Interfacing the Urban Land–Atmosphere System Through Coupled Urban Canopy and Atmospheric Models

Abstract

We couple a single column model (SCM) to a cutting-edge single-layer urban canopy model (SLUCM) with realistic representation of urban hydrological processes. The land-surface transport of energy and moisture parametrized by the SLUCM provides lower boundary conditions to the overlying atmosphere. The coupled SLUCM–SCM model is tested against field measurements of sensible and latent heat fluxes in the surface layer, as well as vertical profiles of temperature and humidity in the mixed layer under convective conditions. The model is then used to simulate urban land–atmosphere interactions by changing urban geometry, surface albedo, vegetation fraction and aerodynamic roughness. Results show that changes of landscape characteristics have a significant impact on the growth of the boundary layer as well as on the distributions of temperature and humidity in the mixed layer. Overall, the proposed numerical framework provides a useful stand-alone modelling tool, with which the impact of urban land-surface conditions on the local hydrometeorology can be assessed via land–atmosphere interactions.

1 Introduction

The world is undergoing rapid urbanization with the percentage of the world’s population in urban areas increasing from 30 % in 1950 to 47 % in 2000, and projected to rise to 60 % by 2030 (Collier 2006; UN 2012). The rapid growth and the associated landscape modification of urban areas necessarily modify the surface energy and moisture balance by altering key physical and biophysical properties, with impacts on local and regional hydroclimate (Arnfield 2003), leading to potential global climate responses via a cascade of land–atmosphere interactions (Niyogi et al. 2009). Urban land-use land-cover (LULC) changes, with modified surface geometric and hydrothermal properties compounded by anthropogenic heat and moisture sources, contribute to numerous urban environmental features such as the urban heat island (UHI), air pollution, and convective rainfall initiation (Taha 1997; Collier 2006).

To assess the impacts of LULC changes, land–atmosphere coupling has been utilized in atmospheric general circulation models (e.g. Yang 1995; Bonan et al. 2002) and regional climate models, such as the Weather Research and Forecasting (WRF) model (Skamarock and Klemp 2008; Trier et al. 2011; Chen et al. 2011). In these models, a major source of uncertainty exists in the parametrization of the surface heat and moisture budgets (Chen and Avissar 1994; Chen and Dudhia 2001; Trier et al. 2011). In addition, due to the unique characteristics of built terrains, e.g. spatial heterogeneity, a modified turbulence field and modified surface energy balance (Wang et al. 2011b), a scalable urban land–atmosphere coupling framework, analogous to the “big canyon” representation for building arrays (Nunez and Oke 1977), is appropriate for studying the direct impact of urban LULC changes.

To parametrize urban land-surface processes, numerous urban canopy models (UCMs) have been developed and widely used in the last decade. Broadly, there are two groups of UCMs, including the single-layer urban canopy model (SLUCM) (Masson 2000; Kusaka et al. 2001; Wang et al. 2013) that focuses on the parametrization of the surface energy budget, and the more complex multilayer UCMs that also capture the momentum transport in urban canopies (Martilli et al. 2002; Dupont et al. 2004; Kondo et al. 2005). Up to date, most UCMs are capable of resolving the vertical transport of energy and sensible heat and predicting surface temperatures realistically, but are inadequate in representing the water transport due to the oversimplification of urban hydrological processes (Grimmond et al. 2010, 2011). Recently, an improved SLUCM including an urban hydrologic model has been developed by Wang et al. (2013), which enables a more realistic representation of evapotranspiration, infiltration, irrigation, and soil moisture states in urban areas and improves the prediction of latent heat fluxes.

