Parameterizing the aerodynamic effect of trees in street canyons for the street network model MUNICH using the CFD model Code_Saturne

Trees provide many ecosystem services in cities such as urban heat island reduction, water runoff limitation, and carbon storage. However, the presence of trees in street canyons reduces the wind velocity in the street and limits pollutant dispersion. Thus, to obtain accurate simulations of pollutant concentrations, the aerodynamic effect of trees should be taken into account in air quality models at the street level.

The Model of Urban Network of Intersecting Canyons and Highways (MUNICH) simulates the pollutant concentrations in a street network, considering dispersion and physico-chemical processes. It can be coupled to a regional-scale chemical transport model to simulate air quality over districts or cities. The aerodynamic effect of the tree crown is parameterized here through its impact on the average wind velocity in the street direction and the vertical transfer coefficient associated with the dispersion of a tracer. The parameterization is built using local-scale simulations performed with the computational fluid dynamics (CFDs) code Code_Saturne. The two-dimensional CFD simulations in an infinite street canyon are used to quantify the effect of trees, depending on the tree characteristics (leaf area index, crown volume fraction, and tree height to street height ratio) using a drag porosity approach. The tree crown slows down the flow and produces turbulent kinetic energy in the street, thus impacting the tracer dispersion. This effect increases with the leaf area index and the crown volume fraction of the trees, and the average horizontal velocity in the street is reduced by up to 68 %, while the vertical transfer coefficient by up to 23 % in the simulations performed here.

A parameterization of these effects on horizontal and vertical transfers for the street model MUNICH is proposed. Existing parameterizations in MUNICH are modified based on Code_Saturne simulations to account for both building and tree effects on vertical and horizontal transfers. The parameterization is built to obtain similar tree effects (quantified by a relative deviation between the cases without and with trees) between Code_Saturne and MUNICH. The vertical wind profile and mixing length depend on leaf area index, crown radius, and tree height to street height ratio. The interaction between the trees and the street aspect ratio is also considered.

1 Introduction

Cities are, by definition, areas with high densities of people, infrastructure, and activities, and this urbanization causes many issues. First, air quality is poor because of the numerous air pollutants emitted by anthropic activities such as traffic, industries, or residential activities, and the reduction in air flow by high buildings limits the dispersion of these pollutants (Faiz, 1993; Akimoto, 2003; Yuan et al., 2014; Zhang et al., 2020). In addition to air pollution issues, radiative and water budgets are strongly modified in cities compared to the countryside (Bozonnet et al., 2015). Temperatures are, on average, higher than in the countryside because of the urban heat island created by additional anthropogenic energy released, storage of radiative energy in dark materials, radiation multi-reflection, and lack of vegetation and associated evapotranspiration (Oke, 1982; Pigeon et al., 2007; Stewart, 2011; Hebbert and Jankovic, 2013). Impervious soils also decrease water infiltration and intensify runoff (Leopold, 1968). In addition, growing urbanization and increasing extreme events (due to climate change) such as pollution peaks, heat waves, and floods have negative consequences on the environment and human health (Robine et al., 2007; Angel et al., 2011; Pascal et al., 2013; West et al., 2016; IPCC, 2021).

One nature-based solution is to green the city by planting vegetation as lawns, planting trees in streets or in parks, and growing green walls and roofs (Livesley et al., 2016; Revelli and Porporato, 2018). Vegetation and especially trees contribute to improve human thermal comfort by creating a favorable micro-climate with lower air temperature (through solar radiation interception and creation of shade) and higher evaporation (through ground and vegetation evapotranspiration; Taha et al., 1991; Bowler et al., 2010; Gillner et al., 2015; Klemm et al., 2015; Lobaccaro and Acero, 2015; Gunawardena et al., 2017). This positive effect of trees is significant in particular during heat wave episodes, which will be more frequent in the future due to climate change (IPCC, 2021). Trees and vegetated areas also favor infiltration in soils that contributes to offset water runoff induced by soil artificialization (Armson et al., 2013; Berland et al., 2017). Besides, vegetation is known to store carbon (Nowak and Crane, 2002; Svirejeva-Hopkins et al., 2004) and to enhance human well-being (van Dillen et al., 2012; Bertram and Rehdanz, 2015; Krekel et al., 2015). For all these ecosystem services in urban areas, city greening is often promoted and, for example, the city of Paris has about 205 000 trees of which 52 % are roadside trees (Direction des Espaces Verts et de l'Environnement – Mairie de Paris, 2021).

Many studies have tried to figure out the impact of trees on air pollution in a street canyon, and they have shown that trees are an important parameter to take into account if we want to understand and accurately simulate the pollutant concentrations in the streets (Beckett et al., 1998; Nowak et al., 1998; Jayasooriya et al., 2017). Vegetation, and especially trees, represent surfaces available for pollutant dry deposition and, hence, can contribute to reducing air pollutant concentrations. However, pollutant removal and its impact on air quality vary greatly, depending on tree characteristics, tree species, and pollutant type (Nowak et al., 2006; Hwang et al., 2011; Selmi et al., 2016; Xue and Li, 2017; Ozdemir, 2019). Trees may also affect atmospheric chemistry by emitting biogenic volatile organic compounds (BVOCs), which may lead to the formation of ozone and secondary organic aerosols (Calfapietra et al., 2013; Préndez et al., 2019; Gu et al., 2021). Furthermore, trees may alter air quality by influencing aerodynamic processes and limiting the pollutant dispersion (Buccolieri et al., 2011; Wania et al., 2012; Vos et al., 2013; Gromke and Ruck, 2007, 2009, 2012; Gromke and Blocken, 2015). The aerodynamic effect of trees is defined as the drag force resulting from the friction between the air and the leaves. Since the tree crown can be seen as a porous medium as air passes through it but is slowed down, and the drag force increases with the leaf surface.

For wind perpendicular to a street, the air recirculates inside the street canyon (Harman et al., 2004). Pollutants emitted at the bottom of the street (by traffic) accumulate on the leeward side of the street, inducing higher local concentrations (Vardoulakis et al., 2003; Cai et al., 2008; Huang et al., 2019). Obstacles in the street, such as trees, can strongly impact the air flow and intensify the pollutant accumulation (Vos et al., 2013). This effect has been studied using computational fluid dynamics (CFD) models where the aerodynamic influence of trees on the flow is represented by a porosity model (Buccolieri et al., 2009; Zaïdi et al., 2013; Wei et al., 2016; Jeanjean et al., 2017; Santiago et al., 2017). Vos et al. (2013) showed that pollutant concentrations may increase by 20 % in a street because of the presence of two rows of trees in the street. The effects of vegetation depend on the height, width, and density (leaf area index) of trees (Vos et al., 2013; Janhäll, 2015), as well as the height to width ratio of streets (Wania et al., 2012). Finally, it is necessary to accurately assess the effect of trees on pollutant dispersion in order to find what configurations are more effective in reducing air pollution in street canyon and to guide urban development policy (Janhäll, 2015).

