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Observational study of power-law approximation of wind profiles within an urban boundary layer for various wind conditions
Journal of Wind Engineering & Industrial Aerodynamics 164 (2017) 13–21
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Observational study of power-law approximation of wind profiles within an urban boundary layer for various wind conditions
MARK
⁎
Hideki Kikumotoa, , Ryozo Ookaa, Hirofumi Sugawarab, Jongyeon Limc a b c
Institute of Industrial Science, The University of Tokyo, 4-6-1 Komaba, Meguro-ku, Tokyo 153-8505, Japan Earth and Ocean Sciences, National Defense Academy of Japan, 1-10-20 Hashirimizu, Yokosuka, Kanagawa 239-8686, Japan School of Engineering, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan
A R T I C L E I N F O
A BS T RAC T
Keywords: Wind profile Urban boundary layer Doppler lidar Power law Low wind speed
This paper investigates the accuracy and limitations of wind profile modeling using the power-law (PL), especially for low speed conditions in which air and thermal pollution can prevail. A Doppler lidar system and ultrasonic anemometer were installed to measure wind profiles and turbulence statistics in the urban boundary layer of Tokyo, Japan over seven months. The wind speeds at a height of 67.5 m (ub) at average intervals of 10 min were < 6 m/s for 80% of the observation period. For low wind speeds, the difference in wind direction with height is significant, making it difficult to determine the prevailing wind direction. The PL could be used to model the wind profiles for high wind speeds (ub > 6 m/s), whereby the power-law index (PLI) converges to 0.25. Although the PL model can be used for an ensemble-averaged profile composed of all profiles from the observed period, the accuracy of the PL decreases for profiles with low speeds and short average time intervals. The PLI on average decreases to ~0.21 for low speeds and shows diurnal changes with small PLIs during the daytime. This research quantitatively discusses the application limits of the PL for wind profiles under low speed conditions.
1. Introduction
common methods in wind engineering for expressing the relationship between wind speed and height above ground (z):
Modeling the behavior of a flow approaching an area of interest is one of the most important aspects of wind engineering. The analysis is either based on physical or numerical modeling of features such as the profile of mean velocity, as well as other turbulence statistics, which significantly influence wind characteristics in the analysis domain. For this reason, wind engineers and climatologists seek a model that can illustrate wind behavior in the atmospheric boundary layer (Counihan, 1975; Davenport, 1960; Holtslag, 1984; Hsu, 1982; Hsu et al., 1994; Panofsky and Dutton, 1984). Several theoretical and empirical models such as logarithmic law, power law, and the Deaves-Harris model have been applied to describe mean velocity profiles in atmospheric boundary layers (Davenport, 1960; Deaves, 1981; Drew et al., 2013; Li et al., 2010). Because the roughness of terrain and atmospheric stability are major factors affecting the characteristics of wind profiles (Monin and Obukhov, 1954), previous research has investigated the relationships between wind profile model parameters and terrain roughness or atmospheric stability (Counihan, 1975; Irwin, 1979; Kanda et al., 2013; Tamura et al., 2001). The power law (PL), described in Eq. (1), is one of the most
⎛ z ⎞α UPL (z ) = Un ⎜ ⎟ , ⎝ zn ⎠
⁎
(1)
where zn is the reference height, Un is the reference speed at zn, and α is the power-law index (PLI). Although the theoretical foundation is not as clear as for the logarithmic law, past observations have shown the potential of PL in modeling wind profiles in the atmospheric boundary layers above urban terrains (Counihan, 1975; Li et al., 2010; Tamura et al., 1999, 2001, 2007). In wind engineering, PL was originally proposed for wind profiles of extremely high speed for use in designing the wind load in structural engineering (Davenport, 1960). High speed and neutrality were thus prerequisites for the use of this model. In this respect, a unique parameter in the PL, the PLI, is recommended and determined according to the surface roughness of the terrain (Architectural Institute of Japan, 2004). However, because the PL is a simple mathematical expression and can be applied to a relatively large range of heights compared to the logarithmic law (Counihan, 1975), the PL has been employed in many research fields and under various condi-
Corresponding author. E-mail address: [email protected] (H. Kikumoto).
http://dx.doi.org/10.1016/j.jweia.2017.02.003 Received 25 March 2016; Received in revised form 26 January 2017; Accepted 3 February 2017 0167-6105/ © 2017 Elsevier Ltd. All rights reserved.
Journal of Wind Engineering & Industrial Aerodynamics 164 (2017) 13–21
H. Kikumoto et al.
ub U Un UPL U wdb z zn α
Nomenclature ECM DLS PL PLI UA L R2
eddy covariance method Doppler lidar system power law power-law index ultrasonic anemometer the Monin-Obukhov length L coefficient of determination
wind speed for the lowest DLS level (z=67.5 m) measured horizontal wind speed reference speed in the PL speed expressed by the PL averaged speed along the DLS height levels wind direction for the lowest DLS level (z=67.5 m) height above the ground reference height above the ground in the PL value of the power-law index
conditions related to the speed and averaging time.
tions. For example, the PL has been used in the analysis of environmental problems such as wind environment, air pollution (Li and Meroney, 1983; Pavageau and Schatzmann, 1999; Tominaga et al., 2008), and wind power potential (Emeis, 2014; Farrugia, 2003; Peterson and Hennessey, 1977; Wharton and Lundquist, 2012). In such cases, the high speed and neutrality of the boundary layer is not assured and the accuracy of the PL can change depending on the conditions. Previous studies have revealed the dependence of the PLI on the wind speed, atmospheric stability, and the height and time of the day at which the PL is evaluated (Farrugia, 2003; Hanafusa et al., 1986; Hussain, 2002; Irwin, 1979; Touma, 1977; Zoumakis, 1993; Zoumakis and Kelessis, 1991). Nevertheless, the applicability of the PL is typically discussed for ensemble-averaged profiles derived from a large number of samples. Even though the ensemble-averaged profile can reflect an average wind condition, the profile is idealistic and sometimes differs significantly from an instantaneous wind profile under actual wind conditions that are affected by many disturbances. With the development of techniques related to wind engineering, the demand is increasing for the simulation of more realistic wind situations with a high accuracy. Although new methods have been developed for analyzing realistic wind situations, such as multi-scale modeling (Baklanov and Nuterman, 2009; Schlünzen et al., 2011; Yamada and Koike, 2011), the conventional method that uses the PL or other empirical profile models is still valid due to high computational efficiency. This study therefore focuses on the following questions: firstly, how well can the PL model instantaneous and real (not idealistic ensemble-averaged) wind profiles; and secondly, how accurate is the PL for low speed conditions in the absence of a dominant force driving the wind. Because the accuracy of the PL for realistic wind conditions has not been sufficiently discussed, it is necessary to investigate the applicability and accuracy of the PL for modeling profiles under various different speeds and intervals of averaged time. Conventional wind measurements are conducted using anemometers located on a tower (Hanafusa et al., 1986; Holtslag, 1984; Li et al., 2010). However, it can be very difficult to find an appropriate location for towers in urban areas and their construction can be very expensive. Following recent developments in remote sensing techniques, remote measurements using Doppler sodar and Doppler lidar are now being applied in wind engineering (Davies et al., 2004; Drew et al., 2013; Gryning et al., 2013; Li et al., 2014; Post and Neff, 1986; Tamura et al., 1999). Doppler sodar can measure wind speed, wind direction, and turbulent structure at high spatial resolutions and was used prior to Doppler lidar for lower atmosphere measurements (Lang and McKeogh, 2011). However, Doppler lidar also enables wind profile measurement for lower atmosphere without sound noise, unlike Doppler sodar, which can annoy residents in nearby urban areas. In this research, we measured wind profiles in the urban boundary layer under various wind conditions in Tokyo, Japan. A Doppler lidar system (DLS) was used for the measurement of wind profiles. An ultrasonic anemometer (UA) was simultaneously used to measure turbulent statistics in the boundary layer using the eddy covariance method (ECM). Using the observed data, we quantitatively discuss the approximation accuracy of the PL under a variety of real wind
2. Observation site and instrumentation Fig. 1 shows aerial maps of the observation site and its surroundings. The DLS was installed on a building rooftop at the Institute of Industrial Science (University of Tokyo, Meguro-ku, Tokyo, Japan; latitude: 35°40′N, longitude: 139°41′E, altitude: 40 m). Velocities were measured between 67.5 and 527.5 m high at 20 m intervals (24 levels). The UA was positioned on a tower at Tokai University (Shibuya-ku, Tokyo, Japan) at a ground height of 52 m. The distance between the sites was about 600 m, with no undulating terrain between them, and their difference in altitude was 2 m. The buildings surrounding the site are mainly residential areas with a homogeneous geometry. However, there are two large commercial areas with a high density of skyscrapers, located 2 km east-southeast (Shibuya area) and 3 km northnortheast (Shinjuku area) of the DLS site. A large green area is also
Fig. 1. Aerial photo map of the observation site (Tokyo, Japan; modified from Google Maps).
