Abstract
 Top of page
 Abstract
 1. Similarity theory in the convective urban roughness sublayer
 2. Site description
 3. Results and discussion
 4. Conclusions
 Acknowledgements
 Appendix
 References
Similarity theory using local scales was applied to the normalized standard deviation of the vertical wind component, w, and potential temperature, θ, σ_{w}/u_{*loc} and σ_{θ}/θ_{*loc}, where u_{*loc} and θ_{*loc} are the friction velocity and temperature and ‘loc’ refers to variables that are locally measured. These data were obtained in a tropical city under convective atmospheric conditions within the roughness sublayer. The following parameters were assessed based on the upwind characteristics of the site, denoted as ‘sectors’: the nondimensional height, z/h (1.16 and 1.20), and the nondimensional vertical heterogeneity, σ_{h}/h (0.56 and 0.32). The results obey a semiempirical relation of the form σ_{w}/u_{*loc} = Φ(−ζ_{loc})^{1/3} (ζ_{loc} = z′/L_{loc}, where z′ is the effective measurement height, L_{loc} is the local Obukhov length). The resultant extrapolated nearneutral constants depend on σ_{h}/h: (σ_{w}/u_{*loc})_{neutral} = 1.04 and 1.27 for σ_{h}/h = 0.56 and 0.32, in the range − 0.01 > ζ_{loc} > − 20. Spectral analysis of w reveals a separation of the spectral power at low nondimensional frequencies (f < 0.03) with increasing deviation as the atmosphere approaches neutral conditions, ζ_{loc} > − 0.6 for σ_{h}/h = 0.32. The term σ_{θ}/u_{*loc} follows the form Φ(−ζ_{loc})− ^{1/3} with constants of (σ_{θ}/θ_{*loc})_{neutral} = − 1.31 and − 1.34 for the two sectors, thus being independent of both σ_{h}/h and z/h. These findings show that similarity theory using local scales is applicable for determining σ_{w} and σ_{θ} in the convective roughness sublayer but depends on the vertical heterogeneity of the upwind sectors for σ_{w}, which only affects the nearneutral σ_{w}/u_{*loc} constants. Copyright © 2012 Royal Meteorological Society
1. Similarity theory in the convective urban roughness sublayer
 Top of page
 Abstract
 1. Similarity theory in the convective urban roughness sublayer
 2. Site description
 3. Results and discussion
 4. Conclusions
 Acknowledgements
 Appendix
 References
The urban roughness sublayer (RS), which lies within the urban boundary layer (UBL), is the atmospheric layer directly above an urban surface, which is nonnegligible in height (Rotach, 1999) and depends on atmospheric stability. The RS is also known as the ‘transition’, ‘interfacial’, or ‘wake’ layer and is mechanically and thermally influenced by scales related to surface heterogeneity (Roth, 2000). The height of this layer also depends on the ‘roughness’ of its surface and can even extend until it covers the entire atmospheric surface layer (ASL) (Christen et al., 2009). This height has been reported to range from 1.5 to 4 times the mean obstacle height and can widen to tens of metres in some urban areas (Rotach et al., 2005). Roth (2000) stated that the average building (or obstacle) height, h, is one of the most important characteristics of an urban surface, aside from the distance between the influential buildings, D, aerodynamic roughness length, z_{0}, and zeroplane displacement height, z_{d}.
The absence of a theoretical framework to simplify turbulence behaviour in the RS due to its threedimensional nature leads most researchers to resort to using the Monin–Obukhov similarity theory (MOST) (Roth, 2000). Based on this theory, the mean gradients and turbulence characteristics of the stratified ASL are only influenced by the following (Foken, 2006): height, z; kinematic surface stress, u′w′; kinematic heat flux, w′θ′; and buoyancy, g/θ, where u and w are the longitudinal and vertical wind velocity, respectively, θ is the temperature, and g is the gravitational acceleration (an apostrophe denotes the deviation from the mean, represented by overbars, e.g., w = w̄ + w′). Aside from the condition that turbulence must be statistically and directionally stationary, Panofsky and Dutton (1984) concluded that the applicability of MOST is confined to simple and homogeneous terrain and is only valid for some scalars and vectors, e.g., temperature and vertical wind component, respectively, while the horizontal wind components (u and v) depend on the height of the planetary boundary layer (PBL) (Arya, 1999). The velocity and temperature scales, which are generally valid in the ASL, are the friction velocity, u_{*}, and friction temperature, θ_{*}.