The effect of trees on aerodynamic processes should also be considered at the street level in air quality models. As discussed previously, CFD models including trees are used to study wind fields and pollutant transport in street canyons (Li et al., 2006). However, as the street is discretized with a fine mesh, the computational cost is high, and simulations at the city scale are too expensive today. Fast-running codes, such as simplified street network or street-in-grid models, are developed to simulate street pollutant concentrations over neighborhoods or cities, but they do not take into account the effect of trees in the streets. The objective of this study is to parameterize the effect of trees on air flow in the Model of Urban Network of Intersecting Canyons and Highways (MUNICH; Kim et al., 2018; Lugon et al., 2020, http://cerea.enpc.fr/munich/, last access: 17 December 2021). To build this parameterization, simulations in street canyons are performed with Code_Saturne (Archambeau et al., 2004, https://www.code-saturne.org/, last access: 17 December 2021), a CFD code, which can represent the tree aerodynamic effect with a drag porosity approach (Katul et al., 2004) and has previously been compared with field measurements (Zaïdi et al., 2013). MUNICH parameterizations have already been compared with Code_Saturne results in a treeless canyon, and a new parameterization for horizontal and vertical transfers has been developed in MUNICH based on Wang (2012, 2014) and Code_Saturne simulations (Maison et al., 2022). In the present study, the tree aerodynamic effect is added to this parameterization, and Code_Saturne (version 6.0) is used as a reference to parameterize the aerodynamic effect of trees in the street network model MUNICH. CFD simulations are performed in three streets of aspect ratios varying from 0.3 to 1.0, and a large range of tree leaf area index, crown radius, and heights is tested. The tree aerodynamic effect quantified with Code_Saturne is analyzed depending on these street and tree characteristics.

The approach to modeling the dispersion of pollutants in MUNICH and Code_Saturne is fundamentally different due to their physical modeling and discretization in space and time. MUNICH is a street network model that simulates air pollutant concentration in an urban canopy. Street dimensions and pollutant concentrations are assumed to be homogeneous in each street segment. Air flow is divided into a horizontal flux from one street to another and a vertical flux between the street and the background (Maison et al., 2022). Background concentrations above the street can be computed by 3D chemistry transport models (CTMs), such as Polair3D (Sartelet et al., 2018; Lugon et al., 2020, 2021). To build the tree parameterization, the CFD simulation setup is adapted for the comparison with MUNICH, and several simulations are performed with Code_Saturne, considering a range of street and tree characteristics.

The structure of the paper is as follows. The MUNICH and Code_Saturne models are presented in Sect. 2. Then, the tree effect on horizontal wind speed and vertical transfer coefficient is quantified with Code_Saturne simulations in Sect. 3 and parameterized in MUNICH in Sect. 4. Conclusions are presented in Sect. 5.

2 Materials and methods

2.1 Description of MUNICH

In MUNICH, each street segment is assumed to be homogeneous, i.e., with uniform building height H and street width W and of length L (m). The street is characterized by its height-to-width ratio called the aspect ratio, $a_{\mathrm{r}}=H/W$ (–). Only the average pollutant concentrations over the street are considered. Pollutants are transported by the horizontal wind speed (advection) in the street network and by a vertical transfer coefficient between the streets and the background. Several parameterizations of the horizontal wind speed and of the vertical transfer coefficient exist in MUNICH. The ones recently developed in Maison et al. (2022) and based on Code_Saturne simulations are used and detailed here. The vertical profile of the wind speed in the street direction is calculated as an attenuation of the wind speed in the street direction and at the roof level, U_H,φ (m s⁻¹), as follows (Maison et al., 2022):

$$\begin{array}{}\text{(1)}& \begin{array}{rl}U\left(z\right)& =U_{H,\mathit{\phi }}\left[C_{\mathrm{1}}{I}_{\mathrm{0}}\left(g\right(z\left)\right)+C_{\mathrm{2}}{K}_{\mathrm{0}}\left(g\right(z\left)\right)\right]\\ & \text{with}{U}_{H,\mathit{\phi }}=U_H\left|\cos \left(\mathit{\phi }\right)\right|,\end{array}\end{array}$$

where C₁ and C₂ are integration coefficients, and I₀ and K₀ are the first and second type modified Bessel functions of order 0. Besides, the wind speed at the roof level U_H has to be multiplied by $\left|\cos \right(\mathit{\phi }\left)\right|$ to select the component of the wind speed in the street direction, where φ is the angle between the wind direction and the street orientation (^∘). This vertical wind profile is then integrated between the soil roughness $z_{{\mathrm{0}}_{\mathrm{s}}}$ (m) and H to compute the average horizontal wind speed in the street direction. The function g is calculated as follows (Wang, 2012, 2014):

$$\begin{array}{}\text{(2)}& g\left(z\right)=\mathrm{2}\sqrt{\mathit{\alpha }{\displaystyle \frac{z}{H}}},\end{array}$$

and

$$\begin{array}{}\text{(3)}& \begin{array}{rl}{C}_{\mathrm{1}}& ={\displaystyle \frac{\mathrm{1}}{I_{\mathrm{0}}\left(g\right(H\left)\right)-I_{\mathrm{0}}\left(g\right(z_{\mathrm{0}s}\left)\right)K_{\mathrm{0}}\left(g\right(H\left)\right)/K_{\mathrm{0}}\left(g\right(z_{\mathrm{0}s}\left)\right)}}\\ & \text{and}{C}_{\mathrm{2}}=-{\displaystyle \frac{C_{\mathrm{1}}{I}_{\mathrm{0}}\left(g\right(z_{\mathrm{0}s}\left)\right)}{K_{\mathrm{0}}\left(g\right(z_{\mathrm{0}s}\left)\right)}},\end{array}\end{array}$$

where α is a dimensionless coefficient expressing the effects of wind angle on wind attenuation in the street. It is computed as follows:

$$\begin{array}{}\text{(4)}& {\displaystyle }\mathit{\alpha }={\displaystyle \frac{C_{\mathrm{B}}{a}_{\mathrm{r}}}{\mathit{\kappa }{s}_H}}\text{with}{C}_{\mathrm{B}}=\mathrm{0.31}\left[\mathrm{1}-\exp (-\mathrm{1.6}{a}_{\mathrm{r}})\right]f_{\mathit{\phi }}\text{(5)}& {\displaystyle }\mathrm{\text{and}}{f}_{\mathit{\phi }}=\left\{\begin{array}{ll}\left|\cos \right(\mathrm{2}\mathit{\phi }){|}^{\mathrm{3}}& \text{if}\mathit{\phi }\in [\mathrm{0},\mathrm{45}{}^{\circ }]\cup [\mathrm{135},\mathrm{225}{}^{\circ }]\\ & \cup [\mathrm{315},\mathrm{360}{}^{\circ }]\\ \mathrm{0}& \text{if}\mathit{\phi }\in ]\mathrm{45},\mathrm{135}{}^{\circ }[\cup ]\mathrm{225},\mathrm{315}{}^{\circ }[,\end{array}\right.\end{array}$$

where $s_H=s\left(z=H\right)$ is a dimensionless factor describing the effect of canopy on the mixing length l_m (Wang, 2012, 2014). The mixing length is calculated as follows:

$$\begin{array}{}\text{(6)}& {\displaystyle \frac{\mathrm{1}}{l_{\mathrm{m}}}}={\displaystyle \frac{\mathrm{1}}{\mathit{\kappa }z}}+{\displaystyle \frac{\mathrm{1}}{l_{c_{\mathrm{b}}}}}\Rightarrow l_{\mathrm{m}}=\mathit{\kappa }z{\displaystyle \frac{l_{c_{\mathrm{b}}}}{l_{c_{\mathrm{b}}}+\mathit{\kappa }z}}=\mathit{\kappa }zs\left(z\right),\end{array}$$

where κz corresponds to the mixing length over a rough bare soil (without canopy), and $l_{c_{\mathrm{b}}}$ is the characteristic length (m) in the street canyon, corresponding to the mixing length of the urban canopy alone ($l_{c_{\mathrm{b}}}=\mathrm{0.5}$ W).