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same data were not available for the DLS. The PDF shapes are very similar and indicate that the lack of data from the DLS did not have significant influence on the statistics of the mean speeds.
located in the northeast of the site. These complex ground surface configurations can cause changes in wind speed profile due to wind direction. The DLS used in this study emits a pulsed laser with a wavelength of 1.54 µm in four directions and calculates wind speed in the viewing line in each direction. The angle of scanning cone is about 15°. One round of scanning takes about 40 s. Vectors are synthesized from data on the four directions to calculate wind direction and speed at each height. The threshold value of signal-to-noise ratio (SNR) was set to −27 dB (manufacturer recommended value). Observations were recorded for seven months in September– December 2013 and April–June 2014. Although DLS data was recorded every 40 s, 10 and 30 min averages were mainly used.
3.3. Wind direction during the observation period
3. Results and discussion
Frequency distributions of the wind direction measured by the DLS at the highest and lowest levels are shown in Fig. 5. The results include the distributions for all speeds and high speeds (ub > 6 m/s). During the observed period, southern and northern winds were the most common at the lower level; however, the distribution was slightly different for each height. The peaks observed at 0° and 60° at 67.5 m disappear at the higher level. This difference in the frequency distribution of the wind direction between the height levels became small for higher speeds, and the south wind was dominant for the all heights.
3.1. Wind speed during the observation period
3.4. Wind speed and wind direction deviation along height
As DLS uses remote sensing of light scattered by atmospheric aerosols, it sometimes fails to obtain data in unfavorable atmospheric conditions such as clean air or in cloud cover. The data loss rate tended to be large at the highest and lowest altitudes of observations. To control data quality and ensure that data at each altitude were measured under the same atmospheric conditions, we only used profiles for which speeds could be measured at all observed heights. Table 1 shows the number of 10 min averaged profiles available under this condition. The data acquisition ratio tended to be lower in the winter because of low concentrations of aerosols in the atmosphere. During the whole observation period, the data acquisition percentage for the 10 min averaged profiles was 58.9%. In contrast, the data acquisition ratio by the UA was nearly 100%. Fig. 2 shows probability density functions (PDF) and cumulative distribution functions (CDF) of horizontal wind speed for the lowest DLS level (ub). The PDF has a Weibull distribution (Holtslag, 1984; Kelly et al., 2014) and has a long tail towards high wind speed. However, the mean ub for the 10 min averages was 4.3 m/s. Because 80% of the speeds were < 6 m/s, moderate and weak wind conditions dominated most of the observed period.
Fig. 6 shows the scatter plots between the wind speed and the deviation in wind direction in relation to the direction measured at the lowest DLS level (wdb) for four observed heights. The deviation tended
Fig. 2. Probability density and cumulative distribution functions of horizontal wind speeds measured by the DLS (67.5 m high) for 10 min averages.
3.2. Comparison of wind speed measured by the DLS and UA The velocities of the western and southern wind components measured by the DLS and UA are compared in Fig. 3. Even though there was a 16 m difference in the measuring height and 600 m distance between the sites, the data for both directions were significantly correlated for 30 min averages. The correlation coefficients are 0.93 and 0.99 for the west-east and south-north wind components, respectively. Because of this strong correlation, the two sites for the DLS and UA measurements were assumed to be in the same wind environment. Fig. 4 compares PDFs and CDFs of horizontal wind speed for the 30 min averages measured by the DLS and UA. In this figure, all of the available data from the UA were used to derive the PDF even if the Table 1 Number of available data on 10-min average basis. Month/Year
Number of available data
Data acquisition ratio [%]
9/2013 10/2013 11/2013 12/2013 4/2014 5/2014 6/2014 Total
3441 2707 2200 1080 2907 3070 2523 17,928
83.5 60.6 50.9 24.2 68.1 68.8 58.4 58.9
Fig. 3. Correlation of the wind velocities from UA (52 m high) and DLS (67.5 m high). The west-east and south-north wind components for the 30 min averages are compared between the observed data types.
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> 6 m/s and > 12 m/s) conditions. Plots are separated by north or south wind direction at z=67.5 m. Although the variation of the wind profiles remained relatively large, the mean speed could be fitted using the PL for wind speeds > 6 m/s, with a PLI of 1/3 and 1/4 for north and south wind, respectively, below 200 m high. These values coincided with the PLI categorized in the “rough” or “very rough” used by Counihan (1975). There is a large commercial area including skyscrapers on the north side of the DLS site (Fig. 1), and this difference in land use may have resulted in a difference in PLI due to wind direction. In higher wind cases ( > 12 m/s), the variation of the wind profiles became small and the PLI difference due to the direction decreased. A unique PLI of 1/4 can be used for both north and south wind profiles. However, in Fig. 8, the PL profiles are drawn with the lowest DLS level as the reference height. As a result, with increasing measurement height, the wind speed tended to move away from the wind profile based on the PL. Even though the modeling of approaching flow using unique wind direction is inappropriate because of the large deviation in the wind direction (Fig. 6 and Fig. 7), we made ensemble-averaged profiles using data under all wind conditions and present them in Fig. 9. Because those profiles are composed of larger ensemble members than those used in Fig. 8, they were assumed to reflect more realistic average shape of the wind profiles. For the all speed conditions, there were large variations in the shape of wind profiles. However, the averaged profiles for the all measurement heights coincided well with the PL with a PLI of 1/4.
Fig. 4. Probability density and cumulative distribution functions of the horizontal wind speed measured by the DLS (67.5 m high) and UA for the 30 min averages.
to increase with an increase in height difference. However, it decreased with the increase in wind speed for all heights. For low wind speeds < 5 m/s, the deviation was very large, e.g., about 180° for a 100 m height difference. For these cases, the determination of the representative wind direction was very difficult, and it is inappropriate to assume a unique wind direction along the height for the modeling of approaching flow. Fig. 7 shows the vertical profiles of the average of the wind direction deviation with their standard deviation based on the wind speed and direction at z=67.5 m. As shown in Fig. 6, the wind direction deviation increased dramatically for 10 min averages when all wind speeds were included, and the average deviation of the wind direction was meaningless because of its large standard deviation. However, when using only high wind speed data (ub≥6 m/s), the standard deviation decreased and the average deviation of wind direction at each measurement height was < 7°. Even though the average deviation tended to become positive with an increase in observation height, which agrees with the Ekman spiral predictions, the change of wind direction along the height seems to be insignificant when ub > 6 m/s.
3.6. Wind speed and the PLI When we assume a wind profile obeys the PL, its PLI (α) can be evaluated using the following equation, rewriting Eq. (1) in the following form in order to derive α:
α=
ln(U (z2 )/ U (z1)) , ln(z2 / z1)
(2)
where U(zi) is the wind speed (m/s) at height zi (i=1, 2, in m). Fig. 10 shows the relationship between the wind speed and the PLI from the 30 min averaged speeds, setting z1 and z2 in Eqs. (2) to 67.5 and 167.5 m, respectively. For this analysis, we did not consider the variation in wind direction. With a decrease in wind speed, the PLI exhibited a larger variability. Sometimes the PLI reaches negative values when ub < 8 m/s and indicates that the speed at the lower level (z1) can be larger than that at the higher level (z2). However, for high wind speeds, the PLI converged to around 1/4 (Fig. 8). Fig. 11 presents the probability density functions of the PLI derived for Fig. 10. Although there was a large variation in the PLI, it tended to be smaller for low speeds. The mode of the PLI for all speeds was about 0.15 with the distribution having a longer tail towards a larger PLI. The average PLIs for all, high, and low wind speeds are 0.22, 0.25, and 0.21, respectively.