Above a built environment, a single column model (SCM) (Troen and Mahrt 1986; Holtslag and Moeng 1991; Noh et al. 2003) can be used to predict the time evolution and spatial distribution of temperature and humidity in the overlying atmospheric boundary layer (ABL). The SCM is a relatively simple representation of the convective boundary layer (CBL) and can faithfully capture the growth of the boundary-layer. It has been found that the boundary-layer growth is sensitive to land-surface characteristics such as evaporation (Troen and Mahrt 1986), which necessitates the need for a realistic representation of the surface states (temperature, soil moisture, sensible and latent heat fluxes). In addition, the entrainment process at the top of the ABL also plays an important role in regulating boundary-layer growth and depends largely on the buoyancy flux, with significant contributions from, e.g. the latent heat flux (Hong et al. 2006).

In this study, we integrate a single column atmospheric model to the latest SLUCM including an improved urban hydrological module (Wang et al. 2011a, 2013; Sun et al. 2013a). The coupled SLUCM–SCM framework is capable of predicting the urban surface energy and water budgets with improved accuracy. Using the proposed model, a range of scenarios of urban LULC changes can be simulated, and their impact on the boundary-layer growth and temperature/humidity distribution under convective conditions assessed.

The paper is organized as follows: an introduction to parametrization schemes in the SLUCM, SCM, and the coupled SLUCM–SCM framework is presented in Sect. 2. In Sect. 3, we evaluate the model performance of both the SLUCM and SCM using field measurements of surface energy fluxes and vertical profiles of temperature and humidity in the ABL. The coupled SLUCM–SCM framework is then applied to study the transport of heat and moisture in the integrated urban land–atmosphere system with different landscape characteristics, e.g. by changing urban geometry, surface albedo, vegetation fraction and aerodynamic roughness; results are presented in Sect. 4. The corresponding variation in ABL responses (primarily the growth of the ABL and the evolution of temperature and humidity profiles) helps to demonstrate the effectiveness of different urban planning strategies, particularly for UHI mitigation such as cool and green roofs. Lastly, the main results are summarized in Sect. 5, followed by concluding remarks.

2 Methodology

In this section, the numerical framework of the coupled urban land–atmosphere model is introduced. The urban surface layer is parametrized using the SLUCM while the overlying mixed layer is parametrized using the SCM. A schematic of the coupled SLUCM–SCM framework is shown in Fig. 1. The SLUCM–SCM framework is essentially one-dimensional with assumed horizontal homogeneity, and is appropriate for representing the ABL under free convection conditions.

Fig. 1

Schematic of the coupled SLUCM–SCM framework: urban land-surface processes are parametrized by the SLUCM and the overlying ABL is represented by the SCM

2.1 Land-Surface Processes Represented by the SLUCM

To capture the coupled transport and co-evolution of water and energy budgets in a built environment, here we adopt the latest SLUCM with a realistic urban hydrological model developed by Wang et al. (2011a, 2013). The SLUCM employs the common single-layer street canyon representation for urban areas (Nunez and Oke 1977; Masson 2000; Kusaka et al. 2001) (see Fig. 1). The model explicitly resolves radiative trapping and shading effects inside the street canyon, taking into account the canyon orientation and the diurnal variation of solar azimuth angle. Besides, surface heterogeneity for each urban facet (i.e. roof, wall, and ground) is also included: e.g. the ground can consist of, but is not limited to, engineered (asphalt or concrete) pavements, vegetation or bare soil; similarly, wall materials can be brick or glass; roofs can be paved or vegetated. Furthermore, the urban hydrological module in the SLUCM is capable of predicting water transport over both natural and engineered surfaces, especially evapotranspiration from urban lawns and water retention on porous pavements. Forced by air temperature, humidity, pressure, wind speed and shortwave and longwave radiation fluxes, the SLUCM not only predicts the surface energy balance (i.e. net radiation, surface temperature and sensible and ground heat fluxes), but also hydrological processes (infiltration, evapotranspiration and irrigation) and sub-surface soil moisture states in urban areas.