The vertical transfer coefficient that drives pollutant exchange between the street and the background zone is calculated at z=H as follows:

$$\begin{array}{}\text{(7)}& q_{\mathrm{vert}}={\mathit{\sigma }}_W{l}_{\mathrm{m}}={\mathit{\sigma }}_W\mathit{\kappa }H{s}_H,\end{array}$$

where σ_W (in m s⁻¹) is a velocity scale equal to the standard deviation of the vertical wind speed. It depends on both the friction velocity above the urban canopy, u_*, and on atmospheric stability (Soulhac et al., 2011).

In Sect. 4, the tree effect is parameterized by taking into account the characteristic length of the trees in the mixing length l_m (Eq. 19, which leads to modifications in the coefficient α Eq. 23).

2.2 Description of Code_Saturne

2.2.1 Street and tree modeling setup

In the present study, the tree effect is studied in three street canyons of the following different street aspect ratios: a wide street canyon (WC), an intermediate canyon (IC), and a narrow canyon (NC). Their characteristics are presented in Table 1.

Table 1 Characteristics of the three canyons studied.

A color code is introduced to simplify the reading of the figures, with red symbols for the wide canyon (WC), purple ones for the intermediate canyon (IC), and blue ones for the narrow canyon (NC). Abbreviations, parameters, and variables used are listed in Appendix A.

The k−ε linear production turbulence model is used in Code_Saturne. Stationary simulations are performed with a thermally neutral atmosphere. A 2D infinite street canyon is modeled with periodic condition on the y axis, but the flow and the wind speed vector are 3D. The mesh is composed of hexahedral cells of 1 m in the y-axis direction and 0.5 m on the x and z axes. The vertical profiles of U, k, and ε are set in the inlet (top left border of the domain). The complete description of Code_Saturne simulation setup in treeless canyons can be found in Maison et al. (2022).

The tree geometry and its representation in Code_Saturne are shown in Fig. 1. Here, two rows of trees (n=2) are considered, with one on each side of the street, and whose positions on the x axis are ​​​​​​​x=34.5 m and x=48.0 m (the wall positions are x=27.5 and x=55.0 m). The tree crown centers are located in a position on the x axis so that the largest crowns (CVF≈25 %) do not reach the street walls for the three street canyons studied. r is the tree radius (m), h, h_min, and h_max correspond, respectively, to the middle, minimum, and maximum heights of the tree crown (m). Note that all the simulations verify h_max≤H, i.e., the top of the trees do not exceed the top of the street.

Figure 1 The 3D scheme of the street canyon with cylindrical tree crown dimensions.

In the cells containing the trees, an additional drag term is added to the Navier–Stokes equations as follows (Katul et al., 2004; Zaïdi et al., 2013):

$$\begin{array}{}\text{(8)}& S_{u,i}=-\mathit{\rho }\mathrm{LAD}{C}_{D_{\mathrm{t}}}\mathrm{|}U\mathrm{|}{\mathit{U}}_i,\end{array}$$

where |U| is the modulus of U, U_i is the velocity in the i-direction, LAD is the leaf area density in square meters of leaf surface per cubic meter of crown volume (${\mathrm{m}}_{\mathrm{leaves}}^{\mathrm{2}}{\mathrm{m}}_{\mathrm{tree}\mathrm{crown}}^{-\mathrm{3}}$), ρ is the air density, and $C_{D_{\mathrm{t}}}$ is the tree drag coefficient set to 0.2, which is a representative value for trees (Katul and Albertson, 1998). The leaf area index (LAI) is the one-sided green leaf area per unit ground surface area, and the leaf area density (LAD) is the one-sided green leaf area per unit volume. They are calculated as follows:

$$\begin{array}{}\text{(9)}& \begin{array}{rl}\mathrm{LAI}& ={\displaystyle \frac{\text{surface of leaves}}{\text{soil projected surface area}}}\left({\mathrm{m}}_{\mathrm{leaf}}^{\mathrm{2}}{\mathrm{m}}_{\mathrm{soil}}^{-\mathrm{2}}\right)\\ & \text{and}\mathrm{LAD}={\displaystyle \frac{\text{surface of leaves}}{\text{volume of tree crown}}}\left({\mathrm{m}}_{\mathrm{leaf}}^{\mathrm{2}}{\mathrm{m}}_{\mathrm{crown}}^{-\mathrm{3}}\right).\end{array}\end{array}$$

The source terms for trees are also implemented in the k−ε equations as follows:

$$\begin{array}{}\text{(10)}& {\displaystyle }{S}_k=\mathit{\rho }\mathrm{LAD}{C}_{D_{\mathrm{t}}}\left({\mathit{\beta }}_p\mathrm{|}U{\mathrm{|}}^{\mathrm{3}}-{\mathit{\beta }}_d\mathrm{|}U\mathrm{|}k\right)\text{(11)}& {\displaystyle }{S}_{\mathit{\epsilon }}=\mathit{\rho }\mathrm{LAD}{C}_{D_{\mathrm{t}}}\left(C_{\mathrm{4}\mathit{\epsilon }}{\mathit{\beta }}_p\mathrm{|}U{\mathrm{|}}^{\mathrm{3}}{\displaystyle \frac{\mathit{\epsilon }}{k}}-C_{\mathrm{5}\mathit{\epsilon }}{\mathit{\beta }}_d\mathrm{|}U\mathrm{|}\mathit{\epsilon }\right),\end{array}$$

where $C_{\mathrm{4}\mathit{\epsilon }}=C_{\mathrm{5}\mathit{\epsilon }}=\mathrm{0.9}$, β_p=1.0, and β_d=5.03 are constants of the model (Zaïdi et al., 2013). Note that this type of tree aerodynamic effect modeling is commonly used and evaluated by comparison with experimental results (Buccolieri et al., 2018).

Only the impact of the tree leaves is considered, and the impact of the tree trunk and branches on the flow is not modeled. As the streets are modeled in 2D and an infinite length in the y direction is assumed (see Sect. 2), the LAI considered here is the equivalent cylindrical LAI_2D. Usually, studies consider the tree LAI, which is well defined for a 3D tree. The relation between the tree LAI_3D and the equivalent cylindrical LAI_2D is given in Appendix B. To check that this effect of trees is well modeled using this simplified 2D setup, 3D simulations were performed and compared to some of the 2D simulations presented here, showing very good agreement between the 2D and 3D simulations in terms of the influence of the trees on the flow (Appendix B).

To gain a complete vision of the effect of trees, 45 simulations are performed per canyon, and the impact of three independent tree parameters are studied, including the following:

• The 2D-equivalent LAI, i.e., LAI_2D=0.5, 1, 2, 3, and 4 (see Appendix B, for the conversion between LAI_3D and equivalent LAI_2D).

