• similarity theory;
  • roughness sublayer;
  • convective conditions;
  • turbulence statistics


  1. Top of page
  2. Abstract
  3. 1. Similarity theory in the convective urban roughness sublayer
  4. 2. Site description
  5. 3. Results and discussion
  6. 4. Conclusions
  7. Acknowledgements
  8. Appendix
  9. References

Similarity theory using local scales was applied to the normalized standard deviation of the vertical wind component, w, and potential temperature, θ, σw/u*loc and σθ*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 non-dimensional height, z/h (1.16 and 1.20), and the non-dimensional vertical heterogeneity, σh/h (0.56 and 0.32). The results obey a semi-empirical relation of the form σw/u*loc = Φ(−ζloc)1/3loc = z′/Lloc, where z′ is the effective measurement height, Lloc is the local Obukhov length). The resultant extrapolated near-neutral constants depend on σh/h: (σw/u*loc)neutral = 1.04 and 1.27 for σh/h = 0.56 and 0.32, in the range − 0.01 > ζloc > − 20. Spectral analysis of w reveals a separation of the spectral power at low non-dimensional frequencies (f < 0.03) with increasing deviation as the atmosphere approaches neutral conditions, ζloc > − 0.6 for σh/h = 0.32. The term σθ/u*loc follows the form Φ(−ζloc)− 1/3 with constants of (σθ*loc)neutral = − 1.31 and − 1.34 for the two sectors, thus being independent of both σh/h and z/h. These findings show that similarity theory using local scales is applicable for determining σw and σθ in the convective roughness sublayer but depends on the vertical heterogeneity of the upwind sectors for σw, which only affects the near-neutral σw/u*loc constants. Copyright © 2012 Royal Meteorological Society

1. Similarity theory in the convective urban roughness sublayer

  1. Top of page
  2. Abstract
  3. 1. Similarity theory in the convective urban roughness sublayer
  4. 2. Site description
  5. 3. Results and discussion
  6. 4. Conclusions
  7. Acknowledgements
  8. Appendix
  9. 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 non-negligible 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, z0, and zero-plane displacement height, zd.

The absence of a theoretical framework to simplify turbulence behaviour in the RS due to its three-dimensional 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, uw′; 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 = + 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 non-dimensional 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 non-dimensional equation were provided by Panofsky et al. (1977) and shown in Equation (A.2) for comparison.

It was generally accepted that from intermediate-unstable to near-neutral conditions (0 > ζ > − 0.1), σw/u* is independent of − ζ, whereas σw/u* is proportional to (−ζ)1/3 (hence the 1/3 power slope law) in very unstable atmospheric conditions (ζ > − 1). The parameter σw 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 σw/u* with − ζ to the power of 1/3 is mainly due to the effect of dominating heat fluxes (w′θ′) on σw (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, σθ*, and according to similarity theory, non-dimensional 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 σθ* 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 semi-empirical equation form and empirical constants differ.

It would be interesting to determine the extent to which this theory can be applied in the inferred non-constant heat and momentum flux layer above a heterogeneous surface of the RS using locally scaled turbulent parameters, e.g., σw and σθ, 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 (man-made 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 sites
ResearchersMeasurement heightBuilding height range and/or averageDescriptionSurface coveredzdz0σw/u*aσθ*a
  1. a

    Constant, near-neutral condition.

  2. b

    Qualitatively described by the authors, < 1.3.

  3. c

    Same as Al-Jiboori (2008) and Al-Jiboori et al. (2002).