Based on the assumption that the thermal energy involved in advection, radiative flux divergence, and canyon air temperature change is small in comparison with the energy stored in urban surfaces (Nunez and Oke 1977), the energy balance in the SLUCM for the whole urban canopy layer is given by,

$$\begin{aligned} R_\mathrm{n} +A_\mathrm{F} =H_\mathrm{u} +LE_\mathrm{u} +G_0, \end{aligned}$$

(1)

where $R_\mathrm{n}$ is the net radiation, $A_\mathrm{F}$ is the anthropogenic heat and moisture fluxes, $H_\mathrm{u}$ and $LE_\mathrm{u}$ are the turbulent sensible and latent heat fluxes arising from the entire urban canopy layer with the subscript $u$ denoting the urban canopy, and $G_{0}$ is the ground (conductive) heat flux aggregated over all urban facets, taking into account the actual thickness and thermal mass of roofs, walls and ground.

The net radiation $R_\mathrm{n}$ for a generic urban facet (such as a roof) is calculated as

$$\begin{aligned} R_\mathrm{n} =S^{\downarrow }+L^{\downarrow }-S^{\uparrow }-L^{\uparrow }, \end{aligned}$$

(2)

where $S^{\downarrow }$ and $L^{\downarrow }$ are the downwelling shortwave and longwave radiative fluxes respectively, $S^{\uparrow } = {aS}^{\downarrow }$ is the upwelling shortwave radiative flux with $a$ the surface albedo, and $L^{\uparrow } = \varepsilon \sigma T_\mathrm{s}^{4}$ is the upwelling longwave radiative flux, $\varepsilon$ is the emissivity, $\sigma$ is the Stefan-Boltzmann constant, and $T_\mathrm{s}$ is the surface temperature. The computation of net radiation inside a street canyon involves shading and radiative trapping effects, as detailed in Sect. 4.1.

The total turbulent fluxes $H_\mathrm{u}$ and ${LE}_\mathrm{u}$ from the urban area can be obtained as the areal averages of the fluxes from roof and canyon, viz.

$$\begin{aligned} H_\mathrm{u}&= r\sum _{k=1}^{N_\mathrm{R}} {f_{\mathrm{R},k} H_{\mathrm{R},k} } +wH_\mathrm{can}, \end{aligned}$$

(3)

$$\begin{aligned} LE_\mathrm{u}&= r\sum _{k=1}^{N_\mathrm{R}} {f_{\mathrm{R},k} LE_{\mathrm{R},k} } +wLE_\mathrm{can}, \end{aligned}$$

(4)

while the canyon turbulent fluxes are aggregated from all canyon sub-facets, i.e. walls and ground, viz.

$$\begin{aligned} H_\mathrm{can}&= \frac{2h}{w}\sum _{k=1}^{N_\mathrm{W} } {f_{\mathrm{W},k} H_{\mathrm{W},k} } +\sum _{k=1}^{N_\mathrm{G} } {f_{\mathrm{G},k} H_{\mathrm{G},k}}, \end{aligned}$$

(5)

$$\begin{aligned} LE_\mathrm{can}&= \frac{2h}{w}\sum _{k=1}^{N_\mathrm{W}} {f_{\mathrm{W},k} LE_{\mathrm{W},k} } +\sum _{k=1}^{N_\mathrm{G} } {f_{\mathrm{G},k} LE_{\mathrm{G},k} }, \end{aligned}$$

(6)

where subscripts can, R, W, G denote street canyon, roof, wall, and ground respectively, $r = R \big / ({{R}}+{{W}})$, $h = H \big / (R+W)$ and $w = W \big /(R+W)$ are the normalized (dimensionless) roof width, building height and road width respectively, with $R$, $H$ and $W$ the corresponding physical dimensions, $N_\mathrm{R}$, $N_\mathrm{W}$ and $N_\mathrm{G}$ are the number of sub-facet types of roof, wall and ground, and $f_{\mathrm{R},k}$, $f_{\mathrm{W},k}$, and $f_{\mathrm{G},k}$ are the areal fractions of each sub-facet.