3.5. Average profiles of wind speed Because the variation of wind direction along the height was very large when the wind speed was low, we selected the profiles under high speed conditions. Fig. 8 presents the ensemble averages in high wind (
3.7. Averaging time and asymptotic behavior to the power law In the preceding sections, profiles averaged for 10 and 30 min were used for the analysis. However, averaging time can affect the accuracy of the profile modeling using the PL. To quantitatively evaluate the asymptotic behavior of the PL, we averaged the profiles with different time length (from 10 min to 4 h) and divided them into different groups according to the wind speed at the lowest level of the DLS measurement. Fig. 12 shows the number of profiles for each averaging time and speed category used for the following analysis. The number of samples decreased the longer the averaging time and the larger the wind speed. There is no sample of ub > 15 m/s for an averaging time of 4 h and ub > 20 m/s for an averaging time of 2 h. We fitted the profiles with the PL using the least-squares method with two unknown parameters of the reference speed and the PLI. In
Fig. 5. Frequency distributions of the wind direction for the 10 min averages measured by the DLS at heights of 527.5 and 67.5 m. 0°, 90°, 180° and 270° are corresponding to north, east, south and west wind directions, respectively. The high speed data were extracted using the horizontal wind speed for the lowest DLS level (ub).
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Fig. 6. Relationship between the wind speed at a height of 67.5 m (ub) and deviation in the wind direction for z=107.5, 167.5, 267.5, and 467.5 m from the direction at 67.5 m high (wdb) (10 min average).
this fitting, we limited the lowest value of the PLI to zero. From the results of the fitting, we calculated the coefficient of determination (R2) for each profile, which is defined in Eq. (3): N
R2 = 1 −
∑k =1 (U (zk ) − UPL (zk ))2 N
∑k =1 (U (zk ) − U )2
, (3)
where U(zk) is a speed measured by the DLS at the height zk (k=1, 2, …, N) averaged for a length of time, N is the number of levels used for the fitting by the PL, and U is averaged speed along the N height levels. The value of R2 implies how well the PL can express the change of measured wind speed along the height. Fig. 13 shows the mean value of R2 for the different averaging times and speeds. Regardless of the range of the height, the R2 value was high for all speeds (mean R2 > 0.9), especially when the ub > 15 m/s. It is difficult to fit wind speeds of a large altitude range with one PL profile (Fig. 8), therefore when we used a profile under 200 m, the R2 was also relatively higher in comparison to all heights. The reason is not clear for a drop in the mean R2 for the category with 10-min averages and speeds of 20–25 m/s. It is possible that the data were affected by a specific profile because the number of samples in this category was small compared with those in other categories. The longer
Fig. 7. Vertical profiles of the average of the wind direction deviation in relation to the direction at a height of 67.5 m on 10 min average basis. The error bars show the rootmean-square value of the wind direction deviation. The high wind speed data corresponds to wind speeds (ub)≥6 m/s.
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Fig. 8. Ensemble-averaged profiles of wind speed based on the 10 min averages for (i) ub≥6 m/s and (ii) ub≥12 m/s. Each profile is averaged after normalization using the horizontal wind speed at the lowest DLS level (ub). The error bars show the standard deviation of the profiles. North wind corresponds to 0≤ wind direction (wdb) < 20° or 340≤wdb < 360°, while south wind corresponds to 180≤wdb < 220°.
Fig. 9. Ensemble-averaged profiles of wind speed based on the 10 min averages for the all speed conditions. See Fig. 8 for a description of other details.
Fig. 11. Probability and cumulative distribution functions of the PLI derived from the 30 min averaged wind speeds at 67.5 and 167.5 m high for (i) all speeds and (ii) high (ub > 6 m/s) and low (ub < 6 m/s) speeds.
averaging time tended to result in a higher R2 when the speed was relatively high. However, even though the averaging time was 4 h, the R2 was 0.61 and 0.85 for the all heights and for the heights < 200 m, respectively when the ub was < 5 m/s. The wind speed had a greater effect on the accuracy of the PL than the averaging time.
Fig. 10. Wind speed at the DLS level (ub) and PLI (α) derived from the 30 min averages of wind speed at 67.5 and 167.5 m high.
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H. Kikumoto et al. 100000 10 mins
30 mins
1 hr
2 hrs
4 hrs
Number of samples
10000
1000
100
10
1 0 - 5 m/s
5 - 10 m/s
10 - 15 m/s Wind speed
15- 20 m/s
20 - 25 m/s
Fig. 12. Number of available samples of profiles for each time-averaging period and wind speed category. Profiles were categorized using the horizontal wind speed for the lowest DLS level (ub).
Fig. 14. Diurnal variations of wind speed (DLS and UA) for each height averaged for the entire measurement period.
1 10 mins 1 hr 4 hrs
Mean R [-]
0.9
30 mins 2 hrs
acceleration, and H is upward turbulent heat flux. Fig. 15(i) and (ii) show the average diurnal changes of the turbulent fluxes of momentum and heat from the UA measurement. Using these two results, the Monin-Obukhov length was calculated, and z/L at the height of the UA (52 m) is presented in Fig. 15(iii). The downward turbulent momentum flux increased in the daytime corresponding to the increase in the wind speed in Fig. 14, and the flux value peaked around 15:00. The upward heat flux increased during the daytime with its largest value occurring around the noon, which is earlier than the peak in momentum flux. As a result, the atmosphere tended to be unstable in the daytime and the stability (z/L) had a negative peak from 10:00 to 11:00. Stability showed very large standard deviation at several times in Fig. 15(iii). This is because very small (close to zero) friction velocity was measured in some samples, and z/L is proportional to u*−3. Fig. 16 also presents mean diurnal variation of the PLI, but only uses data with ub > 6 m/s because of the validity of the PL. Because the measurement site can be categorized in terms of the ground roughness into the “rough” or “very rough” categories provided by Counihan (1975), the PLI was expected to reach 0.2–0.4 on these ground surfaces. Although the PLIs in this study were close to this range, the PL had a pattern of diurnal change. In the evening, the PLI gradually increased and peaked in the early morning (from 5:00 to 6:00), and then the PLI dropped to lower than 0.2 during the daytime. The trend of the small PLIs in unstable conditions agrees with the data from Irwin (1979) and Touma (1977). Therefore, the small PLI during the low wind speeds might be explained in terms of the atmospheric stability. However, further discussion of the modeling of the PLI based on the stability is beyond the scope of this paper, and we leave it for future studies.
0.8 0.7 0.6 0.5 0.4 0 - 5 m/s
5 - 10 m/s
10 - 15 m/s Wind speed
15- 20 m/s
20 - 25 m/s
15- 20 m/s
20 - 25 m/s
(i) 1
10 mins 1 hr 4 hrs
Mean R [-]
0.9
30 mins 2 hrs
0.8 0.7 0.6 0.5 0.4 0 - 5 m/s
5 - 10 m/s
10 - 15 m/s Wind speed
(ii) Fig. 13. Mean coefficient of determination (R2) of the fitting of horizontal wind speeds at (i) all levels and (ii) levels less than 200 m using the PL for each time-averaging period and wind speed category. Profiles were categorized using the horizontal wind speed for the lowest DLS level (ub).