 47 m3–13 mResidential houses/buildings22%
 120 m
Al-Jiboori (2008) and Al-Jiboori et al. (2002)280 m70–90 mTall buildings15%5.4 m4.3 m1.22− 2.23
 18 mOld trees
 14 m (geometric mean height)
 14–253 m60 m (average height)Madison Square Garden, Manhattan, New York City1.36− 3.24
Hanna et al. (2007)
 34–229 m15–20 m (average height)Downtown Oklahoma City1.56− 3.63
 50 m8 m (average height)Nanjing University (city centre)0.63 m1.23
Yumao et al. (1997)− 1.86
 16–164 mBaguazhou (rural)0.045 m1.35
Moriwaki and Kanda (2006)29 m7.3 ± 1.3 mLowb
Quan and Hu (2009)cccccc1.33− 1.5
Wilson (2008)8.71–25.69 mDry lakebed, Dugway Proving Grounds, Utah0.8; 1.0
 8.8 ± 3.0 mGreater London (< 10 km radius)0.87 ± 0.48 m
Wood et al. (2010)190.3 m4.3 ± 1.9 m1.31− 1.4
 5.6 ± 1.8 mSuburban Greater London (> 10 km radius)0.27 ± 0.21 m
 Sector A
 15.0 ± 4.8 m• Buildings24%
 • Trees22%1.04− 1.34
Present study4 mSector B14 m0.38 m
 15.4 ± 8.7 m• Buildings24%1.27− 1.31
 • Trees21%

Al-Jiboori (2008) reported that the empirical similarity relationships of various correlation co-efficients, 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, σh/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 Al-Jiboori (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 σw/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 σw/u*loc was high, ( = 1.5), possibly caused by site heterogeneity (σh/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 σw/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 σw/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, z0 and zd, although most studies do not report σh/h values. Furthermore, all researchers have observed the theoretical 1/3 power slope for unstable conditions. However, the neutral constant of σθ* 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 Al-Jiboori 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 urban-atmosphere 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 high-wind-speed 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, z0 and zd) 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’ (Al-Jiboori 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 plane-mixing 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 non-dimensional wind gradient was similar to its inertial sublayer counterpart. However, another paper (Rotach, 1993a) it was stated that the height-dependent 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 σw 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 man-made 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 mid-latitude 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 σh (and their non-dimensional variants, z/h and σh/h). Applications of this work include the use of the derived semi-empirical 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 air-pollution dispersion simulation greatly enhanced the downwind estimation of the ground-level pollutant concentrations.

2. Site description

  1. Top of page
  2. Abstract
  3. 1. Similarity theory in the convective urban roughness sublayer
  4. 2. Site description
  5. 3. Results and discussion
  6. 4. Conclusions
  7. Acknowledgements
  8. Appendix
  9. References

Observations using tower-based 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, zd, and roughness length, z0 (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 rule-of-thumb 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 average-weighted height, termed the vertical heterogeneity, σh/h, and was chosen because σh varies more strongly between sectors than h does. The site surface is clearly complex. Thus, uw′ 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 uw′ increased based on the crosswind/along-wind direction of flow in relation to a street canyon, while Al-Jiboori and Fei (2005) discovered that z0 was considerably direction-dependent over an urban area.

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Figure 1. Schematic of area of study detailing the features of sectors A, B, C, and D

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Table 2. Simple land classification scheme for each of the four upwind sectors surrounding the monitoring station
Sector (z/h)Weighted average height (m) of obstaclesσh/hArea (%)
BuildingsTreesOther (streets, parking lots, etc.)
A (1.20)15.0 ± 4.80.32242254
B (1.16)15.4 ± 8.70.56242155
C (1.24)13.8 ± 5.40.39172459
D (1.16)15.9 ± 5.00.31261856

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.

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Figure 2. Frequency distribution of the wind direction and speed at the monitoring station (rooftop) on sunny days

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2.1. Instrumentation and data analysis

A Young Model 81000 Ultrasonic Anemometer (three axes) was used in this study. This sensor measures the three-dimensional wind velocity (u, v and w, corresponding to the x, y, and z axes of the Cartesian co-ordinates, 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 (wu0 and wv0) 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 H0 is the surface sensible heat flux and cp 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 (Al-Jiboori, 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 non-zero 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, Rl, must be properly chosen to ensure the accurate calculation of w′θ′, wu, and wv and, consequently, u*. The methodology suggested by Vickers and Mahrt (1997) was used to determine Rl. The relative systematic flux error (RSE) test was performed on the data to check for flux underestimation caused by an inappropriate choice of Rl. RSE tests were performed for three record lengths: 5, 30 and 60 min. The results of these tests indicate that Rl = 5 min underestimated the fluxes, while Rl = 60 min produced the same results as Rl = 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 σw/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, along-wind relative non-stationarity (RNu), crosswind relative non-stationarity (RNv), and vector-wind relative non-stationarity (RNS) tests. The collected data were determined to be nonstationary for 11% of the data. However, non-stationary 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).

