C V σ 1 τ Fig. 2 Pdf s with different time lags τ for dataset1 (normalized with sample standard deviation σ ). The dot lines are standardized normal distribution. The broken lines are symmetric and strictly stable distributions with parameters α = and γ = The lines are truncated stable distributions with α = ,γ = and λ = The scaling constants C τ are also shown in the plots set2 are presented in Fig. 3a and the corresponding scaling constants C 0.2s = 0.85 and C 0.3s = The results of dataset3 are presented in Fig. 4a and the corresponding scaling constants C 0.2s = 1andC 0.3s = We also measure the pdf s of V when τ = 1s,

6 248 L. Liu et al. C 1 P(C V τ σ 1 ) (a) τ=0.1s C=1 τ=0.2s C=0.85 τ=0.3s C=0.78 τ=0.4s C=0.72 τ=0.6s C= C V τ σ 1 C 1 P(C V τ σ 1 ) (b) τ=0.1s C=1 τ=1s C=0.6 τ=1min C=0.2 τ=5min C=0.2 τ=30min C= C V τ σ 1 Fig. 3 Pdf s with different time lags τ for dataset2. The dot lines are standardized normal distribution. The broken lines are symmetric and strictly stable distributions with parameters α = and γ = The lines are truncated stable distributions with α = ,γ = and λ = min, 5 min and 30 min. The results are shown in Figs. 2b, 3b and4b for dataset1, dataset2 and dataset3 respectively. It is found that even when τ = 30 min Eq. 3 can also be satisfied by choosing suitable scaling constants.

7 Pdf s of Velocity Increments (a) τ=0.1s C =1 τ τ=0.2s C =1 τ τ=0.3s C =0.92 τ τ=0.4s C τ =0.9 C τ 1 P(Cτ V τ σ 1 ) τ=0.6s C τ = C V σ 1 τ τ C τ 1 P(Cτ V τ σ 1 ) (b) τ=0.1s C 1 =1 τ=1s C 1 =0.8 τ=1min C 1 =0.6 τ=5min C 1 =0.6 τ=30min C 1 = C τ V τ σ 1 Fig. 4 Pdf s with different time lags τ for dataset3. The dot lines are standardized normal distribution. The broken lines are symmetric and strictly stable distributions with parameters α = and γ = The lines are truncated stable distributions with α = ,γ = and λ = It should be noted that n V τ = V δτ,i, (4) i=1

8 250 L. Liu et al. where n = τ δτ, and δτ equals the sampling interval. Since the sampling frequency of the ultrasonic data in our analysis is 10 Hz, δτ equals to 0.1 s in our analysis. In Eq. 4, V δτ,i (i = 1, 2,, n) are the values of n adjacent velocity increments with the same time lag δτ. Therefore, C τ corresponds to C n and Eq. 3 is equivalent to Eq. 8 in the Appendix A. That is, the measured pdf s of wind increments with different time lags τ coincide by using suitable scaling transformations. This is a property of stability that is similar to that of the stable distributions. Because of this similarity, we will fit the data with stable distributions in the next section. 2.3 Failures of Stable Distribution and its Revision The measured probability density functions (pdf s) in Sect. 2.2 are approximately symmetric around 0. Thus, we speculate that the skewness index β 0 and the shift index δ 0 (see the characteristic functions of stable distributions in Appendix A). Other parameters, the characteristic index α and the scale index γ, are estimated by the maximum likelihood method (Nolan 2009; Uchaikin and Zolotarev 1999). Then, it is found that α and γ for dataset1. For dataset2, α and γ For dataset3, α and γ The fitted stable pdf s are shown as broken lines in Figs. 2, 3, 4. Obviously, their tails decay slower than these of the measured ones. We also use the sample fractile technique (Fama and Roll 1968, 1971) to estimate the parameters and the corresponding fitted curves still decay slower. The commonly used stochastic model, standardized normal distribution (with mean value µ = 0 and standard deviation σ = 1), is also shown in these figures. In fact, the normal distribution is a special stable distribution with α = 2 (see Appendix A). Contrary to the fitting results of other stable distributions (with α = 2), the tails of normal distributions decay much faster than the data. As a result, the fitting results show that the measured pdf s decay much faster than the general stable distributions. Besides, the latter have infinite variance that implies infinite average kinetic energy of turbulent eddies. This seems unphysical. Therefore, the stable distributions are not suitable to describe the measured pdf s. It can be proved that a much slower decay of tail distributions (compared with x 3 ) will lead to the divergence of variance (Nolan 2009). Therefore, the problem of infinite variance may be resolved by truncating the tails of the stable distributions. Many truncating methods have been proposed (e.g. Mantegna and Stanley, 1994; Koponen, 1995; Gupta and Campanha, 1999, 2000; Matsushita et al., 2003). The one proposed by Koponen (1995)hasa smooth pdf with the fewest parameters (Koponen 1995). Indeed, Koponen s truncated stable distributions (KTSD) has only one more parameter than the stable distributions. The new parameter is referred to as the cut-off length λ and it represents how the tails of the stable distributions are truncated (see Appendix B). If cut-off length λ = 0, there is no truncation and KTSD is reduced to the stable distribution. We fit the measured pdf s with the KTSD by the non-linear least-squares method. The fitting results are listed as follows: for dataset1, α = , γ = and λ = ; for dataset2, α = , γ = and λ = ; for dataset3, α = , γ = and λ = The fitted KTSD are shown as solid lines in Figs. 2, 3, 4. From these figures, we see that the agreement between the measured pdf s and the KTSD is excellent. In conclusion, we find that the truncated stable distributions are good approximations to the pdf s of wind increments. This result is different from other findings mentioned in Sect. 1.