3.8. Diurnal change of wind speed and the PLI 4. Conclusions
In Section 3.6, we showed that the PLI tended to decrease for the low wind speeds. Fig. 14 shows mean diurnal variations of wind speed. The speeds clearly increase with time during the day (from 8:00 to 18:00), and the difference in speed along the height decreases at the higher measurement heights in the afternoon. Previous studies showed that the PLI depended not only on the roughness of the land surface, but also on the atmospheric stability and can be explained using atmospheric stability parameters, such as the Pasquill stability class and the Monin-Obukhov length (L) (Hanafusa et al., 1986; Irwin, 1979; Monin and Obukhov, 1954; Touma, 1977). The Monin-Obukhov length is given as:
L=−
ρcp Θ0 u*3 κgH
,
Wind profiles in the urban boundary layer under various wind conditions were measured in Tokyo, Japan using a DLS. We simultaneously used a UA to measure turbulent statistics using the ECM in the boundary layer. Measurements were conducted for a total of seven months. We applied the PL to approximate the measured wind profiles and discussed the approximate accuracy for various wind conditions, especially in terms of wind speed and averaging time. Moderate and low wind conditions dominated during the observed period. The 10-min wind speed at 67.5 m high (ub) averages were < 6 m/s during 80% of the observation period. Wind speeds measured at the DLS and UA sites were highly correlated with each other. The data acquisition ratio of 10-min averaged profiles by the DLS was 58.9% because of missing data from unfavorable measurement conditions. However, we did not find significant influence on the statistics of mean speeds due to the lack of samples from the DLS when we compared the
(4)
where ρ is density, cp is specific heat capacity, Θ0 is mean temperature, u* is friction velocity, κ is the von Karman constant, g is gravitational 19
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PDFs of wind speed with the UA data. For low wind speeds (ub < 6 m/s), the large deviation in wind direction with height made it difficult to determine the prevailing wind direction. Therefore, modeling the approaching flow with unique wind direction along the height was inappropriate under low wind conditions. For high wind speeds, the deviation in wind direction decreased and the PL could be used to model the ensemble-averaged wind profiles. When ub was > 12 m/s, the PLI converged to 1/4. However, when we approximate the wind profile based on wind speed at the lowest level of the measurement, the wind speed tended to move away from the PL at greater heights (z > 300 m). When we ignored the wind direction deviation with height and the wind profile was composed of many ensemble-averaged samples, the PL was a good model and the PLI of the profile was around 1/4. However, when we focused on profiles with a 30-min average, the PLI had a large variation for low wind conditions. The mode of the PLI for all speeds was about 0.15, and the PDF has a long-tailed distribution towards a larger PLI. The average PLIs for all, high (ub > 6 m/s), and low (ub < 6 m/s) wind speeds were 0.22, 0.25, and 0.21, respectively. The approximation accuracy of the PL for wind profiles significantly depended on the wind speed and then with the averaging time. When the wind speed was low (ub < 6 m/s), profile fitting using the PL was inappropriate to model the approaching flow even for long averaging times, e.g., 4 h. However, with increasing wind speed, the approximation accuracy significantly improved even for short averaging times. When ub was > 15 m/s, the coefficient of determination by the PL for the profiles was > 0.9 for 10-min averaged data. Although the PL could only be used for high wind conditions, there was a clear diurnal pattern to the PLI. It was larger in the nighttime and early morning, reaching values of 0.4 at night and dropping to < 0.2 during the day. We suggest that this pattern is correlated with wind speed and atmospheric stability. The analysis of thermal and air pollution in urban areas is more relevant at lower wind speeds because this results in higher temperature or pollutant concentrations affecting the daily lives of people. This study revealed the limitation in modeling wind profiles for low speeds, which should be addressed in future work. Acknowledgements This work was partly supported by the Japan Society for the Promotion of Science (JSPS) KAKENHI Grants No. 24226013, 26709041, and 24241008. Fig. 15. Diurnal variations of (i) downward turbulent momentum flux, (ii) upward turbulent heat flux and (iii) z/L (z=52 m, L=Monin-Obukhov length). The data are derived from the UA measurements and averaged for the entire measurement period. Error bars show the magnitude of the standard deviation.
References Architectural Institute of Japan, 2004. Recommendations for loads on buildings. Archit. Inst. Jpn.〈http://www.aij.or.jp/jpn/symposium/2006/loads/loads.htm〉, [WWW Document] (accessed 11.03.15.). Baklanov, A.A., Nuterman, R.B., 2009. Multi-scale atmospheric environment modelling for urban areas. Adv. Sci. Res.. http://dx.doi.org/10.5194/asr-3-53-2009. Counihan, J., 1975. Adiabatic atmospheric boundary layers: a review and analysis of data from the period 1880–1972. Atmos. Environ. 9, 871–905. http://dx.doi.org/ 10.1016/0004-6981(75)90088-8. Davenport, A.G., 1960. Rationale for determining design wind velocities. J. Struct. Div. 86, 39–68. Davies, F., Collier, C.G., Pearson, G.N., Bozier, K.E., 2004. Doppler Lidar measurements of turbulent structure function over an urban area. J. Atmos. Ocean. Technol. 21, 753–761. http://dx.doi.org/10.1175/1520-0426(2004)021 < 0753:DLMOTS > 2.0.CO;2. Deaves, D.M., 1981. Computations of wind flow over changes in surface roughness. J. Wind Eng. Ind. Aerodyn. 7, 65–94. http://dx.doi.org/10.1016/0167-6105(81) 90068-4. Drew, D.R., Barlow, J.F., Lane, S.E., 2013. Observations of wind speed profiles over Greater London, UK, using a Doppler Lidar. J. Wind Eng. Ind. Aerodyn. 121, 98–105. http://dx.doi.org/10.1016/j.jweia.2013.07.019. Emeis, S., 2014. Current issues in wind energy meteorology. Meteorol. Appl. 21, 803–819. http://dx.doi.org/10.1002/met.1472. Farrugia, R.N., 2003. The wind shear exponent in a Mediterranean island climate. Renew. Energy 28, 647–653. http://dx.doi.org/10.1016/S0960-1481(02)00066-6.
Fig. 16. Diurnal variations of the PLI derived from the 30 min wind speed average at 67.5 and 167.5 m high for high speed (ub≥6 m/s). Error bars show the magnitude of standard deviation.
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Engineering Applications. John Wiley and Sons, New York. Pavageau, M., Schatzmann, M., 1999. Wind tunnel measurements of concentration fluctuations in an urban street canyon. Atmos. Environ. 33, 3961–3971. http:// dx.doi.org/10.1016/S1352-2310(99)00138-7. Peterson, E.W., Hennessey, J.P., Jr., 1977. On the use of power laws for estimates of wind power potential. J. Appl. Meteorol. 17, 390–394. http://dx.doi.org/10.1175/ 1520-0450(1978)017 < 0390:OTUOPL > 2.0.CO;2. Post, M.J., Neff, W.D., 1986. Doppler Lidar measurements of winds in a narrow mountain valley. Bull. Am. Meteorol. Soc. 67, 274–281. http://dx.doi.org/10.1175/ 1520-0477(1986)067 < 0274:DLMOWI > 2.0.CO;2. Schlünzen, K.H., Grawe, D., Bohnenstengel, S.I., Schlüter, I., Koppmann, R., 2011. Joint modelling of obstacle induced and mesoscale changes—current limits and challenges. J. Wind Eng. Ind. Aerodyn. 99, 217–225. http://dx.doi.org/10.1016/ j.jweia.2011.01.009. Tamura, Y., Iwatani, Y., Hibi, K., Suda, K., Nakamura, O., Maruyama, T., Ishibashi, R., 2007. Profiles of mean wind speeds and vertical turbulence intensities measured at seashore and two inland sites using Doppler sodars. J. Wind Eng. Ind. Aerodyn. 95, 411–427. http://dx.doi.org/10.1016/j.jweia.2006.08.005. Tamura, Y., Suda, K., Sasaki, A., Iwatani, Y., Fujii, K., Hibi, K., Ishibashi, R., 1999. Wind speed profiles measured over ground using Doppler sodars. J. Wind Eng. Ind. Aerodyn. 83, 83–93. Tamura, Y., Suda, K., Sasaki, A., Iwatani, Y., Fujii, K., Ishibashi, R., Hibi, K., 2001. Simultaneous measurements of wind speed profiles at two sites using Doppler sodars. J. Wind Eng. Ind. Aerodyn. 89, 325–335. http://dx.doi.org/10.1016/S01676105(00)00085-4. Tominaga, Y., Mochida, A., Yoshie, R., Kataoka, H., Nozu, T., Yoshikawa, M., Shirasawa, T., 2008. AIJ guidelines for practical applications of CFD to pedestrian wind environment around buildings. J. Wind Eng. Ind. Aerodyn. 96, 1749–1761. http:// dx.doi.org/10.1016/j.jweia.2008.02.058. Touma, J.S., 1977. Dependence of the wind profile power law on stability for various locations. J. Air Pollut. Control Assoc. 27, 863–866. http://dx.doi.org/10.1080/ 00022470.1977.10470503. Wharton, S., Lundquist, J.K., 2012. Atmospheric stability affects wind turbine power collection. Environ. Res. Lett. 7, 14005. http://dx.doi.org/10.1088/1748-9326/7/1/ 014005. Yamada, T., Koike, K., 2011. Downscaling mesoscale meteorological models for computational wind engineering applications. J. Wind Eng. Ind. Aerodyn. 99, 199–216. http://dx.doi.org/10.1016/j.jweia.2011.01.024. Zoumakis, N.M., 1993. The dependence of the power law exponent on surface roughness and stability in a neutrally and stably stratified surface boundary layer. Atmosfera 6, 79–83. Zoumakis, N.M., Kelessis, A.G., 1991. Methodology for bulk approximation of the wind profile power-law exponent under stable stratification. Bound. Layer Meteorol. 55, 199–203. http://dx.doi.org/10.1007/BF00119335.