3. Results and discussion

  1. Top of page
  2. Abstract
  3. 1. Similarity theory in the convective urban roughness sublayer
  4. 2. Site description
  5. 3. Results and discussion
  6. 4. Conclusions
  7. Acknowledgements
  8. Appendix
  9. References

3.1. General description of processed turbulence data

The observed standard deviations of the u, v and w wind velocity components are summarized in Table 3. Bin Yusup et al. (2008) performed a similar study across the strait from this site in the nearby heavy industrial area of Prai (approximately 10 km away). Higher values of σu and σv but lower values for σw were observed in the present study compared to those by Bin Yusup et al. (2008) due to the presence of more surface obstacles in this site, which mainly affects the horizontal wind components. In addition, the average and median values of σu and σv were equal for this site. As mentioned above, the measurements were taken from mostly low-wind-speed (< 3 m s−1) conditions for which large standard deviations are common. The number of data sets (Ntotal = 574), N, for each atmospheric stability class observed was distributed as follows (where z′/Lloc = ζloc):

  1. − 0.01 > ζloc > − 0.1, N = 65

  2. − 0.1 > ζloc > − 1, N = 421

  3. − 1 > ζloc > − 10, N = 86

  4. ζloc < − 10, N = 2

Table 3. Daytime (convective conditions) arithmetic mean and median of field experiment conditions
ParametersPresent study (N = 574)Bin Yusup et al. (2008) (N = 80)
z′ (m)42
V (m s−1)1.97 ± 0.541981.82 ± 0.851.87
U*loc (m s−1)0.27 ± ± 0.200.36
w′θ′0.11 ± 0.040.11
|Lloc| (m)20 ± 1717
σu (m s−1)1.54 ± 0.411.540.91 ± 0.240.96
σv (m s−1)1.54 ± 0.421.530.80 ± 0.150.82
σw (m s−1)0.41 ± 0.110.400.65 ± 0.180.68
σθ ( °C)0.74 ± 0.170.77

The prevalent unstable atmospheric state was anticipated because the observations were made in the afternoon.

The numbers of data categorized (based on wind direction) into sectors A, B, C and D were 290, 176, 88 and 20, respectively. The data in sectors C and D were not analysed further due to a lack of data and the occurrence of non-stationary conditions (wind meandering caused by very low wind speeds, < 1 m s−1, evident in Figure 2 for sector C). Higher-order moments of θ also indicated that there many occurrences of skewness and kurtosis (soft-flagged) for θ (14 and 4% of the data, respectively) and kurtosis for u (3%) in sector C, whereas the incidences of kurtosis and skewness of u, v and θ for sectors A and B were low (2–3%). Thus, the analysed data describe turbulence originating from upwind characteristics of similar average-weighted heights, h, but different standard deviation of obstacle heights, σh, for sectors A and B (refer to Table 2), for which the effect of the dimensionless heights, z/h and σh/h, and can be tested on σw and σθ.

Because measurements were made on the rooftop of a building, the collected data represent turbulence characteristic within the RS. However, the concept of local scaling (locally measured parameters) is still predicted to hold true in the roughness sublayer (Hogstrom et al., 1982) and was demonstrated by Oikawa and Meng (1995) and other authors (Rotach, 1999; Moriwaki and Kanda, 2006). This concept will be applied to the vertical wind velocity component and temperature and should be adequately described by this concept, in contrast to the horizontal wind component (Rotach, 1993a). Christen et al. (2009) has cautioned that this approach could be inadequate for flow within the RS (in street canyons) and the near-roofs vicinity. However, their study primarily dealt with near-neutral atmospheric conditions.

