Observational Investigation of the Statistical Properties of Surface-Layer Turbulence in a Suburban Area of São Paulo, Brazil: Objective Analysis of Scaling-Parameter Accuracy and Uncertainties

Abstract

Statistical properties of turbulence, specifically variances of velocity components, temperature, water vapor, and carbon dioxide densities, are observationally characterized using turbulence measurements carried out between 2009 and 2017, at 25.4 m above surface in a suburban area in the metropolitan region of São Paulo (MRSP), Brazil. An objective analysis indicated that the best method to evaluate the zero-plane displacement (d), among five morphometric and anemometric methods is the temperature variance method. A new procedure based on the convergence of these methods is proposed to estimate the accuracy of the aerodynamic parameters. Normalized standard deviation of wind components and scalar properties are described by similarity functions based on Monin–Obukhov similarity theory. Uncertainties in d and Obukhov length are propagated to similarity functions derived for the MRSP and indicated uncertainties of up to 12% (22%) for wind components, 15% (23%) for temperature, 6% (23%) for water vapor and 11% (23%) for carbon dioxide densities during stable (unstable) conditions. By hypothesis testing it was demonstrated that the coefficients of normalized standard deviations of wind components for neutral stability conditions found at MRSP can be considered statistically equal to Roth’s urban averages (Q J R Meteorol Soc126:941–990, 2000), revealing the universal character of these functions. This agreement, added to other evidence, indicates that measurements used in the present study were performed in the inertial sublayer.

Appendices

Appendix A: Similarity Function Uncertainties

Uncertainties in the MOST function, $\phi$ = $\phi$(d, L), can be expressed by error propagation as:

$$ {\sigma }_{\phi }^{{2}} \approx \left( {\frac{\partial \phi }{{\partial {{d}}}}} \right)^{{2}} {\sigma }_{{{d}}}^{{2}} + \left( {\frac{\partial \phi }{{\partial {{L}}}}} \right)^{{2}} {\sigma }_{{{L}}}^{{2}} + {\text{2cov}}\left( {{d, L}} \right)\frac{\partial \phi }{{\partial {{d}}}}\frac{\partial \phi }{{\partial {{L}}}}, $$

(3)

where cov(d, L) is the covariance between d and L. σ_d and σ_L are the uncertainty in d and L, respectively.

Although the covariance term in Eq. 3 may be significant (Zilitinkevich et al. 2008), it is unclear how it can be estimated for each 30-min block. Assuming cov(d, L) ≈ 0 (Saleski and Chamecki 2012; Sun et al. 2015), the uncertainty of $\phi$ can be rewritten as:

$$ {\sigma }_{\phi }^{{2}} \approx \left( {\frac{\partial \phi }{{\partial {{d}}}}} \right)^{{2}} {\sigma }_{{{d}}}^{{2}} + \left( {\frac{\partial \phi }{{\partial {{L}}}}} \right)^{{2}} {\sigma }_{{{L}}}^{{2}} . $$

(4)

Hence, the relative global uncertainty in $\phi_{{1}} \left( {{\zeta }^{ ^{\prime}} } \right)$ = ${{A}}\left( {{1} + {{B}}\left| {{\zeta }^{ ^{\prime}} } \right|} \right)^{{{C}}}$ and $\phi_{{2}} \left( {{\zeta }^{ ^{\prime}} } \right)$ = ${{ D}} + {{E}}\left| {{\zeta }^{ ^{\prime}} } \right|^{{{F}}}$ is given respectively by:

$$ \left( {\frac{{{\sigma }_{{\phi_{{1}} }} }}{{\phi_{{1}} }}} \right)^{{2}} \approx \frac{{{{C}}^{{2}} }}{{\left( {{1} + \frac{{1}}{{{{B}}\left| {{\zeta }^{{ ^{\prime}}} } \right|}}} \right)^{{2}} }}\left[ {\left( {\frac{{{\sigma }_{{{d}}} }}{{{{z}} - {{d}}}}} \right)^{{2}} + \left( {\frac{{{\sigma }_{{{L}}} }}{{{L}}}} \right)^{{2}} } \right] $$

(5)

and

$$ \left( {\frac{{{\sigma }_{{\phi_{{2}} }} }}{{\phi_{{2}} }}} \right)^{{2}} \approx \frac{{{{F}}^{{2}} }}{{\left( {{1} + \frac{{{D}}}{{{{E}}\left| {{\zeta }^{{{ }_{{}}^{^{\prime}} }} } \right|^{{{F}}} }}} \right)^{{2}} }}\left[ {\left( {\frac{{{\sigma }_{{{d}}} }}{{{{z}} - {{d}}}}} \right)^{{2}} + \left( {\frac{{{\sigma }_{{{L}}} }}{{{L}}}} \right)^{{2}} } \right]. $$

(6)

The individual impact of σ_d (σ_L) on similarity functions can be assessed by solving Eq. 5 and Eq. 6 for σ_L = 0 (σ_d = 0).