Gryning, S.-E., Batchvarova, E., Floors, R., Peña, A., Brümmer, B., Hahmann, A.N., Mikkelsen, T., 2013. Long-term profiles of wind and weibull distribution parameters up to 600 m in a rural coastal and an inland suburban area. Bound.-Layer Meteorol. 150, 167–184. http://dx.doi.org/10.1007/s10546-013-9857-3. Hanafusa, T., Lee, C.B., Lo, A.K., 1986. Dependence of the exponent in power law wind profiles on stability and height interval. Atmos. Environ. 20, 2059–2066. http:// dx.doi.org/10.1016/0004-6981(86)90348-3. Holtslag, A.A.M., 1984. Estimates of diabatic wind speed profiles from near-surface weather observations. Bound.-Layer Meteorol. 29, 225–250. http://dx.doi.org/ 10.1007/BF00119790. Hsu, S.A., 1982. Determination of the power-law wind profile exponent on a tropical coast. J. Appl. Meteorol. 21, 1187–1190. http://dx.doi.org/10.1175/15200450(1982)021 < 1187:DOTPLW > 2.0.CO;2. Hsu, S.A., Meindl, E.A., Gilhousen, D.B., 1994. Determining the power-law wind-profile exponent under near-neutral stability conditions at sea. J. Appl. Meteorol. 33, 757–765. http://dx.doi.org/10.1175/1520-0450(1994)033 < 0757:DTPLWP > 2.0.CO;2. Hussain, M., 2002. Dependence of power law index on surface wind speed. Energy Convers. Manag. 43, 467–472. http://dx.doi.org/10.1016/S0196-8904(01)00032-2. Irwin, J.S., 1979. A theoretical variation of the wind profile power-law exponent as a function of surface roughness and stability. Atmos. Environ. 13, 191–194. http:// dx.doi.org/10.1016/0004-6981(79)90260-9. Kanda, M., Inagaki, A., Miyamoto, T., Gryschka, M., Raasch, S., 2013. A new aerodynamic parametrization for real urban surfaces. Bound.-Layer Meteorol. 148, 357–377. http://dx.doi.org/10.1007/s10546-013-9818-x. Kelly, M., Troen, I., Jørgensen, H.E., 2014. Weibull-k revisited: “Tall” profiles and height variation of wind statistics. Bound.-Layer Meteor. 152, 107–124. http://dx.doi.org/ 10.1007/s10546-014-9915-5. Lang, S., McKeogh, E., 2011. LIDAR and SODAR measurements of wind speed and direction in upland terrain for wind energy purposes. Remote Sens. 3, 1871–1901. http://dx.doi.org/10.3390/rs3091871. Li, Q.S., Zhi, L., Hu, F., 2010. Boundary layer wind structure from observations on a 325 m tower. J. Wind Eng. Ind. Aerodyn. 98, 818–832. http://dx.doi.org/10.1016/ j.jweia.2010.08.001. Li, S.W., Tse, K.T., Weerasuriya, A.U., Chan, P.W., 2014. Estimation of turbulence intensities under strong wind conditions via turbulent kinetic energy dissipation rates. J. Wind Eng. Ind. Aerodyn. 131, 1–11. http://dx.doi.org/10.1016/ j.jweia.2014.04.008. Li, W.-W., Meroney, R.N., 1983. Gas dispersion near a cubical model building. Part I. Mean concentration measurements. J. Wind Eng. Ind. Aerodyn. 12, 15–33. http:// dx.doi.org/10.1016/0167-6105(83)90078-8. Monin, A.S., Obukhov, A.M., 1954. Basic laws of turbulent mixing in the surface layer of the atmosphere. Contrib. Geophys. Inst. Acad. Sci. USSR 24, 163–187. Panofsky, H.A., Dutton, J.A., 1984. Atmospheric Turbulence: Models and Methods for
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Observational study of power-law approximation of wind profiles within an urban boundary layer for various wind conditions
MARK
⁎
Hideki Kikumotoa, , Ryozo Ookaa, Hirofumi Sugawarab, Jongyeon Limc a b c
Institute of Industrial Science, The University of Tokyo, 4-6-1 Komaba, Meguro-ku, Tokyo 153-8505, Japan Earth and Ocean Sciences, National Defense Academy of Japan, 1-10-20 Hashirimizu, Yokosuka, Kanagawa 239-8686, Japan School of Engineering, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan
A R T I C L E I N F O
A BS T RAC T
Keywords: Wind profile Urban boundary layer Doppler lidar Power law Low wind speed
This paper investigates the accuracy and limitations of wind profile modeling using the power-law (PL), especially for low speed conditions in which air and thermal pollution can prevail. A Doppler lidar system and ultrasonic anemometer were installed to measure wind profiles and turbulence statistics in the urban boundary layer of Tokyo, Japan over seven months. The wind speeds at a height of 67.5 m (ub) at average intervals of 10 min were < 6 m/s for 80% of the observation period. For low wind speeds, the difference in wind direction with height is significant, making it difficult to determine the prevailing wind direction. The PL could be used to model the wind profiles for high wind speeds (ub > 6 m/s), whereby the power-law index (PLI) converges to 0.25. Although the PL model can be used for an ensemble-averaged profile composed of all profiles from the observed period, the accuracy of the PL decreases for profiles with low speeds and short average time intervals. The PLI on average decreases to ~0.21 for low speeds and shows diurnal changes with small PLIs during the daytime. This research quantitatively discusses the application limits of the PL for wind profiles under low speed conditions.
1. Introduction
common methods in wind engineering for expressing the relationship between wind speed and height above ground (z):
Modeling the behavior of a flow approaching an area of interest is one of the most important aspects of wind engineering. The analysis is either based on physical or numerical modeling of features such as the profile of mean velocity, as well as other turbulence statistics, which significantly influence wind characteristics in the analysis domain. For this reason, wind engineers and climatologists seek a model that can illustrate wind behavior in the atmospheric boundary layer (Counihan, 1975; Davenport, 1960; Holtslag, 1984; Hsu, 1982; Hsu et al., 1994; Panofsky and Dutton, 1984). Several theoretical and empirical models such as logarithmic law, power law, and the Deaves-Harris model have been applied to describe mean velocity profiles in atmospheric boundary layers (Davenport, 1960; Deaves, 1981; Drew et al., 2013; Li et al., 2010). Because the roughness of terrain and atmospheric stability are major factors affecting the characteristics of wind profiles (Monin and Obukhov, 1954), previous research has investigated the relationships between wind profile model parameters and terrain roughness or atmospheric stability (Counihan, 1975; Irwin, 1979; Kanda et al., 2013; Tamura et al., 2001). The power law (PL), described in Eq. (1), is one of the most
⎛ z ⎞α UPL (z ) = Un ⎜ ⎟ , ⎝ zn ⎠
⁎
(1)
where zn is the reference height, Un is the reference speed at zn, and α is the power-law index (PLI). Although the theoretical foundation is not as clear as for the logarithmic law, past observations have shown the potential of PL in modeling wind profiles in the atmospheric boundary layers above urban terrains (Counihan, 1975; Li et al., 2010; Tamura et al., 1999, 2001, 2007). In wind engineering, PL was originally proposed for wind profiles of extremely high speed for use in designing the wind load in structural engineering (Davenport, 1960). High speed and neutrality were thus prerequisites for the use of this model. In this respect, a unique parameter in the PL, the PLI, is recommended and determined according to the surface roughness of the terrain (Architectural Institute of Japan, 2004). However, because the PL is a simple mathematical expression and can be applied to a relatively large range of heights compared to the logarithmic law (Counihan, 1975), the PL has been employed in many research fields and under various condi-
Corresponding author. E-mail address: [email protected] (H. Kikumoto).
http://dx.doi.org/10.1016/j.jweia.2017.02.003 Received 25 March 2016; Received in revised form 26 January 2017; Accepted 3 February 2017 0167-6105/ © 2017 Elsevier Ltd. All rights reserved.