3.2. Correlation between uv and w–θ

Several studies have determined that the condition when surface layer scaling or its equivalent is applicable is defined by a stationary, homogeneous terrain and a high correlation between w–θ and u–v of generally r > 0.5. Kaimal and Finnigan (1994) observed that in the surface layer, the correlation was approximately 0.5 (0 > ζ > − 2) between w and θ and approximately − 0.35 (1 > ζ > − 1) between u and v. Our correlation was generally lower than these ranges, e.g., 0.56 > r > 0.13 and 3.32 × 10−2 > − ruv > 1.73 × 10−4. The averages were r = 0.38 and − ruv = 0.011. The low correlation of uv can be attributed to the many obstacles located upwind. Similarly, an even lower average of correlation for w–θ in unstable conditions ( = 0.24) and large scattering of the correlation for uv were obtained by Al-Jiboori (2008), despite similarity theory using local scales being applicable at his site. A check of the turbulent intensity, Iw = σw/V, showed that it had an average of 0.22 where Taylor's frozen hypothesis (Iw < 0.5) still holds (Agarwal et al., 1995). However, it was found that Iu and Iv were larger than 0.5.

3.3. Scaling σw using u*loc in RS

The normalized standard deviations of w and θ throughout the observational period are plotted against the atmospheric stability parameter, ζloc. These plots enable derivation of semi-empirical equations that relate the aforementioned turbulent statistics with ζloc when properly scaled. Moraes (2000) mentioned that significant scatter in these plots indicates that the scaling parameters used are inadequate, causing the turbulence fluxes to depend on other parameters. Simple power law equations were used to fit σw/u*loc, as done by Kader and Yaglom (1990), and the equation in the form made popular by Panofsky and Dutton (1984) was used to fit σθ*loc. The data were fitted using the Matlab® 2008a curve fitting tool, more specifically the least square regression scheme. The form of the equation used was based on the R2 obtained, in favour of a value near unity.

Figure 3(a) and (b), for sectors A and B clearly show that σw/u*loc is a function of (−ζloc)1/3 for very convective conditions starting at approximately ζloc = − 0.3 (equation image for A and equation image for B) (Oikawa and Meng, 1995), which agrees with predictions, although Arya (1999) suggested that the increase should start below − 0.5. Above ζloc = − 0.5, the data were more scattered because upwind surface heterogeneity caused scatter in the variable u*loc despite its generally larger value compared to when ζloc < − 0.5. Oikawa and Meng (1995) found that the magnitude of the normalized standard deviations was systematically higher within the canopy, suggesting that the values obtained for sector B were within the ‘urban layer canopy’. However, this conclusion contradicts that drawn by Roth (2000).

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Figure 3. The σw/u*loc distribution with − ζloc(−0.01 > ζloc > − 20) and sectors using a log-log scale for better visualization of the 1/3 power trend (dashed line). The vertical dashed line denotes ζloc = − 0.3, and the vertical dot-dot line denotes ζloc = − 0.5. The solid lines are based on Equations ((1))–(2) for sectors A (a) and B (b)

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From these results, the semi-empirical equations derived are valid for σw/u*loc and can be written as Equations ((1)) and ((2)), which exhibit a reasonably good fit (Figure 3).

Hogstrom et al. (1982) suggested, and Rotach (1993a) later demonstrated, that locally scaled σw can be related to ζloc in the URS in the same fashion as scaling in the ASL. The equation given by Panofsky et al. (1977) is similar to the equation that describes the data in sectors A and B. A side-by-side comparison of the equations in the form of Equation ((2)) is also provided in Figure 3:

  • equation image(1)
  • equation image(2)

(note: for − 0.01 > ζloc > − 100).

