A Backward-Lagrangian-Stochastic Footprint Model for the Urban Environment

Abstract

Built terrains, with their complexity in morphology, high heterogeneity, and anthropogenic impact, impose substantial challenges in Earth-system modelling. In particular, estimation of the source areas and footprints of atmospheric measurements in cities requires realistic representation of the landscape characteristics and flow physics in urban areas, but has hitherto been heavily reliant on large-eddy simulations. In this study, we developed physical parametrization schemes for estimating urban footprints based on the backward-Lagrangian-stochastic algorithm, with the built environment represented by street canyons. The vertical profile of mean streamwise velocity is parametrized for the urban canopy and boundary layer. Flux footprints estimated by the proposed model show reasonable agreement with analytical predictions over flat surfaces without roughness elements, and with experimental observations over sparse plant canopies. Furthermore, comparisons of canyon flow and turbulence profiles and the subsequent footprints were made between the proposed model and large-eddy simulation data. The results suggest that the parametrized canyon wind and turbulence statistics, based on the simple similarity theory used, need to be further improved to yield more realistic urban footprint modelling.

Appendices

Appendix A: Derivation of ${\phi }/{g_a }$ in Eqs. 7–9

In the 3D Euclidean space, ${\phi }/{g_a }$, based on Thomson’s “simplest” solution (Thomson 1987; Rodean 1996), is given by

$$\begin{aligned} \frac{\phi _1 }{g_a }= & {} \frac{1}{2}\frac{\partial (\sigma _1 \sigma _l )}{\partial x_l }+\frac{\partial U_1 }{\partial t}+U_l \frac{\partial U_1 }{\partial x_l }+\frac{1}{2}(\sigma _l \sigma _j )^{-1}\frac{\partial (\sigma _1 \sigma _l )}{\partial x_k }(u_j -U_j )(u_k -U_k )\nonumber \\&+\left[ {\frac{1}{2}(\sigma _l \sigma _j )^{-1}\left( {\frac{\partial (\sigma _1 \sigma _l )}{\partial t}+U_m \frac{\partial (\sigma _1 \sigma _l )}{\partial x_m }} \right) +\frac{\partial U_1 }{\partial x_j }} \right] (u_j -U_j ), \end{aligned}$$

(21)

$$\begin{aligned} \frac{\phi _2 }{g_a }= & {} \frac{1}{2}\frac{\partial (\sigma _2 \sigma _l )}{\partial x_l }+\frac{\partial U_2 }{\partial t}+U_l \frac{\partial U_2 }{\partial x_l }+\frac{1}{2}(\sigma _l \sigma _j )^{-1}\frac{\partial (\sigma _2 \sigma _l )}{\partial x_k }(u_j -U_j )(u_k -U_k )\nonumber \\&+\left[ {\frac{1}{2}(\sigma _l \sigma _j )^{-1}\left( {\frac{\partial (\sigma _2 \sigma _l )}{\partial t}+U_m \frac{\partial (\sigma _2 \sigma _l )}{\partial x_m }} \right) +\frac{\partial U_2 }{\partial x_j }} \right] (u_j -U_j ),\end{aligned}$$

(22)

$$\begin{aligned} \frac{\phi _3 }{g_a }= & {} \frac{1}{2}\frac{\partial (\sigma _3 \sigma _l )}{\partial x_l }+\frac{\partial U_3 }{\partial t}+U_l \frac{\partial U_3 }{\partial x_l }+\frac{1}{2}(\sigma _l \sigma _j )^{-1}\frac{\partial (\sigma _3 \sigma _l )}{\partial x_k }(u_j -U_j )(u_k -U_k )\nonumber \\&+\left[ {\frac{1}{2}(\sigma _l \sigma _j )^{-1}\left( {\frac{\partial (\sigma _3 \sigma _l )}{\partial t}+U_m \frac{\partial (\sigma _3 \sigma _l )}{\partial x_m }} \right) +\frac{\partial U_3 }{\partial x_j }} \right] (u_j -U_j ). \end{aligned}$$

(23)

With the assumptions of horizontal homogeneity, and no subsidence, we have

$$\begin{aligned} \frac{\partial }{\partial x}= & {} \frac{\partial }{\partial y}=\frac{\partial }{\partial t}=0, \end{aligned}$$

(24)

$$\begin{aligned} V(z)= & {} 0\rightarrow U_2 \frac{\partial }{\partial x_2 }=0\nonumber \\ W(z)= & {} 0\rightarrow U_3 \frac{\partial }{\partial x_3 }=0. \end{aligned}$$

(25)

Note that in Eqs. 21, 22 and 23, the ensemble average is given as

$$\begin{aligned} \sigma _i \sigma _j =\overline{(u_i -U_i )(u_j -U_j )} . \end{aligned}$$

(26)

Further assuming there is no covariance between any two different velocities, Eqs. 21, 22 and 23 can be simplified as

$$\begin{aligned} \frac{\phi _1 }{g_a }= & {} \frac{\partial U}{\partial z}w+\frac{1}{2}\frac{1}{\sigma _1 \sigma _1 }\frac{\partial \sigma _1 \sigma _1 }{\partial z}(u-U)w=\frac{\partial U}{\partial z}w+\frac{1}{2\sigma _u ^{2}}\frac{\partial \sigma _u ^{2}}{\partial z}(u-U)w, \end{aligned}$$

(27)

$$\begin{aligned} \frac{\phi _2 }{g_a }= & {} \frac{1}{2\sigma _2 \sigma _2 }\frac{\partial \sigma _2 \sigma _2 }{\partial z}vw=\frac{1}{2\sigma _v ^{2}}\frac{\partial \sigma _v ^{2}}{\partial z}vw, \end{aligned}$$

(28)

$$\begin{aligned} \frac{\phi _3 }{g_a }= & {} \frac{1}{2}\frac{\partial \sigma _3 \sigma _3 }{\partial z}+\frac{1}{2\sigma _3 \sigma _3 }\frac{\partial \sigma _3 \sigma _3 }{\partial z}w^{2}=\frac{1}{2}\frac{\partial \sigma _w ^{2}}{\partial z}+\frac{1}{2\sigma _w ^{2}}\frac{\partial \sigma _w ^{2}}{\partial z}w^{2}, \end{aligned}$$

(29)

in the proposed model.