9 Pdf s of Velocity Increments Discussion and Conclusion We measured the pdf s of the velocity increments in the atmospheric boundary layer under different weather conditions, including extreme weather conditions such as the dust storm and typhoon. In each case, the measured pdf s with different time lags coincide by suitable scaling transformation. This property is similar to that of the stable distributions. Thus, we then fit the data with the stable distributions using the maximum likelihood method. It is found that the tails of stable distributions decay much slower than the data. Beside, the infinite variance of the stable distributions implies the infinite average kinetic energy. Therefore, we introduce the Koponen s truncated stable distributions to fit the data and the results are excellent. The classic two-thirds law [V (t + τ) V (t)] 2 τ 2/3 implies successive velocity increments are not independent. However, the property of stability Eq. 3 in Sect. 2.2 probably remains unchanged, even though there is some correlation in the successive increments. It has been proved that adding dependent samples from the same stable distribution will also yield a stable law (Nolan 2009). Besides, we have measured the pdf s of the increments of the fractional Brownian motion B(t). This stochastic process has a scaling law [B(t + τ) B(t)] 2 τ 2H (0 < H < 1) and thus dependent successive increments (Mandelbrot 1968). Considering the famous Kolmogorov law [V (t + τ) V (t)] 2 τ 2/3,we choose H = 0.3 and find that Eq. 3 is indeed satisfied in this situation. Our results are different from others who have concluded that pdf s of the velocity increments are similar to laboratory turbulence (Ragwitz and Kantz 2001); i.e., they are exponential at small scales and almost Gaussian at large scales, or this will be so if they are conditioned on an averaged wind speed (Boettcher et al. 2003). Atmospheric turbulence, influenced by many factors, such as the thermal stability, the topography, the geographical position and so on (Burton et al. 2001; Stull 1988), is more complex than laboratory turbulence. Thus, the pdf s of the velocity increments of the atmospheric turbulence may not be similar to that of the laboratory turbulence. If so, the wind gust that is related to the heavy-tailed statistics of wind fluctuations with different time lags, may not be reduced to the well-known intermittence of the local isotropic turbulence. However, this needs a detailed experimental and theoretical comparison between the two types of turbulence. In this study, we have also shown that, for a range of weather conditions, measured pdf s are the truncated stable distributions. This implies that our results may have wide applications in atmospheric boundary-layer research. Moreover, our results provoke some interesting scientific problems. For example, what is the origin of the property of stability of the wind increments? Why are the qualitative statistical properties of velocity increments insensitive to the drastic weather changes? Does laboratory turbulence have the similar property, even though it has simpler boundary conditions than atmospheric turbulence? Acknowledgements Authors thank Professor JP Nolan for his programme STABLE, which is available from the website Many thanks to Guangdong Meteorological Bureau for most kindly supplying the data. This work is supported by the National Nature Science Foundation of China under Grant No , No and No This work is also supported by the National Science and Technology Pillar Program under Grant 2008BAC37B02 and the Beijing Municipal Fund for the Construction of Key Discipline. Appendix A: Stable Distribution In this section, we give a brief review on the definition and the properties of the stable distribution. More details are referred to in Uchaikin and Zolotarev (1999)andNolan (2009).