This phenomenon is well documented in the literature (Panofsky and Dutton, 1984; Krishnan and Kunhikrishnan, 2002; von Randow et al., 2006), although very convective conditions, − 0.01 > − ζ> − 100, are rarely reported, especially in equatorial cities (Krishnan and Kunhikrishnan, 2002; Bin Yusup et al., 2008). The near-neutral constant obtained as the neutral value extrapolated from Equations ((1)) and ((2)) was higher ( = 1.44, an averaged value) than that used in the equation for σw/u* by Bin Yusup et al. (2008), although there is insufficient data in the near-neutral range to confirm this result. However, Roth (2000) stated that ζ> − 0.05 could still be considered near-neutral, at least for sector A. Similar results to Bin Yusup et al. (2008) should be expected because both observational stations were near one another. Any deviation was caused by the sensor and sampling frequency used, which limits the flux-capture capability of the 1 Hz propeller-based anemometer used by Bin Yusup et al. (2008). Another possible factor was the difference in local topography or terrain and the atmospheric stability range, which the authors measured as only 0 > ζ> − 2.

The upwind surface characteristic could play a role in the difference in results between sectors A and B, although Zhang et al. (2001) mentioned that the vertical wind velocity is influenced by small high-frequency eddies and are able to adapt immediately to a new terrain, thus explaining its good relationship with − ζ, even in complex terrains. As shown in Table 2, the average height of obstacles and the area covered in sectors A and B were somewhat similar (< 3% for z/h between sectors) and similarly ‘homogeneous’ given that the percentage of area covered by buildings, trees and other obstacles are identical. Thus, this observation raises the question: why do the best-fit lines for σw/u*loc differ for these sectors?

The answer lies in the dimensionless σh/h, which differs by approximately 75%. Al-Jiboori and Fei (2005) found that changes in the general wind direction (thus, upwind surface characteristics) cause large disparities in the observed values of constants. Hanna et al. (2007) also mentioned that the presence of obstacles (or buildings) would produce an increase in the horizontal fluctuations caused by the wake of these obstacles that differ in size and/or position. In the present study, this difference manifests itself in the different values for σh/h: 1.04 in Equation ((1)) for sector A and 1.27 in Equation ((2)) for sector B. A lower constant value was explained by Moriwaki and Kanda (2006), who attributed it to the rough feature of the urban surface, which better facilitates momentum transport compared to flatter surfaces. Roth (2000) stated that any deviation from equations and constants obtained in the SL are due to smaller non-dimensional dissipation values for turbulent kinetic energy (TKE) because energy is introduced into the TKE budget from local sources (local production > local dissipation), i.e., vertical transport, flux divergence, pressure transport and horizontal advection. Poggi et al. (2004) found that sparse canopies exhibited higher near-neutral σw/u* values (≈1.3) than its denser counterpart (≈1.1), which was observed using a laboratory-scale channelled flow setup on uniformly distributed obstacles (or rods) of equal height. This conclusion is also supported by Raupach et al. (1996). Both studies are for flow over vegetated canopies: these canopies have strong similarities with the urban canopy, while sparse canopies require additional MOS modifications to be interpreted in the same framework due to the increasing importance of turbulence generated by wakes (Roth, 2000). Referring back to sectors A and B, because aerial obstacle densities are similar for both sectors, the changes in σh/h affect these constants (by 22%) just as they would density. Nevertheless, Equations ((1)) and ((2)) show behaviour expected from local scaling and, by extension, surface layer scaling, further supporting the work by Rotach (1993a).