Sensible heat fluxes in the SLUCM are parametrized as (Masson 2000; Wang et al. 2013),

$$\begin{aligned} H=\frac{c_\mathrm{p} \rho _\mathrm{a} \left( {T_\mathrm{s} -T_\mathrm{a} } \right) }{r_\mathrm{a} }, \end{aligned}$$

(7)

for all urban facets, and latent heat fluxes are calculated from

$$\begin{aligned} LE_\mathrm{eng} =\left\{ {\begin{array}{ll} 0, &{} \hbox {if}\;\delta _w =0 \\ \frac{\rho _\mathrm{a} L_\mathrm{v} \left( {q_\mathrm{eng}^*-q_\mathrm{a} } \right) }{r_\mathrm{a}}, &{} \hbox {if}\;\delta _w >0\\ \end{array}} \right. , \end{aligned}$$

(8)

for engineered surfaces, and

$$\begin{aligned} LE_{nat} =\left\{ {\begin{array}{ll} \frac{L_\mathrm{v} \rho _\mathrm{a} \left( {q_\mathrm{veg}^*-q_\mathrm{a} } \right) }{r_\mathrm{a} +r_\mathrm{s}}, &{}\hbox {for vegetation} \\ \\ \beta _\mathrm{e} \frac{L_\mathrm{v} \rho _\mathrm{a} \left( {q_{soil}^*-q_\mathrm{a} } \right) }{r_\mathrm{a} }, &{} \hbox {for bare soil } \\ \end{array}} \right. , \end{aligned}$$

(9)

for natural surfaces, where $\rho _\mathrm{a}$ is the density of the air, $c_\mathrm{p}$ is the specific heat capacity of the air, $T_\mathrm{a}$ is air temperature, $\delta _\mathrm{w}$ is the actual depth of water retention on the engineered surface, $L_\mathrm{v}$ is the latent heat of water vaporization, $q_\mathrm{a}$ is the specific humidity of the air, $q^{*}$ is the saturated specific humidity, $r_\mathrm{a}$ is the aerodynamic resistance calculated using Monin–Obukhov similarity theory, $r_\mathrm{s}$ is the stomatal resistance, and $\beta _\mathrm{e}$ is a potential evaporation reduction factor. Here, we adopt a closed-form relation proposed by Mascart et al. (1995) for $r_\mathrm{a}$ as functions of surface roughness and atmospheric stability. The factor $\beta _\mathrm{e}$ reflects the constraint on actual evaporation by soil water availability and can be parametrized as (Brutsaert 2005)

$$\begin{aligned} \beta _\mathrm{e} =\frac{W-W_\mathrm{r}}{W_\mathrm{s} -W_\mathrm{r}}, \end{aligned}$$

(10)

where $W$ is the volumetric soil water content, and $W_\mathrm{s}$ and $W_\mathrm{r}$ are the saturated and residual soil water content respectively. The actual soil water content can be computed by solving the Richards equation,

$$\begin{aligned} \frac{\partial W}{\partial t}=\frac{\partial }{\partial z}\left( {D\frac{\partial W}{\partial z}+K+F_\mathrm{W}} \right) , \end{aligned}$$

(11)

where $D$ and $K$ are the hydraulic diffusivity and hydraulic conductivity for unsaturated soils respectively, estimated using Van Genuchten (1980), and $F_\mathrm{W}=P+Q_\mathrm{F} - Ro - E_\mathrm{T}$ is the water availability term with precipitation $P$, anthropogenic water $Q_\mathrm{F}$, surface run-off $Ro$, and evapotranspiration $E_\mathrm{T}$.

In addition, the thermal fields in solid media, i.e. temperatures and soil heat fluxes, are computed by solving the heat conduction equation based on the Green’s function approach (Wang et al. 2011a), where thermal conductivity $k$ and heat capacity $C$ are needed. The anthropogenic heat and water budgets are ignored due to the lack of experimental data, but are recommended to be included whenever data are available.