A nondimensional Equation (A.1), solely dependent on a stability parameter, is expected to describe the ASL (refer to the Appendix for a list of equations). The constants needed in the nondimensional equation were provided by Panofsky et al. (1977) and shown in Equation (A.2) for comparison.
It was generally accepted that from intermediateunstable to nearneutral conditions (0 > ζ > − 0.1), σ_{w}/u_{*} is independent of − ζ, whereas σ_{w}/u_{*} is proportional to (−ζ)^{1/3} (hence the 1/3 power slope law) in very unstable atmospheric conditions (ζ > − 1). The parameter σ_{w} is the standard deviation of the vertical wind component, and ζ is the atmospheric stability parameter, z/L, where L is the Obukhov length. Physically, the increase of σ_{w}/u_{*} with − ζ to the power of 1/3 is mainly due to the effect of dominating heat fluxes (w′θ′) on σ_{w} (Hurk and Bruin, 1995). This theory is supported by Agarwal et al. (1995) and Hicks (1981), who stated that it is commonly observed in the convective regime.
In reference to the normalized standard deviation of temperature with friction temperature, σ_{θ}/θ_{*}, and according to similarity theory, nondimensional equations that depend solely on ζ also apply in both the ASL and the RS (to our knowledge, based on several studies by Rotach (1994) and Oikawa and Meng (1995)). These relations are expressed by Equations (A.3) and (A.4).
In the ASL, the σ_{θ}/θ_{*} with − ζ is well known to obey the − 1/3 power law under unstable or convective conditions (e.g., Tillman (1972), Rotach (1994), Quan and Hu (2009)), although the semiempirical equation form and empirical constants differ.
It would be interesting to determine the extent to which this theory can be applied in the inferred nonconstant heat and momentum flux layer above a heterogeneous surface of the RS using locally scaled turbulent parameters, e.g., σ_{w} and σ_{θ}, as these have been well described in an homogeneous ASL (Panofsky and Dutton, 1984). Some researchers have found evidence that this theory is applicable in the RS (Oikawa and Meng, 1995; Rotach, 1999; Roth, 2000): however, little has been published regarding the very convective atmosphere, where u_{*} is deemed to be at its least influential to turbulence, close to the urban canopy (rooftops) with this context in mind. Furthermore, Roth (2000) concluded that transfer processes near the latter canopy (slightly above and within it) require further study to investigate ‘organized’ motions produced by the wakes of the miscellaneous shapes of structures (manmade or otherwise, e.g., trees) common to the urban area. Thus, a discussion of recently published works that dealt with urban turbulence and similarity theory employing local scales is warranted, as discussed in detail in the following paragraph, in addition to a summary of the relevant information produced by these studies and the current study listed in Table 1 (all researchers employ ‘local’ scales, defined in the following sections, in their analysis).