3.3.1. Spectral analysis of w

The spectral analysis of the normalized w component for a data block of 30 min following the procedure detailed in Kaimal et al. (1972) revealed that the difference between sectors A and B became discernible in the normalized low-frequency range f( = nz/V)< 0.03, while the spectral-peak-normalized frequency (f = 0.2;f = 0.1 from Roth (2000)) remained the same. However, other researchers have observed that the w-spectra have lower frequency offsets at neutral and unstable conditions (Roth, 2000), the overall curves look similar to the ASL predictions (Kaimal et al., 1972). The latter author mentioned that a displacement of peak frequencies would be expected for significant wake length scales, but in the context of the present paper, the spectral power is displaced (upwards for sector B) and only for f < 0.03 and ζloc > − 0.6 due to the different σh/h. The − 2/3 power slope in the inertial subrange is obvious for both sectors and all instability conditions measured. A trend was also observed in which the spectral power increases in value as atmospheric stability decreases (Figure 4). The results of spectral analysis corroborate the lower constant values from the best-fit curves obtained for sector A (with lower σh/h) compared to sector B until beyond ζloc = − 0.6, where the difference becomes less apparent (see Figure 5). Under very unstable conditions, the turbulence in the urban canopy would also involve eddies created by ‘thermal plumes’ or ‘downdraughts’, which are more prevalent than wake contributions or σh/h, originating from heat sources and sinks on the urban surface. Al-Jiboori (2008) showed that when ζloc < − 0.7, the correlation co-efficient, rwθ, increased to 0.24, revealing the atmospheric condition when the transition from mechanically to convectively driven (free convection) turbulence occurs in the urban roughness sublayer, which is also seen in this study but in the layer directly above the urban canopy. However, it is important to note that at higher unstable conditions, the momentum and heat sources and sinks in the immediate vicinity have a stronger influence on its fluxes (Roth, 2000).

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Figure 4. Normalized spectra of w against normalized frequency (f = nz/u) for different stability conditions in sectors A and B. The vertical dashed lines represent f = 0.03 and the vertical solid lines f = 0.2; (a) A: 10/03/2011, 1100; B: 17/03/2011, 1100; (b) A: 01/04/2010, 0950; B: 13/04/2010, 1130; (c) A: 09/03/2010, 1000; B: 18/03/2010, 1000; (d) A: 17/03/2011, 1100; B: 15/03/2011, 1100; and (e) A: 15/03/2011, 1100; B: 18/03/2011, 1100. The listed measurement times are local times

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Figure 5. σw/u*loc plotted as a function of ζloc with error bars for sectors A and B with best-fit curves. The dashed lines represent ζloc = − 0.7

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A figure by Oikawa and Meng (1995) (fig. 6 in their paper) also displays this trend for different z values, where the spectral power decreases more rapidly when lower than f = 0.03 at z = 18 m than at z = 5.4 m. The authors noticed ‘dips’ and concluded that it was due to ‘organized motions’, more specifically, sweeps and ejection. Using quadrant analysis similar to that employed by Oikawa and Meng (1995), it was determined that ejection motion dominates for both sectors A and B (results not reported here). This figure also showed a systematic separation of spectral power directly after the dip. In the present study, the natural frequency of 0.03 Hz (where the dip for ζloc < − 0.6 and separation occur) corresponds to 33 s, which refers to large-scale motions associated with Reynolds stress production. Thus, the vertical heterogeneity (σh/h) of the different sectors A (less heterogeneous) and B affects the normalized σw by increasing its overall value at low atmospheric instability ζloc > − 0.20, just as it would at different measurement heights above the rooftop canopy due to dips and separation in the low-frequency range (large-scale motions).

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Figure 6. σu/u*loc distribution with − ζloc (−0.01 > ζloc > − 20) for sectors A and B

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3.4. Turbulence statistics of σu and σv with ζ in the RS

Similarity theory normally does not apply for u and v (even in the homogeneous surface layer), unlike w (please refer to Figures 6 and 7: the data are more scattered) (Rotach, 1993a; Moraes et al., 2007). In addition, the fact that turbulent intensity for u and w exceeded 0.5 and the low correlation between u and v, as mentioned in Section '3.2. Correlation between uv and w–θ', indicates that Taylor's frozen hypothesis was not applicable for this situation. Thus, interpreting the data in the framework of similarity theory would be even more unfounded. Nevertheless, the observations suggest that surface roughness and terrain complexity influence the turbulent fluxes and their relationship to the stability parameter for σw when σh/h is high, causing wake developments that affect the empirical constants by increasing their values when approaching near-neutral conditions from ζloc > − 0.6 onwards.