2.2 Atmospheric Boundary-Layer Processes Parametrized by the SCM

The evolution of the CBL, including the boundary-layer height and spatial distributions of temperature and humidity in the mixed layer, is parametrized in the SCM. Here the thermal field in the atmosphere is described using the virtual potential temperature $\theta _\mathrm{v}$, accounting for the effect of water vapour and pressure on boundary-layer stability (Ouwersloot and Vilà-Guerau de Arellano 2013). For the surface layer, mean profiles of the virtual potential temperature $\theta _\mathrm{v}$ and the specific humidity $q$ follow approximately logarithmic law distributions, based on Monin–Obukhov similarity theory (Businger et al. 1971; Stull 1988). In the CBL, mean profiles of $\theta _\mathrm{v}$ and $q$ are governed by the following diffusion equation (Troen and Mahrt 1986),

$$\begin{aligned} \frac{\partial X}{\partial t}=\frac{\partial }{\partial z}(-\overline{{w}'{X}'} ), \end{aligned}$$

(12)

where $X=\theta _\mathrm{v}$ or $q$ is a generic atmospheric state variable, $w$ is the vertical wind speed, and $\overline{{w}'{X}'}$ is the vertical kinematic eddy flux, with the overbar denoting the ensemble average.

The kinematic heat and moisture fluxes at the lower boundary of the CBL (or equivalently, at the top of the surface layer) can be derived from $H_\mathrm{u}$ and ${LE}_\mathrm{u}$, as predicted by the SLUCM,

$$\begin{aligned} \left( {\overline{{w}'\theta ^\prime } } \right) _\mathrm{s}&= \frac{H_\mathrm{u} }{\rho _\mathrm{a} c_\mathrm{p} }, \end{aligned}$$

(13)

$$\begin{aligned} \left( {\overline{{w}'{q}'} } \right) _\mathrm{s}&= \frac{LE_\mathrm{u} }{\rho _\mathrm{a} L_\mathrm{v} }, \end{aligned}$$

(14)

where the subscript $s$ denotes the atmospheric surface layer. From the definition of virtual potential temperature, we have

$$\begin{aligned} \left( {\overline{w^\prime \theta _\mathrm{v} ^\prime } } \right) _\mathrm{s} =0.61\overline{\theta }\left( {\overline{w^\prime q^\prime } } \right) _\mathrm{s} +\left( {1+0.61\overline{q} } \right) \left( {\overline{w^\prime \theta ^\prime } } \right) _\mathrm{s}. \end{aligned}$$

(15)

The upper boundary conditions at the top of the CBL are prescribed by Kim et al. (2006),

$$\begin{aligned} \left( {\overline{{w}'\theta _\mathrm{v} ^{\prime }} } \right) _{z_\mathrm{h} }&= -0.15\left( {\frac{\theta _\mathrm{v} }{g}} \right) w_\mathrm{m}^3/z_\mathrm{h}, \end{aligned}$$

(16)

$$\begin{aligned} \left( {\overline{{w}'q^\prime } } \right) _{z_\mathrm{h} }&\approx 0. \end{aligned}$$

(17)

A conventional method used to determine $z_\mathrm{h}$ involves a bulk Richardson number formulation based on the assumption that continuous turbulence vanishes beyond $z_\mathrm{h}$ (Troen and Mahrt 1986; Zilitinkevich and Baklanov 2002),

$$\begin{aligned} z_\mathrm{h} =\frac{Ri_\mathrm{Bc} U^{2}}{\beta \Delta \theta _\mathrm{v} }, \end{aligned}$$

(18)

where ${Ri}_\mathrm{Bc}$ is the critical bulk Richardson number, $\beta =g/T_{0}$ is the buoyancy parameter with $g$ the acceleration due to gravity and $T_{0}$ is the reference temperature, $U$ is the horizontal wind speed at $z_\mathrm{h}$, and $\Delta \theta _\mathrm{v}=\theta _\mathrm{v}(z_\mathrm{h})-\theta _\mathrm{s}$ with $\theta _\mathrm{s}$ calculated from

$$\begin{aligned} \theta _\mathrm{s} =\theta _\mathrm{v} (z_1 )+\theta _\mathrm{T}. \end{aligned}$$