Table 1. Relevant results reported from other urban sitesResearchers  Measurement height  Building height range and/or average  Description  Surface covered  z_{d}  z_{0}  σ_{w}/u_{*}a  σ_{θ}/θ_{*}a 


 47 m  3–13 m  Residential houses/buildings  22%  —  —  —  — 
 120 m  —  —  —  —  —  —  — 
AlJiboori (2008) and AlJiboori et al. (2002)  280 m  70–90 m  Tall buildings  15%  5.4 m  4.3 m  1.22  − 2.23 
 —  18 m  Old trees  —  —  —  —  — 
 —  14 m (geometric mean height)  —  —  —  —  —  — 
 14–253 m  60 m (average height)  Madison Square Garden, Manhattan, New York City  —  —  —  1.36  − 3.24 
Hanna et al. (2007)  —  —  —  —  —  —  —  — 
 34–229 m  15–20 m (average height)  Downtown Oklahoma City  —  —  —  1.56  − 3.63 
 50 m  8 m (average height)  Nanjing University (city centre)  —  —  0.63 m  1.23  — 
Yumao et al. (1997)  —  —  —  —  —  —  —  − 1.86 
 16–164 m  —  Baguazhou (rural)  —  —  0.045 m  1.35  — 
 —  —  Buildings  32.6%  —  —  —  — 
Moriwaki and Kanda (2006)  29 m  7.3 ± 1.3 m  —  —  —  —  Lowb  — 
 —  —  Vegetation  20.6%  —  —  —  — 
Quan and Hu (2009)  c  c  c  c  c  c  1.33  − 1.5 
Wilson (2008)  8.71–25.69 m  —  Dry lakebed, Dugway Proving Grounds, Utah  —  —  —  0.8; 1.0  — 
 —  8.8 ± 3.0 m  Greater London (< 10 km radius)  —  —  0.87 ± 0.48 m  —  — 
Wood et al. (2010)  190.3 m  —  —  —  4.3 ± 1.9 m  —  1.31  − 1.4 
 —  5.6 ± 1.8 m  Suburban Greater London (> 10 km radius)  —  —  0.27 ± 0.21 m  —  — 
 —  —  Sector A  —  —  —  —  — 
 —  15.0 ± 4.8 m  • Buildings  24%  —  —  —  — 
 —  —  • Trees  22%  —  —  1.04  − 1.34 
Present study  4 m  —  Sector B  —  14 m  0.38 m  —  — 
 —  15.4 ± 8.7 m  • Buildings  24%  —  —  1.27  − 1.31 
 —  —  • Trees  21%  —  —  —  — 
AlJiboori (2008) reported that the empirical similarity relationships of various correlation coefficients, r, between u, v, w, and θ were functions of a single local stability parameter under different atmospheric stabilities at multiple heights measured over a wide wind speed range (2–15 m s^{−1}), adding support to the theoretical framework used in the data analysis in the present paper. Because the measurement tower was high (325 m), the author was able to collect data in the ASL (or inertial sublayer) of a city with a mixture of low (3–13 m) and tall buildings (70–90 m), making the vertical heterogeneity, σ_{h}/h, likely to be high. In a more recent study, Wood et al. (2010) determined that similarity theory was also applicable at their site high above the rooftops of London (190.3 m above ground level), placing their measurement within the surface layer. Using the same infrastructure as AlJiboori (2008), Quan and Hu (2009) discovered that surface characteristics were influenced by the relationship between momentum fluxes and velocity variances, while atmospheric stratification affected the relationship between sensible heat fluxes and temperature variances. These researchers derived relationships in the form given by Equations (A1)–(A4), which are similar to the literature values (1.25–1.30) for σ_{w}/u_{*loc}, but they fitted their equations to conform to the free convective limit of power of 1/3. Interestingly, the researchers mentioned that large fluxes and variances were caused by strong winds, even when normalized by u_{*}. In another study (Hanna et al., 2007) measurements were made on top of buildings and at street level in the major North American cities of Oklahoma City and Manhattan, with extremely variable building configurations, it was found that the neutral value of σ_{w}/u_{*loc} was high, ( = 1.5), possibly caused by site heterogeneity (σ_{h}/h was also inferred to be high: refer to Table 1) and large data scatter. Reviewing the universality of similarity theory (or, specifically, MOST) in the surface layer, Wilson (2008) argued that the neutral value of σ_{w}/u_{*loc} was much lower than canonical values, suggesting that it could be as low as 0.8–1, and that the constants of best fit depend on the data scatter at the site, a relatively flat area. The parameter, u_{*}, was suspected of causing most of the scatter (u_{*} > 0.15 m s^{−1} was observed in his study). In a suburban area of Tokyo, Japan, Moriwaki and Kanda (2006) found that heat was transferred by thermal and organized motions (mentioned previously), where its ejections (upward motion) were greater than sweep (downward motion), thus creating a net upward air movement. This phenomenon was caused by the active role of temperature and heat source heterogeneity.