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Figure 7. σv/u*loc distribution with − ζloc (−0.01 > ζloc > − 20) for sectors A and B

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3.5. Scaling σθ using θ*loc in the RS

As stated in the Introduction, Equations (A.3) and (A.4) suggest that σθ* with − ζ will obey the − 1/3 power law under unstable or convective condition. This research supports this theory. The data were fitted using the form of the expression shown in Equation (A.4) (Tillman, 1972), where C1 and C2 are constants, based on the R2 value. It is important to note that extrapolation to near-neutral values is solely dependent on the best-fit equation obtained and not the measured data, thus creating the possibility that other equation forms could be more suitable for 0>ζ> − 0.01.

Figure 8(a) and (b) shows less scatter for all upwind sectors in very unstable regimes. This behaviour is expected because the heat fluxes recorded during the measurement period were high (>0; net upward flux) and constant. Scatter becomes evident when measurements are made in near-neutral conditions. The derived Equations ((3)) and ((4)) for each sector are given below:

  • equation image(3)
  • equation image(4)

(note: for − 0.01 > ζloc > − 100).

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Figure 8. The σθ*loc distribution with − ζloc(−0.01 > ζloc > − 20) and sectors using a log-log scale for better visualization of the 1/3 power trend (dashed line). The vertical dashed line denotes ζloc = − 0.3, and the vertical dot-dot line denotes ζloc = − 0.5. The solid lines are drawn based on Equations ((3))–(4) for sectors A (a) and B (b)

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The fits to the data are better than σw/u*loc in sectors A and B, with a power slope of − 0.33 (sector A), as predicted by theory. The relatively low correlation between w and θ (shown in Section '3.2. Correlation between uv and w–θ') contributed to the better fit for θ than for w, at least at this site. Equation (11) is only valid for ζloc < − 0.0212 because the computed values of σθ*loc for more than the former value would be mathematically indefinable.

The influence of z/h or σh/h (wakes) in different sectors does not seem to affect the scatter of data, which is consistent with what has been concluded by Rotach (1994) and Roth (2000) (C1 = − 3) and Yumao et al. (1997) (C1 = − 1.86). Hanna et al. (2007) found that the range − 2 to − 5 is common for street canyons in near-neutral conditions, this study addresses moderately to very unstable conditions. Interestingly, lower C1 values are usually obtained when the stability range measured falls within this category. This trend is also evident in the studies by Yumao et al. (1997) and Al-Jiboori et al. (2002), albeit a slightly higher value of − 2.23, although Quan and Hu (2009) obtained a value of − 1.5 despite both experimental setups being the same.

Equations ((3)) and ((4)) suggest that a simpler semi-empirical equation of the form σθ*loc = constant × (ζloc)−1/3 would be more appropriate for ζloc < − 0.5. This finding also presents further evidence that the temperature scale would be more useful in the RS than the more common velocity-based scale (Bruin and Bink, 1994) or the zd estimation technique suggested by Rotach (1994). Wakes (σh/h), which were shown to occur in sectors A and B (at relative magnitudes), affected σw/u*loc but not σθ*loc.

4. Conclusions

  1. Top of page
  2. Abstract
  3. 1. Similarity theory in the convective urban roughness sublayer
  4. 2. Site description
  5. 3. Results and discussion
  6. 4. Conclusions
  7. Acknowledgements
  8. Appendix
  9. References