Appendix B: Morphometric Methods

The simplest morphometric method for estimating d and z₀ is the Rule-of-thumb (Rt). Grimmond and Oke (1999) recommended ${{d}}/\overline{{H}}$ = 0.5, 0.6 and 0.7 for low-, medium- and high-density urban sites and ${{z}}_{0}/\overline{{H}}$ = 0.1. They verified reasonable values of d and z₀ for North American cities using ${{d}}/\overline{{H}}$ ~ 0.7 and ${{z}}_{0}/\overline{{H}}$ ~ 0.1. Kutzbach (1961), conducting a series of controlled experiments on a frozen lake varying density and distribution of roughness elements (bushel baskets) at the surface, found ${{z}}_{0}/\overline{{H}}$ ~ ${\lambda}_{{P}}^{1.13}$ and ${{d}}/\overline{{H}}$ ~ ${\lambda}_{{P}}^{0.29}$ that holds only for ${\lambda}_{{P}}$ ≤ 0.29. Counihan (1971), performing wind tunnel simulations for various roughness elements (Lego bricks) distributions, derived expressions ${{z}}_{0}/\overline{{H}}$ = 1.08 ${\lambda}_{{P}}$ – 0.08 and ${{d}}/\overline{{H}}$ = 1.4352 ${\lambda}_{{P}}$ – 0.0463, valid for 0.10 ≤ ${\lambda}_{{P}}$  ≤ 0.25 (see Grimmond and Oke 1999). Macdonald et al. (1998), using staggered arrays of cubes in the wind tunnel simulation, derived the expression: ${{d}}/\overline{{H}}$ = 1 + ${4.43}^{-{\lambda}_{{P}}}$(${\lambda}_{{P}}$ – 1) that is valid for 0.0 ≤ ${\lambda}_{{P}}$  ≤ 1.0.

Kanda et al. (2013) simulated numerically turbulent flows within and above buildings in Japanese cities with a large-eddy simulation and found out the maximum building height (H_max) is also a relevant length scale, that combined with $\overline{{H}}$ and σ_H, can accurately describe the behavior of d in urban surfaces with significant building heights variation. They proposed the following expression: ${{d}}/{{H}}_{{max}}$ = c ${{X}}^{ 2}$ + (a ${\lambda}_{{P}}^{{b}}$ − c)X, where X = $\left({\sigma}_{{H}}+\overline{{H}}\right)/{{H}}_{{max}}$ and the empirical constants are a = 1.29, b = 0.36, and c =  − 0.17. Kent et al. (2017a) verified that morphometric methods based on heterogeneous roughness-element heights, such as Kanda et al.’s (2013), yields d values that are systematically two times bigger than those obtained with Rt method. Taking into consideration this latter feature and based on the simplicity of Rt method, Kent et al. (2017a) proposed using 2Rt values to estimate d.

Kent et al. (2017b) included the aerodynamic effect of trees in the method proposed by Kanda, ${\text{Ka}}_{\text{b-v}}$, by considering an effective building-vegetation fraction λ_P(b-v) = λ_P + (1 − p)λ_P(v), where p is the porosity coefficient, λ_P and λ_P(v) are building and wooded fractions, and the mean and standard deviation of the building-vegetation heights (${\overline{{H}}}_{\left({\text{b-v}}\right)}$ and σ_H(b-v) respectively).