(19)

Here $\theta _\mathrm{T}$ is the scaled potential temperature excess in the surface layer, given by

$$\begin{aligned} \theta _\mathrm{T} =C\frac{\left( {\overline{w^\prime \theta ^\prime } } \right) _\mathrm{s} }{w_\mathrm{s}}, \end{aligned}$$

(20)

where $C$ is a coefficient of proportionality (often set as 6.5 according to Troen and Mahrt 1986). The variable $w_\mathrm{s}$ is the velocity scale for the entire ABL, defined as

$$\begin{aligned} w_\mathrm{s} =(u_*^3 +7\varepsilon \kappa w_*^3 )^{1/3}, \end{aligned}$$

(21)

where $u_*$ is the surface friction velocity, $\varepsilon = z_{1}/z_\mathrm{h} \approx 0.1$ is the ratio of the surface-layer height to that of the ABL, $\kappa$ is the von Karman constant, and $w_*$ is the convective velocity scale given by

$$\begin{aligned} w_*=\left[ {\frac{g}{T_0 }\left( {\overline{{w}'\theta ^{\prime }}} \right) _\mathrm{s} z_\mathrm{h} } \right] ^{1/3}. \end{aligned}$$

(22)

Although the bulk Richardson number method outlined above is a widely used approach to estimate the boundary-layer height adopted by many numerical weather prediction models, the method is limited by the uncertainty associated with the selection of the $Ri_\mathrm{Bc}$ value that varies with surface roughness and flow history and has high spatial variability (Zilitinkevich and Baklanov 2002; Jeričević and Grisogono 2006), leading to numerical instability. In this study, we adopt a recently developed analytical solution to determine $z_\mathrm{h}$ including the effect of moisture (Ouwersloot and Vilà-Guerau de Arellano 2013),

$$\begin{aligned} z_\mathrm{h} =\left\{ {\begin{array}{l} z_{h0}^2 +\frac{\left( {2+4w_\mathrm{e}} \right) }{\gamma _{\theta \mathrm{v}} }\left[ {\Delta \theta _{\mathrm{v},0} z_{h0}^{\frac{1+w_\mathrm{e}}{w_\mathrm{e}}} -\left( {\frac{w_\mathrm{e}}{1+2w_\mathrm{e}}} \right) \gamma _{\theta \mathrm{v}} z_{h0}^{\frac{1+2w_\mathrm{e}}{w_\mathrm{e}}} } \right] \left( {\hat{{z}}_\mathrm{h} ^{-\frac{1}{w_\mathrm{e}}}-z_{h0}^{-\frac{1}{w_\mathrm{e}}} } \right) \\ +\left( {\frac{2+4w_\mathrm{e}}{\gamma _{\theta \mathrm{v}} }} \right) \int \limits _{t_0 }^t {\left( {\overline{w^\prime \theta _\mathrm{v}^\prime } } \right) _\mathrm{s}} {\text {d}}t \\ \end{array}} \right\} ^{1/2}, \end{aligned}$$

(23)

where $z_{h0}$ is the initial CBL height, $w_\mathrm{e}$ is the entrainment rate at the inversion, $\gamma _{\theta \mathrm{v}}$ is the lapse rate in the free atmosphere, $\Delta \theta _\mathrm{s}$ is the potential temperature difference across the inversion, and $\hat{{z}}_\mathrm{h}$ is a correction term given by

$$\begin{aligned} \hat{{z}}_\mathrm{h} =\left[ {z_{h0}^2 +\left( {\frac{2+4w_\mathrm{e}}{\gamma _{\theta \mathrm{v}} }} \right) \int \limits _{t_0 }^t {\left( {\overline{w^\prime \theta _\mathrm{v}^\prime } } \right) _\mathrm{s}} {\text {d}}t} \right] ^{1/2}. \end{aligned}$$