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ub U Un UPL U wdb z zn α
Nomenclature ECM DLS PL PLI UA L R2
eddy covariance method Doppler lidar system power law power-law index ultrasonic anemometer the Monin-Obukhov length L coefficient of determination
wind speed for the lowest DLS level (z=67.5 m) measured horizontal wind speed reference speed in the PL speed expressed by the PL averaged speed along the DLS height levels wind direction for the lowest DLS level (z=67.5 m) height above the ground reference height above the ground in the PL value of the power-law index
conditions related to the speed and averaging time.
tions. For example, the PL has been used in the analysis of environmental problems such as wind environment, air pollution (Li and Meroney, 1983; Pavageau and Schatzmann, 1999; Tominaga et al., 2008), and wind power potential (Emeis, 2014; Farrugia, 2003; Peterson and Hennessey, 1977; Wharton and Lundquist, 2012). In such cases, the high speed and neutrality of the boundary layer is not assured and the accuracy of the PL can change depending on the conditions. Previous studies have revealed the dependence of the PLI on the wind speed, atmospheric stability, and the height and time of the day at which the PL is evaluated (Farrugia, 2003; Hanafusa et al., 1986; Hussain, 2002; Irwin, 1979; Touma, 1977; Zoumakis, 1993; Zoumakis and Kelessis, 1991). Nevertheless, the applicability of the PL is typically discussed for ensemble-averaged profiles derived from a large number of samples. Even though the ensemble-averaged profile can reflect an average wind condition, the profile is idealistic and sometimes differs significantly from an instantaneous wind profile under actual wind conditions that are affected by many disturbances. With the development of techniques related to wind engineering, the demand is increasing for the simulation of more realistic wind situations with a high accuracy. Although new methods have been developed for analyzing realistic wind situations, such as multi-scale modeling (Baklanov and Nuterman, 2009; Schlünzen et al., 2011; Yamada and Koike, 2011), the conventional method that uses the PL or other empirical profile models is still valid due to high computational efficiency. This study therefore focuses on the following questions: firstly, how well can the PL model instantaneous and real (not idealistic ensemble-averaged) wind profiles; and secondly, how accurate is the PL for low speed conditions in the absence of a dominant force driving the wind. Because the accuracy of the PL for realistic wind conditions has not been sufficiently discussed, it is necessary to investigate the applicability and accuracy of the PL for modeling profiles under various different speeds and intervals of averaged time. Conventional wind measurements are conducted using anemometers located on a tower (Hanafusa et al., 1986; Holtslag, 1984; Li et al., 2010). However, it can be very difficult to find an appropriate location for towers in urban areas and their construction can be very expensive. Following recent developments in remote sensing techniques, remote measurements using Doppler sodar and Doppler lidar are now being applied in wind engineering (Davies et al., 2004; Drew et al., 2013; Gryning et al., 2013; Li et al., 2014; Post and Neff, 1986; Tamura et al., 1999). Doppler sodar can measure wind speed, wind direction, and turbulent structure at high spatial resolutions and was used prior to Doppler lidar for lower atmosphere measurements (Lang and McKeogh, 2011). However, Doppler lidar also enables wind profile measurement for lower atmosphere without sound noise, unlike Doppler sodar, which can annoy residents in nearby urban areas. In this research, we measured wind profiles in the urban boundary layer under various wind conditions in Tokyo, Japan. A Doppler lidar system (DLS) was used for the measurement of wind profiles. An ultrasonic anemometer (UA) was simultaneously used to measure turbulent statistics in the boundary layer using the eddy covariance method (ECM). Using the observed data, we quantitatively discuss the approximation accuracy of the PL under a variety of real wind
2. Observation site and instrumentation Fig. 1 shows aerial maps of the observation site and its surroundings. The DLS was installed on a building rooftop at the Institute of Industrial Science (University of Tokyo, Meguro-ku, Tokyo, Japan; latitude: 35°40′N, longitude: 139°41′E, altitude: 40 m). Velocities were measured between 67.5 and 527.5 m high at 20 m intervals (24 levels). The UA was positioned on a tower at Tokai University (Shibuya-ku, Tokyo, Japan) at a ground height of 52 m. The distance between the sites was about 600 m, with no undulating terrain between them, and their difference in altitude was 2 m. The buildings surrounding the site are mainly residential areas with a homogeneous geometry. However, there are two large commercial areas with a high density of skyscrapers, located 2 km east-southeast (Shibuya area) and 3 km northnortheast (Shinjuku area) of the DLS site. A large green area is also
Fig. 1. Aerial photo map of the observation site (Tokyo, Japan; modified from Google Maps).
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same data were not available for the DLS. The PDF shapes are very similar and indicate that the lack of data from the DLS did not have significant influence on the statistics of the mean speeds.
located in the northeast of the site. These complex ground surface configurations can cause changes in wind speed profile due to wind direction. The DLS used in this study emits a pulsed laser with a wavelength of 1.54 µm in four directions and calculates wind speed in the viewing line in each direction. The angle of scanning cone is about 15°. One round of scanning takes about 40 s. Vectors are synthesized from data on the four directions to calculate wind direction and speed at each height. The threshold value of signal-to-noise ratio (SNR) was set to −27 dB (manufacturer recommended value). Observations were recorded for seven months in September– December 2013 and April–June 2014. Although DLS data was recorded every 40 s, 10 and 30 min averages were mainly used.
3.3. Wind direction during the observation period
3. Results and discussion
Frequency distributions of the wind direction measured by the DLS at the highest and lowest levels are shown in Fig. 5. The results include the distributions for all speeds and high speeds (ub > 6 m/s). During the observed period, southern and northern winds were the most common at the lower level; however, the distribution was slightly different for each height. The peaks observed at 0° and 60° at 67.5 m disappear at the higher level. This difference in the frequency distribution of the wind direction between the height levels became small for higher speeds, and the south wind was dominant for the all heights.
3.1. Wind speed during the observation period
3.4. Wind speed and wind direction deviation along height
As DLS uses remote sensing of light scattered by atmospheric aerosols, it sometimes fails to obtain data in unfavorable atmospheric conditions such as clean air or in cloud cover. The data loss rate tended to be large at the highest and lowest altitudes of observations. To control data quality and ensure that data at each altitude were measured under the same atmospheric conditions, we only used profiles for which speeds could be measured at all observed heights. Table 1 shows the number of 10 min averaged profiles available under this condition. The data acquisition ratio tended to be lower in the winter because of low concentrations of aerosols in the atmosphere. During the whole observation period, the data acquisition percentage for the 10 min averaged profiles was 58.9%. In contrast, the data acquisition ratio by the UA was nearly 100%. Fig. 2 shows probability density functions (PDF) and cumulative distribution functions (CDF) of horizontal wind speed for the lowest DLS level (ub). The PDF has a Weibull distribution (Holtslag, 1984; Kelly et al., 2014) and has a long tail towards high wind speed. However, the mean ub for the 10 min averages was 4.3 m/s. Because 80% of the speeds were < 6 m/s, moderate and weak wind conditions dominated most of the observed period.