• The crown volume fraction (CVF), calculated as the tree (2D cylindrical) crown volume divided by the street volume, i.e., $\mathrm{CVF}=\frac{n\mathit{\pi }{r}^{\mathrm{2}}}{HW}$. This ratio is already used in studies quantifying the effect of trees on street pollution such as Gromke and Blocken (2015). The relation between 2D and 3D CVF is shown in Appendix B. Note that, apart from Appendix B, all CVF mentioned in the figures and in the text refer to the 2D CVF. In total, the following three CVF ranges are simulated: CVF≈5, 10, and 25 %.

• The tree-to-street height ratio calculated as the crown middle height divided by the building height ($h/H$). In total, the following three $h/H$ ranges are simulated: $h/H\approx \mathrm{1}/\mathrm{3}$, $\mathrm{1}/\mathrm{2}$, and $\mathrm{2}/\mathrm{3}$.

Note that these three tree characteristics are normalized by street characteristics, and similar values are chosen in the three canyons to be able to compare them and to quantify if the street aspect ratio influences the tree effect (i.e., if there is an interaction between tree and building effects). However, the normalized tree characteristics are not exactly equal in the three canyons due to the street sizes and the 0.5 m mesh cells that limit the possible range of tree sizes. The detailed list of the tree characteristics and the different values used for the tree parameters are presented in Table A4.

2.2.2 Calculation of vertical and horizontal transfers in Code_Saturne for comparison to MUNICH

To evaluate the vertical transfer between the street and the background zone in Code_Saturne, a passive tracer is emitted in each mesh cell of the street, with an arbitrary stationary emission rate e=1000 µg s⁻¹ for a street canyon of length L=1 m. For the tracer to be dispersed only by vertical transfers and not by horizontal winds within the street, simulations are performed with the wind perpendicular to the street at the inlet (φ=90^∘). The initial street and background tracer concentrations are zero. At the end of the simulation (stationary state reached), the tracer concentration is averaged in the street and in the background zone (denoted as C_street and C_bg in µg m⁻³) to reproduce a MUNICH homogeneous street assumption. The background zone corresponds to the area of the same volume as the street but located just above the street (see Maison et al., 2022, for more details). In the absence of other processes (horizontal transport, deposition, and chemical reactions), the tracer mass balance in the street yields the following:

$$\begin{array}{}\text{(12)}& {\displaystyle }& {\displaystyle }{Q}_{\mathrm{vert}}=e\text{(13)}& {\displaystyle }& {\displaystyle }\Rightarrow q_{\mathrm{vert}}\mathrm{WL}{\displaystyle \frac{C_{\mathrm{street}}-C_{\mathrm{bg}}}{H}}=e\text{(14)}& {\displaystyle }& {\displaystyle }\Rightarrow q_{\mathrm{vert}}={\displaystyle \frac{eH}{\mathrm{WL}\left(C_{\mathrm{street}}-C_{\mathrm{bg}}\right)}},\end{array}$$

where Q_vert is the vertical flux of pollutant at the roof level for the whole street (µg s⁻¹), q_vert is the vertical transfer coefficient (m² s⁻¹), and WL is the exchange surface (m²). Thus, the vertical transfer coefficient can be compared between both models. In Code_Saturne, q_vert is calculated from the emission rate e and the concentration gradient $\left(\frac{C_{\mathrm{street}}-C_{\mathrm{bg}}}{H}\right)$, following Eq. (14), and in MUNICH, it is calculated from Eq. (7).

In MUNICH, the horizontal transfer velocity is equal to the street average horizontal wind speed in the street direction U_street, which is calculated by integrating Eq. (1) between $z=z_{{\mathrm{0}}_{\mathrm{s}}}$ and z=H. In Code_Saturne, U_street is estimated from the wind speed in the y direction averaged over the street mesh cells. In addition to U_street, the MUNICH vertical profile of the wind speed (Eq. 1) can also be compared to Code_Saturne by averaging the wind speed in the y direction over the street width (x axis).

As the objective of the study is to parameterize the aerodynamic effect of trees, the chemistry and deposition on built and vegetated surfaces are not considered here. Further details on the simulation setup of MUNICH and Code_Saturne and on the comparison of vertical and horizontal transfers in a treeless canyon are presented in Maison et al. (2022).

3 Quantification of the tree crown effect on horizontal and vertical transfers by Code_Saturne simulations

To quantify tree effect on the horizontal wind speed along the street and on the vertical transfer coefficient, Code_Saturne simulations are performed for a large range of tree characteristics (LAI_2D, CVF, and height ratio $h/H$), as summarized in Table A4. The tree effect is expressed as a relative deviation between the simulations without and with trees.

3.1 Tree effect on horizontal transfer

To quantify and compare the effect of tree crowns in Code_Saturne and MUNICH, and thus to overcome eventual differences between the two models observed in a treeless canyon for the horizontal velocity U_street, a relative deviation (RD) of U_street, between the simulations with and without trees (%), is computed as follows:

$$\begin{array}{}\text{(15)}& {\mathrm{RD}}_{U_{\mathrm{street}}}=\mathrm{100}\times {\displaystyle \frac{U_{\mathrm{street}}-U_{{\mathrm{street}}_{\mathrm{0}}}}{U_{{\mathrm{street}}_{\mathrm{0}}}}},\end{array}$$

where $U_{{\mathrm{street}}_{\mathrm{0}}}$ stands for the average wind velocity in a treeless street, and U_street is the average wind velocity in a street with trees, as computed with Code_Saturne. This RD between the Code_Saturne simulations with and without trees is shown in Fig. 2a for WC, 2b for IC, and 2c for NC. It shows that ${\mathrm{RD}}_{U_{\mathrm{street}}}$ becomes increasingly negative, meaning that U_street decreases as the LAI_2D and the CVF increase. On the opposite, the tree height has either no impact or a small impact compared to LAI_2D and CVF. When the effect of tree height is noticeable, an increase in the tree height induces a decrease in RD.

In the range of the tree characteristics studied, U_street is attenuated from 7.3 % to 62.3 %. The tree effect on U_street can be compared between the three canyons. The tree effect on U_street increases as the canyon is deeper, highlighting the complex interaction between the street dimensions and the tree effect on the velocity. This observation is consistent with the study of Wania et al. (2012).

Figure 2 Relative deviation (RD) of U_street computed from Code_Saturne simulations for different tree LAI_2D, CVF, and height ratio for WC (a), IC (b), and NC (c). The graphic is divided into three columns corresponding to the three height ratios, and higher CVF correspond to darker colors. $\left|{\mathrm{RD}}_{U_{\mathrm{street}}}\right|$ values are specified with data labels.