In summary, the neutral constant of σ_{w}/u_{*loc} is approximately 1.33 ± 0.11 m s^{−1}, which is surprisingly close to its ASL counterpart ( = 1.3, Panofsky et al. (1977)), regardless of urban surface conditions and independent of h, z_{0} and z_{d}, although most studies do not report σ_{h}/h values. Furthermore, all researchers have observed the theoretical 1/3 power slope for unstable conditions. However, the neutral constant of σ_{θ}/θ_{*} is more variable, taking values of − 2.31 ± 0.93, even at the same location, although all of their power slopes are − 1/3. Work by AlJiboori et al. (2002) and Quan and Hu (2009) (refer to Table 1) exemplify this finding particularly well.
Although most of these studies dealt with urban turbulence and similarity theory, their measurements are suspect. These measurements may have actually been within the inertial sublayer (SL), and they exhibited characteristics similar to homogeneous surfaces for which MOST could be applied (e.g., plains, prairies, and oceans). In contrast, the measurements in the present study are quite close to the urbanatmosphere partition (rooftop), where both the heat source and sink and their effect on the variances could be more variable (Moriwaki and Kanda, 2006), in addition to the possible contribution of different urban morphologies to turbulence statistics, i.e., wakes. Furthermore, the wind range measured in the present study is low (0.5–2.0 m s^{−1}), while most of the abovementioned studies collected data in the highwindspeed range (2–15 m s^{−1}), which could create large variances and fluxes even after normalization (Quan and Hu, 2009). In addition, upwind sectors were not considered when investigating the surface characteristics (h, z_{0} and z_{d}) despite their possible significance in the analysis of turbulence variances in the MOS framework (even when employing local scales) given that each sector would have vastly different traits and may even be ‘directionally variable’ (AlJiboori and Fei, 2005), especially in proximity to the urban canopy.
In a review by Roth (2000), the turbulence structure of in the RS resembled the flow above plant canopies and can be described by analogy to a planemixing layer. Other studies address the application of similarity theory in convective boundary layers (CBLs) and/or heterogeneous terrain, such as urban centres (Rotach, 1993a, 1993b; Oikawa and Meng, 1995; Rotach, 1999; Roth, 2000; Moraes et al., 2007; Bin Yusup et al., 2008; Christen et al., 2009). Rotach (1993b) found that, in general, local scaling is relevant in the RS and that the locally scaled nondimensional wind gradient was similar to its inertial sublayer counterpart. However, another paper (Rotach, 1993a) it was stated that the heightdependent characteristic of turbulent momentum flux was able to describe the mean wind speed profile. However, the complexity of urban areas will produce superfluous turbulence, mainly caused by shear stress even under convective conditions, which could affect the wind profile (Zhang et al., 2001; Venkata Ramana et al., 2004; Tsai and Tsuang, 2005). Locally scaling σ_{w} in the RS was found valid because w′θ′ was reported to be independent of height, z/h > 1.2 (Oikawa and Meng, 1995; Christen et al., 2009). However, urban areas are usually associated with complex terrain features and numerous obstructions. Thus, the existence of surface roughness or mechanical production is more significant to turbulence and must be addressed.
The unique climatology feature of the present study is its location in the equatorial and tropical regions, which are exposed to intense solar radiation throughout the year and experience high incidences of low wind conditions of generally < 5 m s^{−1} (Lim and Azizan Abu, 2004). Such meteorological scenarios result in the prevalence of a ‘free convective’ atmosphere for most of the day time in these regions. Urban areas are more prone to experiencing these conditions because of buildings or other manmade structures that enhance convection. Studies of the ‘micro’atmosphere are seldom conducted in tropical and equatorial developing countries, e.g., South East Asian countries, whereas most studies have been performed in western midlatitude regions (Bin Yusup et al., 2008). Thus, the data presented here are rare and describe the convective nature of the atmosphere in a tropical city. This work refers to common similarity theories to assess how the attributes of the urban area vary with upwind direction in terms of h and σ_{h} (and their nondimensional variants, z/h and σ_{h}/h). Applications of this work include the use of the derived semiempirical equations in air pollution models focusing on a very convective atmosphere near the urban canopy. Moreover, Rotach (1999) found that incorporating the influence of the RS in an airpollution dispersion simulation greatly enhanced the downwind estimation of the groundlevel pollutant concentrations.