Fig. 6 shows the scatter plots between the wind speed and the deviation in wind direction in relation to the direction measured at the lowest DLS level (wdb) for four observed heights. The deviation tended
Fig. 2. Probability density and cumulative distribution functions of horizontal wind speeds measured by the DLS (67.5 m high) for 10 min averages.
3.2. Comparison of wind speed measured by the DLS and UA The velocities of the western and southern wind components measured by the DLS and UA are compared in Fig. 3. Even though there was a 16 m difference in the measuring height and 600 m distance between the sites, the data for both directions were significantly correlated for 30 min averages. The correlation coefficients are 0.93 and 0.99 for the west-east and south-north wind components, respectively. Because of this strong correlation, the two sites for the DLS and UA measurements were assumed to be in the same wind environment. Fig. 4 compares PDFs and CDFs of horizontal wind speed for the 30 min averages measured by the DLS and UA. In this figure, all of the available data from the UA were used to derive the PDF even if the Table 1 Number of available data on 10-min average basis. Month/Year
Number of available data
Data acquisition ratio [%]
9/2013 10/2013 11/2013 12/2013 4/2014 5/2014 6/2014 Total
3441 2707 2200 1080 2907 3070 2523 17,928
83.5 60.6 50.9 24.2 68.1 68.8 58.4 58.9
Fig. 3. Correlation of the wind velocities from UA (52 m high) and DLS (67.5 m high). The west-east and south-north wind components for the 30 min averages are compared between the observed data types.
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> 6 m/s and > 12 m/s) conditions. Plots are separated by north or south wind direction at z=67.5 m. Although the variation of the wind profiles remained relatively large, the mean speed could be fitted using the PL for wind speeds > 6 m/s, with a PLI of 1/3 and 1/4 for north and south wind, respectively, below 200 m high. These values coincided with the PLI categorized in the “rough” or “very rough” used by Counihan (1975). There is a large commercial area including skyscrapers on the north side of the DLS site (Fig. 1), and this difference in land use may have resulted in a difference in PLI due to wind direction. In higher wind cases ( > 12 m/s), the variation of the wind profiles became small and the PLI difference due to the direction decreased. A unique PLI of 1/4 can be used for both north and south wind profiles. However, in Fig. 8, the PL profiles are drawn with the lowest DLS level as the reference height. As a result, with increasing measurement height, the wind speed tended to move away from the wind profile based on the PL. Even though the modeling of approaching flow using unique wind direction is inappropriate because of the large deviation in the wind direction (Fig. 6 and Fig. 7), we made ensemble-averaged profiles using data under all wind conditions and present them in Fig. 9. Because those profiles are composed of larger ensemble members than those used in Fig. 8, they were assumed to reflect more realistic average shape of the wind profiles. For the all speed conditions, there were large variations in the shape of wind profiles. However, the averaged profiles for the all measurement heights coincided well with the PL with a PLI of 1/4.
Fig. 4. Probability density and cumulative distribution functions of the horizontal wind speed measured by the DLS (67.5 m high) and UA for the 30 min averages.
to increase with an increase in height difference. However, it decreased with the increase in wind speed for all heights. For low wind speeds < 5 m/s, the deviation was very large, e.g., about 180° for a 100 m height difference. For these cases, the determination of the representative wind direction was very difficult, and it is inappropriate to assume a unique wind direction along the height for the modeling of approaching flow. Fig. 7 shows the vertical profiles of the average of the wind direction deviation with their standard deviation based on the wind speed and direction at z=67.5 m. As shown in Fig. 6, the wind direction deviation increased dramatically for 10 min averages when all wind speeds were included, and the average deviation of the wind direction was meaningless because of its large standard deviation. However, when using only high wind speed data (ub≥6 m/s), the standard deviation decreased and the average deviation of wind direction at each measurement height was < 7°. Even though the average deviation tended to become positive with an increase in observation height, which agrees with the Ekman spiral predictions, the change of wind direction along the height seems to be insignificant when ub > 6 m/s.
3.6. Wind speed and the PLI When we assume a wind profile obeys the PL, its PLI (α) can be evaluated using the following equation, rewriting Eq. (1) in the following form in order to derive α:
α=
ln(U (z2 )/ U (z1)) , ln(z2 / z1)
(2)
where U(zi) is the wind speed (m/s) at height zi (i=1, 2, in m). Fig. 10 shows the relationship between the wind speed and the PLI from the 30 min averaged speeds, setting z1 and z2 in Eqs. (2) to 67.5 and 167.5 m, respectively. For this analysis, we did not consider the variation in wind direction. With a decrease in wind speed, the PLI exhibited a larger variability. Sometimes the PLI reaches negative values when ub < 8 m/s and indicates that the speed at the lower level (z1) can be larger than that at the higher level (z2). However, for high wind speeds, the PLI converged to around 1/4 (Fig. 8). Fig. 11 presents the probability density functions of the PLI derived for Fig. 10. Although there was a large variation in the PLI, it tended to be smaller for low speeds. The mode of the PLI for all speeds was about 0.15 with the distribution having a longer tail towards a larger PLI. The average PLIs for all, high, and low wind speeds are 0.22, 0.25, and 0.21, respectively.
3.5. Average profiles of wind speed Because the variation of wind direction along the height was very large when the wind speed was low, we selected the profiles under high speed conditions. Fig. 8 presents the ensemble averages in high wind (
3.7. Averaging time and asymptotic behavior to the power law In the preceding sections, profiles averaged for 10 and 30 min were used for the analysis. However, averaging time can affect the accuracy of the profile modeling using the PL. To quantitatively evaluate the asymptotic behavior of the PL, we averaged the profiles with different time length (from 10 min to 4 h) and divided them into different groups according to the wind speed at the lowest level of the DLS measurement. Fig. 12 shows the number of profiles for each averaging time and speed category used for the following analysis. The number of samples decreased the longer the averaging time and the larger the wind speed. There is no sample of ub > 15 m/s for an averaging time of 4 h and ub > 20 m/s for an averaging time of 2 h. We fitted the profiles with the PL using the least-squares method with two unknown parameters of the reference speed and the PLI. In
Fig. 5. Frequency distributions of the wind direction for the 10 min averages measured by the DLS at heights of 527.5 and 67.5 m. 0°, 90°, 180° and 270° are corresponding to north, east, south and west wind directions, respectively. The high speed data were extracted using the horizontal wind speed for the lowest DLS level (ub).
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Fig. 6. Relationship between the wind speed at a height of 67.5 m (ub) and deviation in the wind direction for z=107.5, 167.5, 267.5, and 467.5 m from the direction at 67.5 m high (wdb) (10 min average).
this fitting, we limited the lowest value of the PLI to zero. From the results of the fitting, we calculated the coefficient of determination (R2) for each profile, which is defined in Eq. (3): N
R2 = 1 −
∑k =1 (U (zk ) − UPL (zk ))2 N
∑k =1 (U (zk ) − U )2
, (3)
where U(zk) is a speed measured by the DLS at the height zk (k=1, 2, …, N) averaged for a length of time, N is the number of levels used for the fitting by the PL, and U is averaged speed along the N height levels. The value of R2 implies how well the PL can express the change of measured wind speed along the height. Fig. 13 shows the mean value of R2 for the different averaging times and speeds. Regardless of the range of the height, the R2 value was high for all speeds (mean R2 > 0.9), especially when the ub > 15 m/s. It is difficult to fit wind speeds of a large altitude range with one PL profile (Fig. 8), therefore when we used a profile under 200 m, the R2 was also relatively higher in comparison to all heights. The reason is not clear for a drop in the mean R2 for the category with 10-min averages and speeds of 20–25 m/s. It is possible that the data were affected by a specific profile because the number of samples in this category was small compared with those in other categories. The longer
Fig. 7. Vertical profiles of the average of the wind direction deviation in relation to the direction at a height of 67.5 m on 10 min average basis. The error bars show the rootmean-square value of the wind direction deviation. The high wind speed data corresponds to wind speeds (ub)≥6 m/s.
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Fig. 8. Ensemble-averaged profiles of wind speed based on the 10 min averages for (i) ub≥6 m/s and (ii) ub≥12 m/s. Each profile is averaged after normalization using the horizontal wind speed at the lowest DLS level (ub). The error bars show the standard deviation of the profiles. North wind corresponds to 0≤ wind direction (wdb) < 20° or 340≤wdb < 360°, while south wind corresponds to 180≤wdb < 220°.