3.2 Tree effect on vertical transfer

The relative deviation of the vertical transfer coefficient (${\mathrm{RD}}_{q_{\mathrm{vert}}}$) between simulations with and without trees is introduced to quantify the tree effect on the vertical transfer coefficient. Similar to the relative deviation of U_street (Eq. 15), the relative deviation ${\mathrm{RD}}_{q_{\mathrm{vert}}}$ (%) is expressed as follows:

$$\begin{array}{}\text{(16)}& {\mathrm{RD}}_{q_{\mathrm{vert}}}=\mathrm{100}\times {\displaystyle \frac{q_{\mathrm{vert}}-q_{{\mathrm{vert}}_{\mathrm{0}}}}{q_{{\mathrm{vert}}_{\mathrm{0}}}}},\end{array}$$

where $q_{{\mathrm{vert}}_{\mathrm{0}}}$ stands for the vertical transfer coefficient in a treeless street, and q_vert is the vertical transfer coefficient in a street with trees. Code_Saturne ${\mathrm{RD}}_{q_{\mathrm{vert}}}$ is plotted for different tree parameters LAI_2D, CVF, and height ratio $h/H$ in Fig. 3a for WC, 3b for IC, and 3c for NC.

Figure 3 Relative deviation (RD) of q_vert computed from Code_Saturne simulations for different tree LAI_2D, CVF, and height ratio for WC (a), IC (b), and NC (c). The graphic is divided into three columns corresponding to the three height ratios, and higher CVF correspond to darker colors. $\left|{\mathrm{RD}}_{q_{\mathrm{vert}}}\right|$ values are specified with data labels.

For WC, ${\mathrm{RD}}_{q_{\mathrm{vert}}}$ increases with tree LAI_2D, CVF, and height ratio (Fig. 3a). For IC, ${\mathrm{RD}}_{q_{\mathrm{vert}}}$ also increases with CVF and height ratio, but ${\mathrm{RD}}_{q_{\mathrm{vert}}}$ tends to slightly decrease as the LAI increase when the LAI is high and the ratio $h/H$ is small (differences between LAI=3 and 4 for $h/H=\mathrm{0.36}$ and 0.50; Fig. 3b). For NC, a small increase in ${\mathrm{RD}}_{q_{\mathrm{vert}}}$ with LAI_2D and CVF is observed for $h/H=\mathrm{0.65}$. However, for the two other smaller height ratios, the tree effect is very low ($-\mathrm{1.1}\le {\mathrm{RD}}_{q_{\mathrm{vert}}}\le \mathrm{0.5}$; Fig. 3c).

This reduction in the tree effect on q_vert when LAI or CVF increases for small $h/H$ ratios in IC and NC can be explained by micro-scale effects (modified air flow path and turbulent viscosity) and is left out of the scope of the present study due to the corresponding low amplitude. Besides, investigation of such micro-scale effects would probably require more advanced turbulence models, for example, switching from a first-order model, k−ε, to a second-order one (Rij-SSG; Speziale et al., 1991) or a 3D large eddy simulation (LES).

Note that, unlike U_street (Sect. 3.1), the tree effect on q_vert decreases when the canyon becomes deeper. For example, the tree effect, as quantified by ${\mathrm{RD}}_{q_{\mathrm{vert}}}$, ranges between −1.0 % and −20.3 % for WC, −1.0 % and −18.7 % for IC, and 0.5 % and −2.7 % for NC. The averaged tree effect is relatively less strong when the canyon is deeper. In other words, regarding vertical transfers, the street effect dominates over the tree effect.

For vertical transfer, a wind perpendicular to the street (φ=90^∘) is used to focus on the effects on vertical transfers only. In this case of perpendicular wind, the air flow occurring in street canyons is complex and leads to heterogeneous tracer concentration in the street. In general, for the three canyons, the presence of the two tree crowns tends to increase the tracer concentration on the leeward side of the street and to decrease it on the windward side (Buccolieri et al., 2009; Gromke and Ruck, 2012). Depending on the street aspect ratio and, therefore, on the air flow regime (Oke, 1988; Harman et al., 2004) and on the tree characteristics, the tree crown effect on local tracer concentration (leeward versus windward side) is more or less important. For example, for NC, the flow regime is skimming, and on average, the variation in C_street due to the presence of trees compensates between the two sides of the street, explaining why ${\mathrm{RD}}_{q_{\mathrm{vert}}}$ is very low for NC.

This section demonstrated that the tree effect parameterizations should depend on tree characteristics and also on building characteristics to account for the building–tree interactions. The next section aims to parameterize in MUNICH the tree effect on U_street and q_vert observed in Code_Saturne simulations.

4 Parameterization of the aerodynamic effect of tree crowns in MUNICH

4.1 Model description

The MUNICH parameterizations of horizontal and vertical transfers detailed in Maison et al. (2022) are modified to take into account the tree effects. These parameterizations are based on Wang (2012, 2014) equations, which were originally developed for homogeneous vegetated cover. To remain consistent with this hypothesis, the parameterization will depend on the homogeneous leaf area index in the street, denoted as LAI_street. As for the conversion from LAI_3D to LAI_2D, LAI_street is estimated from LAI_2D conserving the leaf surface. LAI_street is calculated by spreading the tree crown cylindrical LAI_2D over the whole street width as follows:

$$\begin{array}{}\text{(17)}& {\displaystyle }{\mathrm{LAI}}_{\mathrm{street}}\times S_{\mathrm{street}}={\mathrm{LAI}}_{\mathrm{2}\mathrm{D}}\times S_{\mathrm{2}\mathrm{D}}\text{(18)}& {\displaystyle }\begin{array}{rl}\Rightarrow & {\mathrm{LAI}}_{\mathrm{street}}={\mathrm{LAI}}_{\mathrm{2}\mathrm{D}}\times {\displaystyle \frac{S_{\mathrm{2}\mathrm{D}}}{S_{\mathrm{street}}}}={\displaystyle \frac{\mathrm{2}rLn{\mathrm{LAI}}_{\mathrm{2}\mathrm{D}}}{\mathrm{WL}}}\\ & ={\displaystyle \frac{\mathrm{2}rn{\mathrm{LAI}}_{\mathrm{2}\mathrm{D}}}{W}},\end{array}\end{array}$$

where S_street is the soil-projected area of the street homogeneous tree crown, and S_2D is the soil-projected area of the Code_Saturne 2D cylindrical tree crown (in m²). Note that, regardless of the tree crown geometry, LAI_street is always equal to the street total leaf surface divided by the street ground area (WL).

To account for the tree effect on the mixing length, the characteristic length of the trees ($l_{c_{\mathrm{t}}}$ in m) is added into the equation describing the mixing length l_m as follows:

$$\begin{array}{}\text{(19)}& {\displaystyle \frac{\mathrm{1}}{l_{\mathrm{m}}}}={\displaystyle \frac{\mathrm{1}}{\mathit{\kappa }H}}+{\displaystyle \frac{\mathrm{1}}{l_{c_b}}}+{\displaystyle \frac{\mathrm{1}}{l_{c_{\mathrm{t}}}{f}_{b\times t}}},\text{(20)}& {\displaystyle }\text{with}{l}_{c_{\mathrm{t}}}={\displaystyle \frac{E_{\mathrm{t}}H}{C_{D_{\mathrm{t}}}\frac{\mathrm{1}}{\mathrm{2}}{\mathrm{LAI}}_{\mathrm{street}}}},\end{array}$$

where $\frac{\mathrm{1}}{\mathrm{2}}{\mathrm{LAI}}_{\mathrm{street}}$ corresponds to the leaf frontal area density, assuming a random leaf orientation distribution, E_t is a proportionality constant taken equal to 0.054 for vegetated cover, as suggested by Wang (2014), and f_b×t is a function parameterized based on Code_Saturne simulations in Sect. 4.2 and representing the interaction between buildings, trees, and the tree crown height.