(24)

Accounting for the non-local mixing and entrainment effects at the ABL top, the heat and moisture fluxes in the mixed layer can be parametrized as (Noh et al. 2003)

$$\begin{aligned} -\overline{{w}'\theta _\mathrm{v}^{\prime } }&= K_\mathrm{h} \left( {\frac{\partial \theta _\mathrm{v} }{\partial z}-\gamma _\mathrm{h} } \right) -\left( {\overline{{w}'\theta _\mathrm{v}^\prime } } \right) _\mathrm{h} \left( {\frac{z}{z_\mathrm{h} }} \right) ^{3}, \end{aligned}$$

(25)

$$\begin{aligned} -\overline{{w}'{q}'}&= K_\mathrm{h} \left( {\frac{\partial q}{\partial z}-\gamma _q } \right) -\left( {\overline{{w}'{q}'} } \right) _\mathrm{h} \left( {\frac{z}{z_\mathrm{h} }} \right) ^{3}, \end{aligned}$$

(26)

where $K_\mathrm{h}$ is the turbulent diffusivity. The non-local mixing terms $\gamma _\mathrm{h}$ and $\gamma _{q}$ are parametrized by Troen and Mahrt (1986) and Noh et al. (2003)

$$\begin{aligned} \gamma _\mathrm{h}&= C\frac{\left( {\overline{{w}'{\theta }'} } \right) _\mathrm{s}}{w_\mathrm{s} z_\mathrm{h}}, \end{aligned}$$

(27)

$$\begin{aligned} \gamma _q&= C\frac{\left( {\overline{{w}'{q}'} } \right) _\mathrm{s} }{w_\mathrm{s} z_\mathrm{h} }. \end{aligned}$$

(28)

Using the SCM schemes outlined above, we can then estimate the profiles of $\theta _\mathrm{v}$ and $q$ in the mixed layer. In addition, the following assumptions are used to obtain the spatial distributions of $\theta _\mathrm{v}$ and $q$ in the entrainment zone,

• (a) According to Deardorff (1980), the thickness of the entrainment zone $\delta$ is estimated by

  $$\begin{aligned} \frac{\delta }{z_\mathrm{h} }=d_1 +\frac{d_2 }{Ri_*}, \end{aligned}$$

  (29)

  where $d_{1} = 0.02$ and $d_{2} = 0.05$ are empirical constants (Noh et al. 2003), and $Ri_*=(g/T_0 )z_\mathrm{h} \Delta \theta /w_*^2$ is the convective Richardson number.

• (b) The heat or moisture flux in the entrainment zone is proportional to the jump in $\theta _\mathrm{v}$ or $q$ at the inversion (Hong et al. 2006). Specifically

  $$\begin{aligned} \Delta \theta _\mathrm{v} |_{z_\mathrm{h} }&= \overline{\left( {w^\prime \theta _\mathrm{v}^\prime } \right) _{z_\mathrm{h}} }/w_\mathrm{e}, \end{aligned}$$

  (30)
  $$\begin{aligned} \Delta q|_{z_\mathrm{h} }&= \overline{\left( {w^{\prime }q^{\prime }}\right) _{z_\mathrm{h}}}/w_\mathrm{e}, \end{aligned}$$

  (31)

  The entrainment rate $w_\mathrm{e}$ is typically in the range 0.01 to $0.2\hbox { m s}^{-1}$ (Stull 1988).

• (c) The lapse rates $\gamma _{\theta \mathrm{v}}$ and $\gamma _{q}$ are considered to be constant in the free troposphere above the CBL (Kim et al. 2006; Ouwersloot and Vilà-Guerau de Arellano 2013).