Appendix C: Anemometric Methods

Rotach (1994) developed an anemometric method (temperature variance method, TVM) to estimate d based entirely on in situ measurements of turbulence in the surface layer over urban areas. In this method, d is estimated iteratively, using as criterion the value of d that produces a minimum in the root-mean-square error (RMSE) between the observed and estimated values of ${\sigma}_{{T}}/{{T}}_{*}$. In the TVM the estimated values are provided by an expression of ${\sigma}_{{T}}/{{T}}_{*}$ that obeys the MOST and is valid for rural areas under convective conditions. Unfortunately, TVM is valid only under thermal homogeneity condition, limiting further its application in urban areas. Besides, determining d from temperature variance casts some doubt about whether this estimate is genuinely representative of dynamic response of the turbulent flow to roughness elements at the surface (Grimmond et al. 1998).

The wind variance method (WVM) is analogue to TVM. It evaluates d using the RMSE between observed and estimated values of ${\sigma}_{{w}}/{{u}}_{*}$. In the WVM the estimated values are provided by an expression of ${\sigma}_{{w}}/{{u}}_{*}$ that obeys the MOST and is valid for rural areas under convective conditions. Toda and Sugita (2003) recommend comparing the d values obtained simultaneously by WVM and TVM to enhance the reliability of the result and to assess the accuracy of the results.

Toda and Sugita (2003) proposed an anemometric method, named friction velocity method (FVM), to determine z₀ using measurements of turbulence performed at a single z level under unstable conditions. In this method z₀ is estimated iteratively by using as criteria the RMSE between the friction velocity observed, ${{u}}_{*}$ = $\sqrt{{\overline{{{u} }^{^{\prime}}{{w}}^{^{\prime}}}}_{0}}$, and estimated by MOST expression for mean wind speed at z level, $\overline{{U}}$ = ${{u}}_{*}/\kappa\left\{{{\text{ln}}}\left[\left({{z}}-{{d}}\right)/{{z}}_{0}\right]+{\psi}_{{m}}\left({\zeta}^{ {{\prime}}}\right)\right\}$, where κ is the von Kármán constant and ψ_m is the stability correction function for momentum. The interactions are performed varying z₀ and fixing d derived previously from TVM. The best estimate of z₀ corresponds to the minimum RMSE.

Appendix D: Uncertainty in the Geometric Properties of Buildings

The building plan area fraction is defined as ${\lambda}_{{P}}$ = ${{A}}_{{P}}/{{A}}_{{T}}$, where A_P is the total building plan area and A_T is the total surface area (here assumed to be an exact number). Therefore, it follows by error propagation that relative uncertainty in λ_P is:

$$\frac{{\sigma}_{{\lambda}_{{P}}}}{{\lambda}_{{P}}}=\frac{{\sigma}_{{{A}}_{{P}}}}{{{A}}_{{P}}},$$

(7)

where ${\sigma}_{{{A}}_{{P}}}$ is the uncertainty in A_P. In turn, the total building plan area is defined as ${{A}}_{{P}}$ = $\sum_{{{i}}={1}}^{{n}}{{{A}}}_{{{P}}\left({{i}}\right)}$ for a number n of individual plan areas of buildings, A_P(i), independent. Hence, using error propagation:

$${\sigma}_{{{A}}_{{P}}}^{2}=\sum_{{{i}}={1}}^{{n}}{\sigma}_{{{A}}_{{{P}}\left({{i}}\right)}}^{2},$$

(8)

where ${\sigma}_{{{A}}_{{{P}}\left({{i}}\right)}}$ is the A_P(i) uncertainty.

By considering the building plan area A_P(i) as a square ${\left({\mathbf{P}}_{\mathbf{1}}{{\mathbf{P}}}_{\mathbf{2}}{{\mathbf{P}}}_{\mathbf{3}}{{\mathbf{P}}}_{\mathbf{4}}\right)}_{\left({{i}}\right)}$, ${\sigma}_{{{A}}_{{{P}}\left({{i}}\right)}}$ can be evaluated through the uncertainty in planimetric coordinates ${{x}}_{{{j}}\left({{i}}\right)}$ and ${{y}}_{{{j}}\left({{i}}\right)}$, assumed constant in this analysis and indicated by σ_xy, of each corner ${{\mathbf{P}}}_{{{j}}\left({{i}}\right)}$ of the square (for j = 1, …, 4). Note that ${{A}}_{{{P}}\left({{i}}\right)}$ = $\left|\left({{x}}_{{2}\left({{i}}\right)}-{{x}}_{{1}\left({{i}}\right)}\right)\left({{y}}_{{4}\left({{i}}\right)}-{{y}}_{{1}\left({{i}}\right)}\right)-\left({{y}}_{{2}\left({{i}}\right)}-{{y}}_{{1}\left({{i}}\right)}\right)\left({{x}}_{{4}\left({{i}}\right)}-{{x}}_{{1}\left({{i}}\right)}\right)\right|$ by the cross product between vectors ${\mathbf{P}}_{\mathbf{2}\left({{i}}\right)}$ − ${\mathbf{P}}_{\mathbf{1}\left({{i}}\right)}$ and ${\mathbf{P}}_{\mathbf{4}\left({{i}}\right)}$ − ${\mathbf{P}}_{\mathbf{1}\left({{i}}\right)}$, with constraint ${\left|{\mathbf{P}}_{\mathbf{2}\left({{i}}\right)}-{\mathbf{P}}_{\mathbf{1}\left({{i}}\right)}\right|}^{2}$ = ${\left|{\mathbf{P}}_{\mathbf{4}\left({{i}}\right)}-{\mathbf{P}}_{\mathbf{1}\left({{i}}\right)}\right|}^{2}$. Therefore, the uncertainty in A_P(i) is given by:

$$\sigma _{{A_{{P\left( i \right)}} }}^{2} \approx 4A_{{P\left( i \right)}} \sigma _{{xy}}^{2},$$

(9)

for ${{x}}_{{1}\left({{i}}\right)}$, ${{x}}_{{2}\left({{i}}\right)}$, ${{x}}_{{4}\left({{i}}\right)}$, ${{y}}_{{1}\left({{i}}\right)}$, ${{y}}_{{2}\left({{i}}\right)}$ and ${{y}}_{{4}\left({{i}}\right)}$ statistically independent.

According to Brazilian legislation (Oliveira and Paradella 2009), the Brazilian Map Accuracy Standards requires at least 90% of the points ${\mathbf{P}}_{{{j}}\left({{i}}\right)}$ (where i = 1, …, n and j = 1, …, 4) with uncertainty in planimetric coordinates less than 1 m at 1:1000 scale used in the plan area restitution of buildings (GeoSampa 2021). Hence, it is plausible to assume σ_xy = 0.5 m and, by substituting Eq. 9 in Eq. 8, the uncertainty in A_P can be evaluated as ${\sigma}_{{{A}}_{{P}}}$ ≈ $\sqrt{{{A}}_{{P}}}$. Applying this result in Eq. 7, a rough value of the relative uncertainty in λ_P can be evaluated as:

$$\frac{{\sigma}_{{\lambda}_{{P}}}}{{\lambda}_{{P}}}\approx \frac{\sqrt{{{A}}_{{P}}}}{{{A}}_{{P}}}.$$

(10)

Appendix E: Uncertainty of the Anemometric Method Results

Let r² be the squared errors between observed and estimated values of a set of n measurements. Then, RMSE² = $\overline{{{r}}^{2}}$ and the best estimate of uncertainty in the mean squared error, $\overline{{{r}}^{2}}$, will be the standard error SE = $\sqrt{{{\text{var}}}\left({{r}}^{2}\right)/{{n}}}$, whose term var(${{r}}^{2}$) ≡ variance of r². Therefore, by error propagation, the uncertainty in the RMSE is:

$${\sigma}_{{RMSE}}=\frac{1}{{2}{{RMSE}}}\sqrt{\frac{{{\text{var}}}\left({{r}}^{2}\right)}{{n}}.}$$

(11)

On the other hand, since r depends on the fitted function, $\phi$, the uncertainty of r² is given by ${\sigma}_{{{r}}^{2}}$ = ($\partial{{r}}^{2}/\partial\phi$)${\sigma}_{\phi }$ = 2r ${\sigma}_{\phi }$. By propagating the uncertainty in r² to RMSE, the uncertainty in RMSE can be expressed by:

$${\sigma}_{{RMSE}}^{2}\approx \frac{1}{{{n}}^{2}{{{RMSE}}}^{2}}\sum_{{{i}}={1}}^{{n}}{{{r}}}_{{i}}^{2}{{\sigma}_{\phi }^{2}}_{{i}}.$$

(12)

If $\phi$ = $\phi$(x₁, …, x_m) and ${{\sigma}_{{x}}}_{{j}}$ is the uncertainty in x_j, given that all variables x_j are statistically independent, then:

$${{\sigma}_{\phi }^{2}}_{{i}}\approx \sum_{{{j}}={1}}^{{m}}{\left(\frac{\partial{\phi }_{{i}}}{\partial{{x}}_{{j}}}\right)}^{2}{{\sigma}_{{x}}^{2}}_{{j}}$$