The dimensionless factor s_H expressing the effects of the tree canopy on the mixing length l_m is calculated using its definition in Eq. (6) and the expression of l_m in Eq. (19) as follows:

$$\begin{array}{}\text{(21)}& {\displaystyle }& {\displaystyle }{l}_{\mathrm{m}}=\mathit{\kappa }H{s}_H\text{at roof level}\text{(22)}& {\displaystyle }& {\displaystyle }\text{with}{s}_H=\left\{\begin{array}{ll}\frac{l_{c_{\mathrm{b}}}}{l_{c_{\mathrm{b}}}+\mathit{\kappa }H}& \text{without tree}\\ \frac{l_{c_{\mathrm{b}}}{l}_{c_{\mathrm{t}}}{f}_{b\times t}}{\mathit{\kappa }H\left(l_{c_{\mathrm{b}}}+l_{c_{\mathrm{t}}}{f}_{b\times t}\right)+l_{c_{\mathrm{b}}}{l}_{c_{\mathrm{t}}}{f}_{b\times t}}& \text{with trees}.\end{array}\right.\end{array}$$

The simulations with and without trees have to be distinguished to avoid any convergence issue since $l_{c_{\mathrm{t}}}$ tends to +∞ when LAI_street tends to 0. This s_H factor now includes rough soil, buildings, and tree effects on the mixing length and is then used in the calculation of the attenuation coefficient α (Eq. 4). In the numerator, the expressions for buildings and trees are added as follows:

$$\begin{array}{}\text{(23)}& \mathit{\alpha }={\displaystyle \frac{C_{\mathrm{B}}{a}_{\mathrm{r}}+C_{D_{\mathrm{t}}}{C}_{\mathrm{u}}\frac{\mathrm{1}}{\mathrm{2}}{\mathrm{LAI}}_{\mathrm{street}}}{\mathit{\kappa }{s}_H}},\end{array}$$

where $C_{D_{\mathrm{t}}}$ is the tree drag coefficient (dimensionless) taken equal to the one, as used in Code_Saturne ($C_{D_{\mathrm{t}}}=\mathrm{0.2}$). Wang (2014) presents C_u as a dimensionless coefficient homogeneous on the vertical axis, but that can depend on canopy features. This coefficient has to be determined based on experimental or simulated observation data and will be parameterized in Sect. 4.2 with CFD simulations.

4.2 Parameter determination based on Code_Saturne simulations

There are two parameters introduced in l_m and α equations, i.e., f_b×t and C_u (Eqs. 19 and 23), that have to be determined based on Code_Saturne ${\mathrm{RD}}_{q_{\mathrm{vert}}}$ and ${\mathrm{RD}}_{U_{\mathrm{street}}}$. The function f_b×t depends on a_r to account for the interaction between trees and buildings observed in Fig. 3 and also on h_max to account for the effect of the tree height. Note that, since tree crowns are assumed to be homogeneous within the canopy in the original Wang formulation, the tree crown height h_max is not taken into account (see Eq. 20), and therefore, it needs to be included in the function f_b×t. The f_b×t expression is determined by maximizing the fit between Code_Saturne ${\mathrm{RD}}_{q_{\mathrm{vert}}}$ and MUNICH ${\mathrm{RD}}_{q_{\mathrm{vert}}}$ as follows:

$$\begin{array}{}\text{(24)}& \begin{array}{rl}{f}_{b\times t}& ={\displaystyle \frac{a_{\mathrm{0}}+a_{\mathrm{1}}\exp \left(a_{\mathrm{2}}{a}_{\mathrm{r}}\right)}{{\left(h_{\max }/H\right)}^{\mathrm{2}}}}\\ & \text{with}{a}_{\mathrm{0}}=\mathrm{3.26},a_{\mathrm{1}}=\mathrm{0.0256}\text{and}{a}_{\mathrm{2}}=\mathrm{6.70}.\end{array}\end{array}$$

A comparison of Code_Saturne ${\mathrm{RD}}_{q_{\mathrm{vert}}}$ and the ${\mathrm{RD}}_{q_{\mathrm{vert}}}$ parameterized with Eqs. (7), (19), (20), and (24) is presented for the three canyons in Fig. 4a, c, e. The normalized mean absolute error (NMAE) and bias (NMB) are calculated to compare Code_Saturne and MUNICH results (see Appendix C for the definition of the statistical indicators). Figure 4a, c, e show a good agreement between Code_Saturne ${\mathrm{RD}}_{q_{\mathrm{vert}}}$ and the parameterized ${\mathrm{RD}}_{q_{\mathrm{vert}}}$ because the f_b×t function was determined to minimize bias. The NMAE can be high for ${\mathrm{RD}}_{q_{\mathrm{vert}}}$ (up to 63 % in NC) because the parameterization does not reproduce the slight decrease in ${\mathrm{RD}}_{q_{\mathrm{vert}}}$ when LAI_2D increases from LAI_2D=3 to 4 when $h/H\approx \mathrm{1}/\mathrm{3}$ and $\mathrm{1}/\mathrm{2}$ (see Sect. 3.2 and Fig. 3b and c). But this is not an issue for NC since the q_vert values are low.

For U_street, the parameter C_u also needs to be determined. A constant value of C_u=6.7 is sufficient to obtain a good fit between Code_Saturne ${\mathrm{RD}}_{U_{\mathrm{street}}}$ and MUNICH ${\mathrm{RD}}_{U_{\mathrm{street}}}$. No dependency of C_u on building or tree features is needed, as shown in Fig. 4b, d, f, which compares Code_Saturne ${\mathrm{RD}}_{U_{\mathrm{street}}}$ and the parameterized ${\mathrm{RD}}_{U_{\mathrm{street}}}$. Note that the parameterizations of $l_{c_{\mathrm{t}}}$ and f_b×t impact not only the vertical transfers but also the horizontal wind speed because they are involved in the calculation of the s_H factor and, hence, of the α coefficient. The good comparisons between Code_Saturne ${\mathrm{RD}}_{U_{\mathrm{street}}}$ and MUNICH ${\mathrm{RD}}_{U_{\mathrm{street}}}$ also show that the parameterized s_H reproduces the horizontal wind speed well and, therefore, also the interactions between trees and building effects and the influence of h_max.

Figure 4 Comparison of (a, c, e) ${\mathrm{RD}}_{q_{\mathrm{vert}}}$ and (b, d, f) ${\mathrm{RD}}_{U_{\mathrm{street}}}$ computed from Code_Saturne simulations and parameterized in MUNICH for different tree LAI_2D, CVF, and height ratio and for (a, b) WC, (c, d) IC, and (e, f) NC.

4.3 Comparison of q_vert, U_street, and wind profiles

Figure 5a and b present, respectively, a comparison of Code_Saturne and parameterized q_vert and U_street. The statistical indicators are presented in Table 2. As MUNICH was parameterized to reproduce the tree effect observed in Code_Saturne well, and as the two models agree well in a treeless canyon (Maison et al., 2022), the vertical transfer coefficient and wind speed with trees are close between the two models, as expected (Fig. 5 and Table 2). The parameterized vertical transfer coefficients with trees agree well with the Code_Saturne ones, with normalized mean absolute errors ranging from 2.2 % to 4.1 % and bias from −4.1 % to 2.8 %. The parameterized average wind speed with trees agree well with Code_Saturne ones, with normalized mean absolute errors ranging from 3.4 % to 6.8 % and a normalized mean bias from −2.8 % to −1.1 %.

