On the representation of urban heterogeneities in mesoscale models

Abstract

The size and arrangement of the obstacles and the presence of a source of heat (anthropogenic heat flux) are distinctive characteristics of an urban area. These two elements, together with the specific applications oriented to improve citizen’s comfort, determine the way urban heterogeneities are represented in mesoscale models. In this contribution two examples are presented. In the first a microscale fluid dynamics model is used to investigate the role of organized motions (dispersive fluxes) of a passive tracer emitted at the surface in a staggered and in an aligned array of cubes. The impact of the dispersive flux, that can reach 90 % of the total flux in the staggered array, is then assessed in a column model. The second example deals with the representation of anthropogenic heat fluxes and the estimation of thermal comfort by means of an urban canopy parameterization with a simple building energy model, implemented in a mesoscale model. The simulation of a typical summer day over the city of Madrid (Spain) shows that the anthropogenic heat fluxes have the largest impact on the air temperature in the evening-night, and that the presence of the city prolongs to the late evening the period of thermal discomfort, compared with the rural areas surrounding the city. The paper is concluded by pointing out that future work must be devoted to deep on the relationship between the real morphology of a city and the simplified morphology adopted in the urban canopy parameterizations.

Appendices

Appendix 1

The column model used in this study solves the following equations, in addition to the conservation equation for the scalar (Eq. 7):

$$\begin{aligned} \begin{array}{l} \frac{\partial \rho \left\langle {\overline{u}} \right\rangle }{\partial t}=\frac{\partial }{\partial z}\left( {\rho K_M \frac{\partial \left\langle {\overline{u}} \right\rangle }{\partial z}} \right) -\alpha C_D \left\langle {\overline{u}} \right\rangle \left| {\left\langle {\overline{u}} \right\rangle } \right| +\frac{u_\tau ^{2}}{4H} \\ \frac{\partial \rho \left\langle {\overline{e}} \right\rangle }{\partial t}=\frac{\partial }{\partial z}\left( {\rho K_M \frac{\partial \left\langle {\overline{e}} \right\rangle }{\partial z}} \right) +K_M \left( {\frac{\partial \left\langle {\overline{u} } \right\rangle }{\partial z}} \right) ^{2}+\alpha C_D \left| {\left\langle {\overline{u} } \right\rangle } \right| ^{3}-\varepsilon \\ \end{array} \end{aligned}$$

The first equation is the mean and spatially averaged conservation equation for the x-component of momentum, where the dispersive stress has been neglected. The first term on the right hand side, represents the gradient of the vertical turbulent flux of momentum (Reynolds stress), the second is the drag force, different than zero only within the canopy, and the third term $\frac{u_\tau ^{2}}{4H}$ is added to represent a pressure gradient along the column to maintain the flow. In the simulations, $u_\tau =1\,\hbox {m}/\hbox {s}$, and $4H$ is the height of the domain.

The second equation is the conservation equation for the spatially averaged turbulent kinetic energy. The first term on the right hand side is the gradient of the vertical turbulent flux of TKE, the second is the gradient of the shear production, the third is the TKE production due to the drag force, and the 4th is the dissipation, modelled as $\varepsilon =c_\varepsilon \frac{\left\langle {\overline{e}} \right\rangle ^{3/2}}{l_\varepsilon }$. The turbulent coefficient is computed as $K_M =c_k l_k \sqrt{tke}$.

The length scales are based on the paper of [22],

$$\begin{aligned} l_\varepsilon&= c_\varepsilon a_1 \left( {H-d} \right) \quad \hbox {if}\; z\le H\\ l_\varepsilon&= c_\varepsilon a_1 \left( {z-d} \right) \quad \,\,\ \! \hbox {if}\; H<z\le \frac{3}{2}H\\ l_\varepsilon&= c_\varepsilon a_2 \left( {z-d_2 } \right) \quad \hbox {if}\; z>\frac{3}{2}H \end{aligned}$$

where $d_2 =\left( {1.-\frac{a_1 }{a_2 }} \right) \frac{3}{2}H+\frac{a_1 }{a_2 }d$ is computed to ensure continuity in $z=\frac{3}{2}H$.

The displacement height is computed as $\frac{d}{H}=\lambda _p ^{0.13}$. The value of $l_k$ is $l_k =\frac{c_\mu }{c_\varepsilon c_k }l_\varepsilon$. The values of the numerical constants are as follows: $c_\varepsilon =0.71,c_k =0.4,c_\mu =0.09,a_1 =2.19,a_2 =1.2$. The air density is assumed constant and equal to one. The model is run until steady state is reached (time derivative become zero).

Appendix 2

The mean radiant temperature for a person 1.80 m tall in the middle of the canyon is computed as follows:

$$\begin{aligned} T_{mr} =\left( {\frac{R_{long} +\frac{a_b }{\varepsilon _p }R^{*}_{short} }{\sigma }} \right) ^{1/4} \end{aligned}$$

(15)

where $a_b$ is the absorption coefficient of the human body (taken as 0.7), $\varepsilon _p$ is the emission coefficient of the human body (0.97), $\sigma$ is the Stefan Boltzmann constant, $R_{long}$ is the long wave contribution, and $R^{*}_{short} \quad$ is the shortwave component.

The human body has been simplified by a vertical surface, 1.80 m height, infinite in the direction of the canyon, and equidistant from the two walls.