10 252 L. Liu et al. For a Gaussian distribution G N(µ, σ 2 ), it can be easily derived that n d G i = an + b n G, (5) i=1 where G i (i = 1, 2,...,n) are independent random variables each having the same distribution as G, n is an positive integral and = d denotes equality in probability. In above equation, a n and b n are two constants where a n = (n n 1/2 )µ, b n = n 1/2. That is to say, the distribution of the sum of independent random variables each having the same Gaussian distribution will be invariant under the linear transformation. This property can be generalized to define the stable distributions. Definition of Stable Random Variables A random variable X is referred to as stable if for any n there exist constants b n > 0anda n such that n d X i = an + b n X, (6) i=1 where X 1, X 2,...are independent random variables each having the same distribution as X. If (6) holds with a n = 0, that is n d X i = bn X, (7) i=1 the stable random variable X is called strictly stable. If f (x) is the probability density function of X, then Eq. 7 is equal to the following equation: Cn 1 f n (x) = f (C n x), (8) where f n (x) is the probability density function of n i=1 X i and C n = bn 1. The stable random variable X has the characteristic function exp( γ α u α [1 iβ(tan πα )(sign u)]+iδu) α = 1 E exp(iux) = 2 exp( γ u [1 + iβ 2 (sign u) log( u )]+iδu) α = 1 π where the sign function is defined by 1 u > 0 sign u = 0 u = 0 1 u < 0. The four parameters α, β, γ and δ are called characteristic index, skewness index, scale index and shift index respectively. The scale and shift index, γ and δ, are similar to the standard deviation and mean of a Gaussian distribution. The stable distributions with γ = 1 and δ = 0 are called standardized distributions. Other distributions X can be standardized by (X δ)/γ. Characteristic index α is the most important characteristic of the stable distribution determining the rate of decrease of its tails. Skewness index β characterizes the

11 Pdf s of Velocity Increments 253 degree of asymmetry of the distribution being different from the Gaussian. Considering the standardized stable distributions, when β = 0 the distributions are symmetric around 0. If β>0, the distributions are skewed with the right tails of the distributions heavier than the left tails, i.e. P(X > x) >P(X < x) for large x > 0. The behaviour of β<0 cases, are the reflection of the β>0 cases, with the left tail being heavier (Nolan 2009). Here are some properties of the stable distributions as follows: (P1) a random variables is stable as long as Eq. 6 is true for n = 2and3(Zolotarev 1986; Uchaikin and Zolotarev 1999); (P2) 0 <α 2, 1 β 1, γ 0andδ R;(P3)whenα = 2, the corresponding stable distribution is just the Gaussian distribution that is the only stable distribution with finite variance. Moreover, when α>1 the expectation of the stable distribution is finite. When α 1, E X =+, so the expectation is undefined; (P4) b n = n 1/α or C n = n 1/α. Thus, it follows that C 2n = C 2 = constant. (9) C n For (P5) the tails of the stable distribution are asymptotically power laws (Nolan 2009); i.e., the right tail of the stable distribution for x is f (x α, β, γ, δ) αγ α c α (1 + β)x (1+α) (10) where 0 <α<2, 1 <β 1andc α = sin(πα/2)ɣ(α)/π, and the left tail for x is f ( x α, β, γ, δ) αγ α c α (1 β)x (1+α) (11) where 0 <α<2, 1 β<1andc α = sin(πα/2)ɣ(α)/π; Appendix B: Truncated Stable Distribution The characteristic function of the truncated stable random variable Y (α = 1) is (Koponen 1995; Nakao 2000) when 0 <α<1, ln 1 (u) = when 1 <α<2, ln 2 (u) = γ α { [ ( λ α (u 2 + λ 2 ) α/2 cos α arctan u λ cos πα/2 [ 1 iβ(sign u) tan γ α ( α arctan u )] } + iδu, λ { [ ( λ α (u 2 + λ 2 ) α/2 cos α arctan u λ cos πα/2 [ 1 iβ(sign u) tan )] )] ( α arctan u )] + iαβλ α 1 u + iδu λ }. Compared with the stable distribution, the truncated stable distribution has a new parameter λ, which is called the cut-off length. When λ 0, the stable distribution (α = 1) is recovered. The right tail of the truncated stable distribution for y is (Nakao 2000) f (y α, β, γ, δ) αγ α c α (1 + β)y (1+α) e λ y 2Ɣ( α)c α(γ λ) α, (12)

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