Appendix B: Equations for Stability, Turbulent Flow and Statistics

The stability correction function $\psi _{m}$ (Horst and Weil 1994) in Eq. 10 is given by

$$\begin{aligned} \psi _m =\left\{ {\begin{array}{ll} 2\ln \left[ {(1+\xi )/2} \right] +\ln \left[ {(1+\xi ^{2})/2} \right] -2\tan ^{-1}\xi +\pi /2&{}\hbox { for }L<0 \\ -5-5\ln \left[ {(z-d)/L} \right] &{}\hbox { for }L\ge 0\hbox { and (}z-d)/L>1 \\ -5(z-d)/L&{}\hbox { for }L\ge 0\hbox { and }(z-d)/L\le 1 \\ \end{array}} \right. , \end{aligned}$$

(30)

where L is the Obukhov length, and

$$\begin{aligned} \xi =\left[ {1-16(z-d)/L} \right] ^{1/4}. \end{aligned}$$

(31)

The velocity gradient in Eq. 7 can be expressed as (Businger et al. 1971; Kormann and Meixner 2001)

$$\begin{aligned} \frac{\partial U}{\partial z}=\left\{ {\begin{array}{ll} \frac{u_{*} }{\kappa (z-d)}\left[ {1-16(z-d)/L} \right] ^{-1/4}&{}\hbox { for }L<0 \\ \frac{6u_{*} }{\kappa (z-d)}&{}\hbox { for }L\ge 0\hbox { and (}z-d)/L>1 \\ \frac{u_{*} }{\kappa (z-d)}\left[ {1+5(z-d)/L} \right] &{}\hbox { for }L\ge 0 \hbox { and }(z-d)/L\le 1 \\ \end{array}} \right. , \end{aligned}$$

(32)

and in the urban canyon, Eq. 32 can be rewritten as

$$\begin{aligned} \frac{\partial U}{\partial z}=\frac{U_h \alpha }{h}\exp \left[ {\alpha \left( {z/h-1} \right) } \right] . \end{aligned}$$

(33)

The general equation for the Lagrangian decorrelation time scale under various stability conditions is (Flesch et al. 1995)

$$\begin{aligned} t_L =\left\{ {\begin{array}{ll} \frac{0.5(z-d)}{\sigma _w }\left[ {\frac{1}{1+5(z-d)/L}} \right] &{}\hbox { for }L<0 \\ \frac{0.5(z-d)}{\sigma _w }\left[ {1-6(z-d)/L} \right] ^{1/4}&{}\hbox { for }L\ge 0 \\ \end{array}} \right. , \end{aligned}$$

(34)

where $\sigma _{w}$ is the standard deviation of the vertical velocity component (Hsieh and Katul 2009)

$$\begin{aligned} \sigma _w =\left\{ {\begin{array}{ll} 1.25u_{*} \left( {1-3\frac{z-d}{L}} \right) ^{{1/3}}&{}\hbox { for }z/L<0 \\ 1.25u_{*} &{}\hbox { for }z/L\ge 0 \\ \end{array}} \right. . \end{aligned}$$

(35)

The standard deviation of the streamwise velocity component is given by (Hsieh and Katul 2009)

$$\begin{aligned} \sigma _u =\left\{ {\begin{array}{ll} 2.5u_{*} \left( {1-3\frac{z-d}{L}} \right) ^{\hbox {1/3}}&{}\hbox { for }z/L<0 \\ 2.5u_{*} &{}\hbox { for }z/L\ge 0 \\ \end{array}} \right. . \end{aligned}$$

(36)

After taking the boundary-layer height (Cai et al. 2008; Hsieh and Katul 2009) into consideration (as an upper boundary), instead of Eq. 36 we use,

$$\begin{aligned} \sigma _u= & {} \left\{ {\begin{array}{ll} (0.35w_{*} ^{2}+2.0u_{*} ^{2})^{1/2} &{}\hbox { for }z/L<0 \\ 2.5u_{*} &{}\hbox { for }z/L\ge 0 \\ \end{array}} \right. , \end{aligned}$$

(37)

$$\begin{aligned} w_{*}= & {} \left( {-\frac{u_{*} ^{3}h_b }{L\kappa }} \right) ^{1/3}, \end{aligned}$$

(38)

where $w_*$ is the convective velocity, and $h_{b}$ is the boundary-layer height that varies with stability (Cai et al. 2008).

The standard deviation of the lateral velocity component is given by (Flesch et al. 1995; Cai et al. 2008)

$$\begin{aligned} \sigma _v =\left\{ {\begin{array}{ll} \sigma _u &{} \hbox { for }z/L<0 \\ \sigma _w &{}\hbox { for }z/L\ge 0 \\ \end{array}} \right. . \end{aligned}$$

(39)

Appendix C: Flow Chart of Particle Trajectory Tracking Process and Touchdown Criteria