For the long wave contribution, we have, for a North–South oriented street canyon.

$$\begin{aligned} R_{long} \!&= \! \underbrace{\sum _{i=1,nz} {\left( {\begin{array}{l} \sigma \varepsilon _{wall} \left( {T^{wall}e_i +T^{wall}w_i } \right) \left( {1.\!-\!pwin} \right) \!+\!\sigma \varepsilon _{window} \left( {T^{window}e_i +T^{window}w_i } \right) pwin\!+\! \\ \left( {1.-\varepsilon _{wall} \left( {1-pwin)\!-\!\varepsilon _{window} pwin} \right) } \right) \left( {Rle^{wall}_i \!+\!Rlw^{wall}_i } \right) \\ \end{array}} \right) \Psi _i \Gamma _{i+1} } }_{walls}\\&+\underbrace{\sum _{i=1,nz} {2Rl_{sky} \Psi _i \left( {1.-\Gamma _{i+1} } \right) } \!+\!2Rl_{sky} \Psi _{sky} }_{sky}\\&+\underbrace{2\left( {\sigma \varepsilon _{street} T^{street}+(1.-\varepsilon _{street} )Rl_{street} } \right) \Psi _{street} }_{street} \end{aligned}$$

The meaning of the symbols is as follows:

$\varepsilon _{wall}$ :

is the emissivity of the wall (windows when the subscript is window)

$T^{wall}e_i ,T^{wall}w_i$ :

is the temperature of the east and west walls at the level $i$ (of the windows when the superscript is window)

$pwin$ :

is the fraction of the wall occupied by the windows

$Rle^{wall}_i ,Rlw^{wall}_i$ :

is the long wave radiation reaching the east and west wall at the level $i$

$\Gamma _{i+1}$ :

is the probability to have a building at level i

$\Psi _i$ :

is the view factor from the wall at level i to the human body in the middle of the street (see below)

$Rl_{sky}$ :

is the long wave radiation from the sky to the human body in the middle of the street

$\Psi _{sky}$ :

is the view factor from the sky to the human body in the middle of the street

$\varepsilon _{street}$ :

is the emissivity of the street

$T^{street}$ :

is the temperature of the street

$Rl_{street}$ :

is the long wave radiation reaching the street

$\Psi _{street}$ :

is the view factor from the street to the human body

The short wave component is divided in direct and reflected, and treated as:

$$\begin{aligned} R^{*}_{short} =f_p R_{short}^{direct} +R_{short}^{reflected} \end{aligned}$$

The reflected component is calculated using the view factors in a similar way as for the long wave radiation:

$$\begin{aligned} R_{short}^{reflected}&= \underbrace{\sum _{i=1,nz} {\left( {\left( {albedo_{wall} \left( {1-pwin)+albedo_{window} pwin} \right) } \right) \left( {Rse^{wall}_i +Rsw^{wall}_i } \right) } \right) \Psi _i \Gamma _{i+1} } }_{walls}\\&+\underbrace{2\left( {albedo_{street} Rs_{street} } \right) \Psi _{street} }_{street} \end{aligned}$$

where $Rse^{wall}_i ,Rsw^{wall}_i$ is the short wave radiations reaching the east and west wall at level i and $Rs_{street}$ is the short wave radiation reaching the street.

For the direct short wave, the formula used is the following:

$$\begin{aligned} R_{short}^{direct} =\frac{R_{solar} }{z_{man} }\sum _{i=1,mz} {\left[ {\max (0,x1-x2)\gamma _{i+1} } \right] } \end{aligned}$$

where $R_{solar}$ is the solar radiation (in W/m$^2$) reaching an unobstructed horizontal surface, $z_{man}$ is the height of the human body (1.80 m), and $\gamma _{i+1}$ is the probability to have a building of height $z_{i+1}$ (where $z_{i+1}$ is the full height of the numerical level i+1— see more about the urban grid definition in [4]), and

$$\begin{aligned} x1&= \min \left( {\left( {z_{i+1} } \right) \tan \left( {Zr} \right) ,W} \right) \\ x2&= \max \left( {0,\left( {z_{i+1} -z_{man} } \right) \tan \left( {Zr} \right) } \right) \\ \end{aligned}$$

For the physical meaning of x1 and x2, see Fig. A1 in [4]. This is valid for a canyon perpendicular to the sun direction. When it is not, it can be easily corrected (see again the technique used in [4]).

The values of fp are computed as:

$$\begin{aligned} fp&= 0.308\cos \left( {\phi \left( {1-\frac{\phi ^{2}}{48402}} \right) } \right) \\ \phi&= \frac{\pi }{2}-Zr \end{aligned}$$

and Zr solar zenith angle.

For the view factor calculation, the formulas used are the same as in [4], but with different parameters. In particular, from A15 of [4]:

$$\begin{aligned} \Psi _i =\frac{1}{2z_{man} }\left( {\begin{array}{l} \left| {z_{man} -z_i } \right| f_{prl} \left( {D,\left| {z_{man} -z_i } \right| ,W/2} \right) -\left| {z_{man} -z_{i+1} } \right| f_{prl} \left( {D,\left| {z_{man} -z_{i+1} } \right| ,W/2} \right) - \\ z_i f_{prl} \left( {D,z_i ,W/2} \right) +z_{i+1} f_{prl} \left( {D,z_{i+1} ,W/2} \right) \\ \end{array}} \right) \end{aligned}$$

from A16

$$\begin{aligned} \Psi _{street} =\left( {f_{nrm} \left( {z_{man} ,D,W/2} \right) -f_{nrm} \left( {0,D,W/2} \right) } \right) \frac{W/2}{z_{man}} \end{aligned}$$

and from A18

$$\begin{aligned} \Psi _{stky} =\left( {f_{nrm} \left( {H,D,W/2} \right) -f_{nrm} \left( {H-z_{man} ,D,W/2} \right) } \right) \frac{W/2}{z_{man} } \end{aligned}$$