2. Site description
 Top of page
 Abstract
 1. Similarity theory in the convective urban roughness sublayer
 2. Site description
 3. Results and discussion
 4. Conclusions
 Acknowledgements
 Appendix
 References
Observations using towerbased meteorological instruments (at a combined height, z = 18 m) were made on the rooftop of the School of Industrial Technology, Universiti Sains Malaysia (USM) (5°21′28″N; 100°18'06.5″E), Penang, Malaysia. The sensors were positioned such that they were not obstructed by buildings or trees. The sonic anemometer was installed at z = 18 m from ground level, and the morphometric methods given by Grimmond and Oke (1999) were used to determine the displacement height, z_{d}, and roughness length, z_{0} (14 and 0.38 m, respectively), which are summed to yield the effective measurement height, z′ = 4 m. Turbulence data were recorded during the 3 month intermittent observational period from November 2009 to March 2011, during the winter monsoon. The fetch distance was also estimated to be 370 m using the ruleofthumb equation given in Roth (2000).
This sampling location was surrounded by various facilities, including lecture hall complexes and student hostels (Figure 1). Because different sectors have different building structures and topographies, the surrounding area was divided into four sectors, labelled A, B, C and D, depending on the 30 min averaged wind direction. The four sectors were also chosen to maximize the number of data points in each sector. The heights of the observation site in relation to other buildings for all sectors in the surrounding area were as follows: (z/h)_{A} = 1.20; (z/h)_{B} = 1.16; (z/h)_{C} = 1.24; and (z/h)_{d} = 1.16, where h is the weighted average of building and tree heights in the sector (refer to Table 2 for details). Even though the sectors seemed similar, they differed by the standard deviation of the obstacle heights (between sector A and B). Thus, B can be considered more heterogeneous than A (A has higher obstacle density), while C had a sparser distribution of buildings (but more trees) than A and B. Because z/h is somewhat similar for all sectors, another dimensionless parameter is introduced that varies more significantly between different sectors. This parameter is the standard deviation of obstacle height to the averageweighted height, termed the vertical heterogeneity, σ_{h}/h, and was chosen because σ_{h} varies more strongly between sectors than h does. The site surface is clearly complex. Thus, u′w′ in part depends on the upwind surface characteristics in this case. Furthermore, the division of a heterogeneous area into sectors is almost standard practice and is recommended near an urban canopy, such as in the work of Christen et al. (2009) and Grimmond and Oke (1999). Christen et al. (2009) found that u′w′ increased based on the crosswind/alongwind direction of flow in relation to a street canyon, while AlJiboori and Fei (2005) discovered that z_{0} was considerably directiondependent over an urban area.
Table 2. Simple land classification scheme for each of the four upwind sectors surrounding the monitoring stationSector (z/h)  Weighted average height (m) of obstacles  σ_{h}/h  Area (%) 

Buildings  Trees  Other (streets, parking lots, etc.) 

A (1.20)  15.0 ± 4.8  0.32  24  22  54 
B (1.16)  15.4 ± 8.7  0.56  24  21  55 
C (1.24)  13.8 ± 5.4  0.39  17  24  59 
D (1.16)  15.9 ± 5.0  0.31  26  18  56 
From the initial analysis of the meteorological data, the mean wind speed was 1.98 ± 0.54 m s^{−1} and temperature was 32.08 ± 1.08 °C within the observational period. Figure 2 includes a wind rose indicating the general wind direction at this site.
2.1. Instrumentation and data analysis
A Young Model 81000 Ultrasonic Anemometer (three axes) was used in this study. This sensor measures the threedimensional wind velocity (u, v and w, corresponding to the x, y, and z axes of the Cartesian coordinates, respectively). The speed of sound was measured using the transmission time of ultrasonic acoustic signals. The sonic temperature (corrected by the potential temperature, θ, using the methodology given by Schotanus et al., 1983) is then calculated from the speed of sound, which is corrected for crosswind effects. Data were collected at a sampling frequency of 10 Hz, directly connected to a personal computer (PC) via serial output (RS232) using HyperTerm as an interface. A similar setup was also used by Mizoguchi et al. (2009).