Generally, similarity theory using local scaling is applicable in form (although not in values for w) in the near-roof vicinity of the urban canopy layer for σw/u*loc and σθ*loc, provided that the upwind surface characteristics are relatively homogeneous and depend on σh/h but not z/h for σw/u*loc and for ζloc > − 0.6 due to wakes. However, it should be noted that ejection (organized motion) is also prevalent in this locality. The constants obtained from best-fit curves are systematically lower in the latter range for the lower σh/h (1.04 vs 1.27) but converge when ζloc > − 0.6, caused by the increasing influence of heat flux on turbulence variance compared on surface momentum transfer. In addition, spectral analysis of the w component showed an overall lower spectral power in the low-frequency range for moderately unstable conditions (f < 0.03 and ζloc > − 0.6) even though the peak spectral power and f were the same. This behaviour is attributed to wakes created by obstacles that are different in the two sectors in moderately unstable to neutral conditions, where higher σh/h values result in higher wake production. The parameter σθ*loc, however, is independent of both z/h and σh/h. The constants in the derived semi-empirical equations were in relative accordance with other works done in more homogeneous locations, where the proportionality to the exponent one third and − 1/3 was observed for σw/u*loc and σθ*loc for ζloc < −0.3 and ζloc < − 0.5, respectively. A simpler semi-empirical equation of the form σθ*loc = constant × (ζloc)−1/3 would be more appropriate for ζloc < − 0.5.


  1. Top of page
  2. Abstract
  3. 1. Similarity theory in the convective urban roughness sublayer
  4. 2. Site description
  5. 3. Results and discussion
  6. 4. Conclusions
  7. Acknowledgements
  8. Appendix
  9. References

We thank Universiti Sains Malaysia (USM) for their infrastructure and financial contributions, including their short-term grant for the project titled ‘Turbulent fluxes and its relationship to stability atmospheric parameter in an industrialized equatorial area’ (Grant no.: 304/PTEKIND/639014). We also gratefully acknowledge the assistance of M. Roth when writing this paper.


  1. Top of page
  2. Abstract
  3. 1. Similarity theory in the convective urban roughness sublayer
  4. 2. Site description
  5. 3. Results and discussion
  6. 4. Conclusions
  7. Acknowledgements
  8. Appendix
  9. References

A list of symbols and equations used in this paper is provided below.

List of symbols
Bi, Ci

Constants, i = 1, 2.


Specific heat capacity, kJ/kg·K


Distance between influential buildings, m


Normalized frequency, f = nz/V


Gravitational acceleration, m s−2


Average building or obstacle height, m


Surface heat flux, kJ m−2


von Karman constant, = 0.4


Obukhov length, m


Local Obukhov length, m


Number of data points


Natural frequency, Hz

nSw(n)/equation image

Normalized w-spectral power


Record length, min

equation image

Co-efficient of determination, i = w, θ and j = sector

r, ruv

Correlation co-efficients of w–θ and uv, respectively


Longitudinal wind component, m s−1


Friction velocity, m s−1


Local friction velocity, m s−1


Mean wind speed, m s−1


Lateral wind component, m s−1


Vertical wind component, m s


Kinematic wu (subscript ‘0’ denotes surface) stress, m2 s−2


Kinematic wv (subscript ‘0’ denotes surface) stress, m2 s−2


Kinematic (subscript ‘0’ denotes surface) heat flux, m·K s−1


Height from the surface, m


Zero displacement height, m


Aerodynamic roughness length, m


Effective height, m


Standard deviation of building heights, m


Standard deviation of the longitudinal wind component fluctuations, m s−1


Standard deviation of the lateral wind component fluctuations, m s−1


Standard deviation of the vertical wind component fluctuations, m s−1


Standard deviation of the potential temperature fluctuations, K


Air density, kg m−3


Potential temperature, K


Friction temperature, K


Local friction temperature, K


Surface Reynolds stress, kg m s−2


Empirical non-dimensional function of ζ, i = w, θ.


Non-dimensional atmospheric stability parameter, = z/L


Local non-dimensional atmospheric stability parameter, = z/Lloc

List of equations

  • equation image(A.1)
  • equation image(A.2)
  • equation image(A.3)
  • equation image(A.4)
  • equation image(A.5)
  • equation image(A.6)
  • equation image(A.7)


  1. Top of page
  2. Abstract
  3. 1. Similarity theory in the convective urban roughness sublayer
  4. 2. Site description
  5. 3. Results and discussion
  6. 4. Conclusions
  7. Acknowledgements
  8. Appendix
  9. References
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