Figure 5 Normalized U_street and q_vert computed from Code_Saturne simulations for different tree LAI_2D, CVF, and height ratios.

Table 2 Statistical indicators (normalized mean absolute error, NMAE, and bias, NMB, in percent) for the comparison of Code_Saturne and MUNICH q_vert and U_street.

For each street canyon, Fig. 6 compares the Code_Saturne and parameterized vertical wind profiles for fixed CVF and $h/H$ ratio but for five different LAI_2D.

Figure 6 Comparison of Code_Saturne (solid lines) and parameterized (dotted lines) vertical wind profiles. The profiles in a treeless canyon are in black and darker colors correspond to increasing LAI_2D. For the three canyons, $h/H\approx \mathrm{0.5}$ and CVF≈25 %.

Figure 6 shows that an increase in the LAI_2D induces a decrease in the wind velocity. Concerning the vertical profile shape, trees induce a lower wind velocity on the entire street and not only in the tree crown. They even slightly impact the velocity just above the street. The maximum of attenuation of the wind is located in the middle of the tree crown and the wind velocity is re-accelerated under the tree crown. In the middle of the tree crown, the parameterized wind speed is close to the one of Code_Saturne. In the lower part and under the tree crown, the parameterized wind speed is underestimated. In fact, the re-acceleration under the tree crown is complex to consider in parameterized models. Besides, in real streets the tree height is not homogeneous, so this re-acceleration under the tree crown might be unrealistic in Code_Saturne simulations. In the parameterized wind profile with trees, the reduction in the wind speed just above the street due to the presence of trees was neglected. So the parameterized wind speed above and in the upper part of the tree crown is overestimated compared to Code_Saturne. Note that Kent et al. (2017) proposed a method to include the vegetation effect in aerodynamic roughness parameters and so to account for vegetation in the above-urban canopy wind profile of Macdonald et al. (1998). This method considers vegetation at the city scale and is not applicable here, since we work at the street scale. For now, as MUNICH assumes a homogeneous street canyon, the profile is averaged to compute U_street, and the shape of the profile is not used.

Moreover, to illustrate the impact of the developed parametrization on pollutant concentrations, MUNICH simulations are performed with carbon monoxide emissions in IC without and with trees of various characteristics. The results are presented in Appendix D.

5 Conclusions

Although the discretizations and the physical modeling used in MUNICH and Code_Saturne are fundamentally different, the setup of the CFD simulations was adapted to compare the two models through the average horizontal wind speed along the street and the calculation of the vertical transfer coefficient at the roof level by averaging a passive tracer concentration in the street and the background (above the streets). To build a parameterization suited to the street network model MUNICH, the CFD simulations were simplified with several hypothesis such as an infinite street canyon with 2D CFD simulations and homogeneous emissions over the street. Furthermore, quantities of interest were averaged over the street in the CFD simulations.

The aerodynamic effect of trees in street canyons is quantified with Code_Saturne simulations. Simulations were performed with two rows of trees and a large range of tree leaf area index, crown radius, and heights. In the range of the simulations performed, the average horizontal velocity in the street is reduced by −7.3 % to −62.3 % and the vertical transfer coefficient by +0.5 % to −20.3 %. This highlights the necessity of adding the tree aerodynamic effects into street models such as MUNICH.

Maison et al. (2022) proposed a parameterization of horizontal and vertical transfers in a treeless canyon, and the present study adds the tree effect to the building effect in this parameterization. The differences in wind speed and vertical transfer coefficient in a treeless canyon are very low between the two models (Maison et al., 2022). However, to overcome any model-specific difference and to build a parameterization for MUNICH of the tree effect on the aerodynamic parameters, relative deviation ratios between cases without and with trees are defined.

First, the Maison et al. (2022) vertical transfer coefficient parameterization is modified by adding a term representing the tree effect and the interaction between the trees and the buildings in the mixing-length expression. This term is a function of two dimensionless parameters to characterize the trees, street leaf area index (LAI_street) and tree-to-street height ratio ($h_{\max }/H$), and also of street characteristics, building height (H), and street width (W). A parameterization of the tree mixing length was defined from Code_Saturne simulations to obtain a tree effect as close as possible in the two models. The parameterized vertical transfer coefficients with trees agree well with those of Code_Saturne, with the normalized mean absolute error ranging from 2.2 % to 4.1 % and bias from −4.1 % to 2.8 %.

Second, this new mixing length expression is also used to compute the vertical wind profile and average wind speed along the street. Only one constant is fixed in the wind attenuation coefficient to maximize the fit between Code_Saturne and MUNICH tree effect on average wind speed. The comparison of the average wind speed gives a normalized mean absolute error ranging from 3.4 % to 6.8 % and a normalized mean bias from −2.8 % to −1.1 %.

MUNICH now includes a relatively simple parameterization of the tree effect on both horizontal and vertical aerodynamic processes, based on commonly used tree parameters that can be easily computed from urban databases and can reproduce the main effects obtained in much more detailed and costly CFD simulations. This parameterization can also be used in urban climate models to compute water and heat transfer in tree-lined streets. The perspectives of this study are to quantify the effect of street trees on air quality from the street level to the scale of the city of Paris. Dry deposition of gaseous pollutants and aerosols on tree leaves and emission of biogenic organic volatile compounds related to tree water stress will be considered in MUNICH. The contribution of biogenic and anthropic precursors to the formation of organic aerosols over an entire city will be compared.

Appendix A: Lists of abbreviations, variable parameters, and tree dimensions

Each line of the Table A4 corresponds to the parameters of a simulation. For each case, simulations with different LAI_2D were performed as follows: 0.5, 1, 2, 3, and 4.

Table A1 List of abbreviations.

Wang (2014)(Katul and Albertson, 1998)

Table A2 List of parameters.

(Soulhac et al., 2011)

Table A3 List of variables.

Table A4 List of tree dimensions simulated with Code_Saturne (r is the radius, h is the middle crown height, h_min is the crown bottom height, and h_max is the crown top height) and calculated tree parameters (CVF is the crown volume fraction, and $h/H$ is the height ratio).