The eddy correlation (EC) method was used to estimate u_{*} and θ_{*} (Krishnan and Kunhikrishnan, 2002). The velocity scale, u_{*}, is a reference wind velocity defined by Equation (A.5). A surface stress representation, in terms of the surface kinematic momentum fluxes (w′u′_{0} and w′v′_{0}) with the subscript ‘0’ representing the surface value, can also be used to estimate u_{*}.
This representation is more useful because it allows the calculation of u_{*} using EC. θ_{*}, the temperature scale, was estimated using Equation (A.6), where H_{0} is the surface sensible heat flux and c_{p} is the heat capacity of air in constant pressure (Agarwal et al., 1995). The length scale, or Obukhov length, L, (see Equation (A.7)) is the height in the surface layer at which surface stresses are likely to influence turbulence under any stability condition. The ratio of height, z, and L is known as the stability parameter and is related to the Richardson number (Ri). This ratio is the measure of the importance of buoyancy in relation to surface stress. MOST states that turbulence quantities, when properly scaled, in the SL are a function of z/L(= ζ, where the negative sign denotes convective or unstable atmospheric conditions).
All scaling parameters presented above apply in the SL, but the ‘local’ versions of the scaling parameters are used in this paper because the effective height of measurement, z′, is less than z/h = 2, which some researchers have found to be high enough above the urban canopy to be classified as ASL (Rotach, 1999; Christen et al., 2009). The subscript ‘loc’ will be used throughout this paper to reflect that the scales used are locally measured.
2.2. Sampling and quality control
The data were collected from 1000 to 1500 local time (0200 to 0700 UTC) daily for 3 months (64 days) to ensure that the data collected only include convective conditions and that turbulence is well developed and stationary, even avoiding diurnal forcing (AlJiboori, 2008). Thus, measurements were only taken during the afternoons of sunny and cloudless days. A sunny day was defined as a day with clear skies (< 1/4 cloud cover) and strong insolation (Pasquill stability class A). Rainy day data were excluded from further analysis due to sensor limitations. The data collected were also inspected visually for instrumental errors. The standard deviation and average of each wind direction velocity and temperature was computed using block averages and has been found to work well in stationary conditions (Culf, 2000), which is in the 30–45 min range (as suggested by Pasquill and Smith (1983)). Horizontal rotation was also performed on the longitudinal wind velocity component before further analysis. The vertical wind component was forced to achieve zero mean, w = 0, (horizontal wind flow) as suggested by Panofsky and Dutton (1984). The mean of the lateral wind component was also forced to be zero, v = 0 (Arya, 1999). However, the obtained values were the same as the values obtained using a nonzero mean for the lateral wind component. Vickers and Mahrt's (1997) data quality control methodology was applied to the collected data to ensure the quality of the calculated turbulent fluxes and variances.
The block averages or record length, R_{l}, must be properly chosen to ensure the accurate calculation of w′θ′, w′u′, and w′v′ and, consequently, u_{*}. The methodology suggested by Vickers and Mahrt (1997) was used to determine R_{l}. The relative systematic flux error (RSE) test was performed on the data to check for flux underestimation caused by an inappropriate choice of R_{l}. RSE tests were performed for three record lengths: 5, 30 and 60 min. The results of these tests indicate that R_{l} = 5 min underestimated the fluxes, while R_{l} = 60 min produced the same results as R_{l} = 30 min. Thus, a record length of 30 min was chosen for this study. This time point was also recommended by Pasquill and Smith (1983) and Panofsky and Dutton (1984) and used in an urban setting by Roth et al. (2006) and Bin Yusup et al. (2008), although some authors have used shorter durations in similar settings (Oikawa and Meng, 1995). It is interesting to note that Wilson (2008) tentatively determined that the averaging time was not important in determining the neutral limit of σ_{w}/u_{*}.
Four types of nonstationarity tests were applied to the data. Nonstationary conditions can affect the fluxes estimated by EC. As outlined by Vickers and Mahrt (1997), the four tests were the wind speed reduction, alongwind relative nonstationarity (RNu), crosswind relative nonstationarity (RNv), and vectorwind relative nonstationarity (RNS) tests. The collected data were determined to be nonstationary for 11% of the data. However, nonstationary data do not significantly increase the scattering of the variances and fluxes within certain ranges of atmospheric stability (2 > ζ > − 3), as shown by Marques Filho et al. (2008).