Fig. 9. Ensemble-averaged profiles of wind speed based on the 10 min averages for the all speed conditions. See Fig. 8 for a description of other details.
Fig. 11. Probability and cumulative distribution functions of the PLI derived from the 30 min averaged wind speeds at 67.5 and 167.5 m high for (i) all speeds and (ii) high (ub > 6 m/s) and low (ub < 6 m/s) speeds.
averaging time tended to result in a higher R2 when the speed was relatively high. However, even though the averaging time was 4 h, the R2 was 0.61 and 0.85 for the all heights and for the heights < 200 m, respectively when the ub was < 5 m/s. The wind speed had a greater effect on the accuracy of the PL than the averaging time.
Fig. 10. Wind speed at the DLS level (ub) and PLI (α) derived from the 30 min averages of wind speed at 67.5 and 167.5 m high.
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H. Kikumoto et al. 100000 10 mins
30 mins
1 hr
2 hrs
4 hrs
Number of samples
10000
1000
100
10
1 0 - 5 m/s
5 - 10 m/s
10 - 15 m/s Wind speed
15- 20 m/s
20 - 25 m/s
Fig. 12. Number of available samples of profiles for each time-averaging period and wind speed category. Profiles were categorized using the horizontal wind speed for the lowest DLS level (ub).
Fig. 14. Diurnal variations of wind speed (DLS and UA) for each height averaged for the entire measurement period.
1 10 mins 1 hr 4 hrs
Mean R [-]
0.9
30 mins 2 hrs
acceleration, and H is upward turbulent heat flux. Fig. 15(i) and (ii) show the average diurnal changes of the turbulent fluxes of momentum and heat from the UA measurement. Using these two results, the Monin-Obukhov length was calculated, and z/L at the height of the UA (52 m) is presented in Fig. 15(iii). The downward turbulent momentum flux increased in the daytime corresponding to the increase in the wind speed in Fig. 14, and the flux value peaked around 15:00. The upward heat flux increased during the daytime with its largest value occurring around the noon, which is earlier than the peak in momentum flux. As a result, the atmosphere tended to be unstable in the daytime and the stability (z/L) had a negative peak from 10:00 to 11:00. Stability showed very large standard deviation at several times in Fig. 15(iii). This is because very small (close to zero) friction velocity was measured in some samples, and z/L is proportional to u*−3. Fig. 16 also presents mean diurnal variation of the PLI, but only uses data with ub > 6 m/s because of the validity of the PL. Because the measurement site can be categorized in terms of the ground roughness into the “rough” or “very rough” categories provided by Counihan (1975), the PLI was expected to reach 0.2–0.4 on these ground surfaces. Although the PLIs in this study were close to this range, the PL had a pattern of diurnal change. In the evening, the PLI gradually increased and peaked in the early morning (from 5:00 to 6:00), and then the PLI dropped to lower than 0.2 during the daytime. The trend of the small PLIs in unstable conditions agrees with the data from Irwin (1979) and Touma (1977). Therefore, the small PLI during the low wind speeds might be explained in terms of the atmospheric stability. However, further discussion of the modeling of the PLI based on the stability is beyond the scope of this paper, and we leave it for future studies.
0.8 0.7 0.6 0.5 0.4 0 - 5 m/s
5 - 10 m/s
10 - 15 m/s Wind speed
15- 20 m/s
20 - 25 m/s
15- 20 m/s
20 - 25 m/s
(i) 1
10 mins 1 hr 4 hrs
Mean R [-]
0.9
30 mins 2 hrs
0.8 0.7 0.6 0.5 0.4 0 - 5 m/s
5 - 10 m/s
10 - 15 m/s Wind speed
(ii) Fig. 13. Mean coefficient of determination (R2) of the fitting of horizontal wind speeds at (i) all levels and (ii) levels less than 200 m using the PL for each time-averaging period and wind speed category. Profiles were categorized using the horizontal wind speed for the lowest DLS level (ub).
3.8. Diurnal change of wind speed and the PLI 4. Conclusions
In Section 3.6, we showed that the PLI tended to decrease for the low wind speeds. Fig. 14 shows mean diurnal variations of wind speed. The speeds clearly increase with time during the day (from 8:00 to 18:00), and the difference in speed along the height decreases at the higher measurement heights in the afternoon. Previous studies showed that the PLI depended not only on the roughness of the land surface, but also on the atmospheric stability and can be explained using atmospheric stability parameters, such as the Pasquill stability class and the Monin-Obukhov length (L) (Hanafusa et al., 1986; Irwin, 1979; Monin and Obukhov, 1954; Touma, 1977). The Monin-Obukhov length is given as:
L=−
ρcp Θ0 u*3 κgH
,
Wind profiles in the urban boundary layer under various wind conditions were measured in Tokyo, Japan using a DLS. We simultaneously used a UA to measure turbulent statistics using the ECM in the boundary layer. Measurements were conducted for a total of seven months. We applied the PL to approximate the measured wind profiles and discussed the approximate accuracy for various wind conditions, especially in terms of wind speed and averaging time. Moderate and low wind conditions dominated during the observed period. The 10-min wind speed at 67.5 m high (ub) averages were < 6 m/s during 80% of the observation period. Wind speeds measured at the DLS and UA sites were highly correlated with each other. The data acquisition ratio of 10-min averaged profiles by the DLS was 58.9% because of missing data from unfavorable measurement conditions. However, we did not find significant influence on the statistics of mean speeds due to the lack of samples from the DLS when we compared the
(4)
where ρ is density, cp is specific heat capacity, Θ0 is mean temperature, u* is friction velocity, κ is the von Karman constant, g is gravitational 19
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PDFs of wind speed with the UA data. For low wind speeds (ub < 6 m/s), the large deviation in wind direction with height made it difficult to determine the prevailing wind direction. Therefore, modeling the approaching flow with unique wind direction along the height was inappropriate under low wind conditions. For high wind speeds, the deviation in wind direction decreased and the PL could be used to model the ensemble-averaged wind profiles. When ub was > 12 m/s, the PLI converged to 1/4. However, when we approximate the wind profile based on wind speed at the lowest level of the measurement, the wind speed tended to move away from the PL at greater heights (z > 300 m). When we ignored the wind direction deviation with height and the wind profile was composed of many ensemble-averaged samples, the PL was a good model and the PLI of the profile was around 1/4. However, when we focused on profiles with a 30-min average, the PLI had a large variation for low wind conditions. The mode of the PLI for all speeds was about 0.15, and the PDF has a long-tailed distribution towards a larger PLI. The average PLIs for all, high (ub > 6 m/s), and low (ub < 6 m/s) wind speeds were 0.22, 0.25, and 0.21, respectively. The approximation accuracy of the PL for wind profiles significantly depended on the wind speed and then with the averaging time. When the wind speed was low (ub < 6 m/s), profile fitting using the PL was inappropriate to model the approaching flow even for long averaging times, e.g., 4 h. However, with increasing wind speed, the approximation accuracy significantly improved even for short averaging times. When ub was > 15 m/s, the coefficient of determination by the PL for the profiles was > 0.9 for 10-min averaged data. Although the PL could only be used for high wind conditions, there was a clear diurnal pattern to the PLI. It was larger in the nighttime and early morning, reaching values of 0.4 at night and dropping to < 0.2 during the day. We suggest that this pattern is correlated with wind speed and atmospheric stability. The analysis of thermal and air pollution in urban areas is more relevant at lower wind speeds because this results in higher temperature or pollutant concentrations affecting the daily lives of people. This study revealed the limitation in modeling wind profiles for low speeds, which should be addressed in future work. Acknowledgements This work was partly supported by the Japan Society for the Promotion of Science (JSPS) KAKENHI Grants No. 24226013, 26709041, and 24241008. Fig. 15. Diurnal variations of (i) downward turbulent momentum flux, (ii) upward turbulent heat flux and (iii) z/L (z=52 m, L=Monin-Obukhov length). The data are derived from the UA measurements and averaged for the entire measurement period. Error bars show the magnitude of the standard deviation.
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