Appendix B: Comparison of 2D and 3D Code_Saturne simulations with trees

Code_Saturne simulations were performed in a periodic 2D canyon of length L=1 m, and the street was therefore considered infinite due to this periodicity. In this case, the tree crown is represented as an infinite cylinder of radius r. To take into account the fact that, in reality, most of the tree crowns are spherical and are spaced from each other, an equivalent cylindrical LAI_2D is calculated in the 2D simulations. For the tree effect to be similar, the surface of the leaves is kept constant between the 2D and 3D simulations, as follows:

$$\begin{array}{}\text{(B1)}& {\displaystyle }{\mathrm{LAI}}_{\mathrm{3}\mathrm{D}}\times S_{\mathrm{3}\mathrm{D}}={\mathrm{LAI}}_{\mathrm{2}\mathrm{D}}\times S_{\mathrm{2}\mathrm{D}}\text{(B2)}& {\displaystyle }{\mathrm{LAI}}_{\mathrm{2}\mathrm{D}}={\mathrm{LAI}}_{\mathrm{3}\mathrm{D}}\times {\displaystyle \frac{S_{\mathrm{3}\mathrm{D}}}{S_{\mathrm{2}\mathrm{D}}}}={\mathrm{LAI}}_{\mathrm{3}\mathrm{D}}\times {\displaystyle \frac{\mathit{\pi }{r}^{\mathrm{2}}}{\mathrm{2}r\left(\mathrm{2}r+\mathit{\delta }\right)}},\end{array}$$

where δ is the spacing between two tree crowns within the row, and 2r+δ is the length of the street section for one spherical tree (m). S_3D and S_2D are, respectively, the soil-projected area of the 3D and 2D tree crowns (m²). In addition to the leaf surface and tree height, the tree radius is also conserved between 2D and 3D simulations; however, the value of the crown volume fraction (CVF) varies, as follows:

$$\begin{array}{}\text{(B3)}& \begin{array}{rl}{\mathrm{CVF}}_{\mathrm{2}\mathrm{D}}& ={\displaystyle \frac{V_{\mathrm{2}\mathrm{D}}}{V_{\mathrm{street}}}}={\displaystyle \frac{n\mathit{\pi }{r}^{\mathrm{2}}L}{HWL}}\\ & \text{and}{\mathrm{CVF}}_{\mathrm{3}\mathrm{D}}={\displaystyle \frac{V_{\mathrm{3}\mathrm{D}}}{V_{\mathrm{street}}}}={\displaystyle \frac{n\frac{\mathrm{4}}{\mathrm{3}}\mathit{\pi }{r}^{\mathrm{3}}}{HW\left(\mathrm{2}r+\mathit{\delta }\right)}}.\end{array}\end{array}$$

V_street is the street volume (m³), and V_3D and V_2D are, respectively, the 3D and 2D tree crown volumes (m³). The relation between 2D and 3D CVF is therefore as follows:

$$\begin{array}{}\text{(B4)}& {\mathrm{CVF}}_{\mathrm{2}\mathrm{D}}={\displaystyle \frac{\mathrm{3}}{\mathrm{4}}}\times {\displaystyle \frac{\mathrm{2}r+\mathit{\delta }}{r}}\times {\mathrm{CVF}}_{\mathrm{3}\mathrm{D}}.\end{array}$$

2D and 3D simulations with different LAI are performed to check the validity of the 2D infinite street canyon hypothesis. 3D simulations are realized with a periodic street with a 10 m length and cell meshes of $\mathrm{0.5}\times \mathrm{0.5}\times \mathrm{0.5}$ m in the x, y, and z directions. In both 2D and 3D simulations, C_street and C_bg are computed with a normal wind to the street (φ=90^∘) and U_street with φ=45^∘. The results are presented in Figs. B1a–d for WC, B1e–h for IC, and B1i–l for NC.

Figure B1 Comparison of C_street (a, e, i), C_bg (b, f, j), q_vert (c, g, k), and U_street (d, h, l) for 2D and 3D simulations in the three canyons without and with trees of LAI_2D=1 and 4. CVF_2D≈10 % and $h/H\approx \mathrm{0.5}$ are fixed. The percentages written in the middle of the histogram bars correspond to the RD between the 2D and 3D cases.

For each canyon, the cases without tree (LAI=0) are equivalent. C_street, C_bg, q_vert, and U_street are similar between the 2D and 3D cases, with a maximum relative difference of 0.1 % for U_street. For the cases with trees, concerning C_street, C_bg, and q_vert, the differences between the 2D and 3D cases are relatively low over the range of simulations performed with a maximum of 3.3 % for C_street and C_bg in WC. For U_street, slightly larger differences are observed (up to −5.4 % in WC for $\mathrm{LAI}=\mathrm{4}\left(\mathrm{2}\mathrm{D}\right)/\mathrm{12.7}\left(\mathrm{3}\mathrm{D}\right)$). These differences can be explained by the distribution of the turbulence around the crown. For each variable and canyon, the relative deviation between the 2D and 3D cases increases with the tree LAI. Given the low differences observed between the 2D and 3D simulations, the hypothesis of a 2D canyon with a cylindrical tree crown associated with an equivalent LAI_2D to represent a spherical tree crown spaced of δ m in a street of length L is reasonable.

Appendix C: Definition of the statistical indicators

Code_Saturne and MUNICH simulations are denoted cs_i and m_i, respectively. In this section, n is the total number of simulations, which is equal to 45 per street canyon.

• Normalized mean absolute error (%):

  $$\begin{array}{}\text{(C1)}& \mathrm{NMAE}=\mathrm{100}\times {\displaystyle \frac{\sum _{i=\mathrm{1}}^n|m_i-{\mathrm{cs}}_i|}{\left|\sum _{i=\mathrm{1}}^n{\mathrm{cs}}_i\right|}}.\end{array}$$

• Normalized mean bias (%):

  $$\begin{array}{}\text{(C2)}& \mathrm{NMB}=\mathrm{100}\times {\displaystyle \frac{\sum _{i=\mathrm{1}}^n\left(m_i-{\mathrm{cs}}_i\right)}{\sum _{i=\mathrm{1}}^n{\mathrm{cs}}_i}}.\end{array}$$

Appendix D: Aerodynamic tree effect on concentration

To illustrate the impact of the developed parameterization on pollutant concentrations, MUNICH simulations are performed in the Intermediate Canyon (IC) with carbon monoxide (CO) emissions in the street. The input parameters are detailed in Table D1.

Table D1 List of parameters fixed in MUNICH simulation.

In the simulations, only the tree aerodynamic effect is considered, and there is no chemistry, no deposition of pollutants on urban or tree surfaces, and no biogenic volatile organic compound (BVOC) emission. One street is modeled without any inflow of pollutant from nearby streets. The following three processes are taken into account: pollutant emission in the street, pollutant outflow by horizontal transfer, Q_outflow (function of the parameterized U_street), and vertical transfer between the street and the background, Q_vert (function of the parameterized q_vert).

The street average CO concentrations (C_street) are compared in Fig. D1 for a canyon without trees (Fig. D1a) and for a canyon with trees of various LAI, CVF, and height ratios $h/H$ (Fig. D1b, c).

Figure D1 shows that the increase in tree LAI and CVF induces higher average street CO concentrations. The effect of the tree height ratio $h/H$ is negligible for low LAI and increases when the LAI increases. The variations in the concentrations are due to both the effect of trees on the horizontal transfer velocity, U_street, and on the vertical transfer coefficient, q_vert.

In this example, the aerodynamical effect of trees leads to a large increase in the CO concentration from 4 % to 45 %. However, to simulate the global effect of trees on air quality at street level, other processes such as dry deposition on leaves and BVOC emissions need to be considered.

Figure D1 Comparison of CO concentration (C_street) simulated with MUNICH in IC for different tree LAI_2D, CVF, and tree height ratios $h/H$. The graphic is divided into three columns corresponding to the treeless canyon (a) and to canyons with trees of height ratios $h/H=\mathrm{0.36}$ (b) and 0.64 (c). The concentrations C_street (in µg m⁻³) are specified with data labels.