(13)

and Eq. 12 can be rewritten as

$${\sigma}_{{RMSE}}^{2}\approx \frac{1}{{{n}}^{2}{{{RMSE}}}^{2}}\sum_{{{j}}={1}}^{{m}}\sum_{{{i}}={1}}^{{n}}{{{r}}}_{{i}}^{2}{\left(\frac{\partial{\phi }_{{i}}}{\partial{{x}}_{{j}}}\right)}^{2}{{\sigma}_{{x}}^{2}}_{{j}}.$$

(14)

Therefore, by substituting Eq. 14 in Eq. 11, the uncertainty of an arbitrary variable x_k can be evaluated by:

$${{\sigma}_{{x}}}_{{k}}\approx \sqrt{\frac{{{n}} \, {\text{var}}\left({{r}}^{2}\right)}{4}-\sum_{{{j}}={1}}^{{{m}}-{1}}\sum_{{{i}}={1}}^{{n}}{{{r}}}_{{i}}^{2}{\left(\frac{\partial{\phi }_{{i}}}{\partial{{x}}_{{j}}}\right)}^{2}{{\sigma}_{{x}}^{2}}_{{j}}} \Bigg /\sqrt{\sum_{{{i}}={1}}^{{n}}{{{r}}}_{{i}}^{2}{\left(\frac{\partial{\phi }_{{i}}}{\partial{{x}}_{{k}}}\right)}^{2}},$$

(15)

when the uncertainties of the m − 1 other different variables of x_k are known.

Hence, the uncertainty of d from anemometric methods TVM and WVM, as well as the uncertainty of z₀ from FVM, evaluated in Sect. 3 is:

$${{\sigma}_{{d}}}_{\left({\text{TVM}}\right)}\approx {0.5}\sqrt{{n \text{var}}\left({{r}}^{2}\right)} \Bigg /\sqrt{\sum_{{{i}}={1}}^{{n}}{{{r}}}_{{i}}^{2}{\left[\frac{0.32}{{{z}}-{{d}}}{\left(-{\zeta}_{{i}}^{{{{\prime}}}}\right)}^{-{1}/{3}}\right]}^{2}},$$

(16)

$${{\sigma}_{{d}}}_{\left({\text{WVM}}\right)}\approx {0.5}\sqrt{{n \text{var}}\left({{r}}^{2}\right)} \Bigg /\sqrt{\sum_{{{i}}={1}}^{{n}}{{{r}}}_{{i}}^{2}{\left[\frac{1.25}{{{z}}-{{d}}}{\left({1}-{{B}}_{{w}}{\zeta}_{{i}}^{ {}^{{{\prime}}}}\right)}^{-{2}/{3}}{\zeta}_{{i}}^{^{{{\prime}}}}\right]}^{2}},$$

(17)

$$\sigma _{{z_{0} \left( \text{FVM} \right)}} \approx \frac{{\sqrt {\frac{{n \, \text{var} \left( {r^{2} } \right)}}{4} - \mathop \sum \nolimits_{{i = 1}}^{n} r_{i}^{2} \left\{ {\frac{{\left( {u_{*}^{{est}} } \right)_{i}^{2} }}{{\kappa \bar{U}_{i} }}\left[ {\frac{1}{{z - d}} + \frac{{\partial \psi _{m} \left( {\zeta _{i}^{\prime } ,\,0} \right)}}{{\partial d}}} \right]} \right\}^{2} \sigma _{d}^{2} } }}{{\sqrt {\mathop \sum \nolimits_{{i = 1}}^{n} r_{i}^{2} \left\{ {\frac{{\left( {u_{*}^{{est}} } \right)_{i}^{2} }}{{\kappa \bar{U}_{i} }}\left[ {\frac{1}{{z_{0} }} - \frac{{\partial \psi _{m} \left( {\zeta _{0} ,\,0} \right)}}{{\partial z_{0} }}} \right]} \right\}^{2} } }},$$

(18)

where κ is the von Kármán constant (0.40), $\overline{{U}}$ the wind speed measured at z level, and ${{u}}_{*}^{{est}}$ the estimated value of friction velocity by MOST.