Based on the above assumptions, the profiles of $\theta _\mathrm{v}$ and $q$ in the entrainment zone and the free atmosphere can be estimated as

$$\begin{aligned} \theta _\mathrm{v}&= \left\{ {\begin{array}{l@{\quad }l} \theta _{vm} +\frac{\Delta \theta _\mathrm{v} }{\delta }\left( {z-z_\mathrm{h}} \right) , &{} z_\mathrm{h} <z<\left( {z_\mathrm{h} +\delta } \right) \\ \theta _{vm} +\Delta \theta _\mathrm{v} +\gamma _{\theta \mathrm{v}} \left[ {z-\left( {z_\mathrm{h} +\delta } \right) } \right] , &{} z>\left( {z_\mathrm{h} +\delta } \right) \\ \end{array}} \right. , \end{aligned}$$

(32)

$$\begin{aligned} q&= \left\{ {\begin{array}{l@{\quad }l} q_m +\frac{\Delta q}{\delta }\left( {z-z_\mathrm{h} } \right) , &{} z_\mathrm{h} <z<\left( {z_\mathrm{h} +\delta } \right) \\ q_m +\Delta q+\gamma _q \left[ {z-\left( {z_\mathrm{h} +\delta } \right) } \right] , &{} z>\left( {z_\mathrm{h} +\delta } \right) \\ \end{array}} \right. , \end{aligned}$$

(33)

where $\theta _{m}$ and $q_{m}$ are the virtual potential temperature and specific humidity at the top of the mixed layer (Kim et al. 2006).

3 Model Evaluation

3.1 Evaluation of the SLUCM

Model predictions by the SLUCM are compared against field measurements from two eddy-covariance (EC) towers located at Phoenix, Arizona and Princeton, New Jersey, USA. Site information of the two EC towers is described in Table 1 (see Chow et al. 2014; Sun et al. 2013b; Ramamurthy et al. 2014 for more details). Meteorological forcing data used to test the SLUCM were collected from June 12–17, 2012 (all clear days) during a pre-monsoon season at Phoenix, and from May 4–9, 2010 for Princeton covering a variety of weather conditions. Results of the comparison between model predictions and field measurements are shown in Fig. 2, which includes net radiation, sensible heat and latent heat fluxes. The root-mean-square errors (RMSE) for $R_{n}$,$H_\mathrm{u}$, ${LE}_\mathrm{u}$ are 20, 34, 20 W m$^{-2}$ respectively for the Phoenix site, and 16, 37, 18 W m$^{-2}$ respectively for the Princeton site. It is clear that the SLUCM is capable of predicting the surface energy budget with reasonable accuracy. The realistic representation of land-surface processes by the SLUCM, especially turbulent sensible and latent heat fluxes arising from urban canopies, then provide reliable boundary conditions to the overlying CBL.

Fig. 2

Comparison of predicted sensible heat, latent heat and net radiative fluxes by the SLUCM and field measurements at a Phoenix, Arizona from June 12 to June 17 2012, and b Princeton, New Jersey from May 4 to May 9 2010

Table 1 Geographic information of the two sites at Phoenix, Arizona and Princeton, New Jersey, USA

3.2 Evaluation of the SCM

To assess the SCM performance, two sets of atmospheric profiling data are used, including, (1) measurements for March 26 2005, at the Point Reyes site ($38^{\circ }5'\,27.6''\hbox {N}$, $122^{\circ }57'\,25.80''\hbox {W}$) in California, USA based on a balloon-borne sounding system (SONDE) (Atmospheric Radiation Measurement Program, 2011), and (2) Day 33 data (August 17 1967) from the Wangara experiment at Hay, New South Wales ($34^{\circ }\,30'\, \hbox {S}$, $144^{\circ }\,56'\, \hbox {E}$) (Clarke et al. 1971). Weather conditions during the measurements at both sites were highly convective, cloudless and free of synoptic frontal influences.

For the Point Reyes site, the surface sensible and latent heat fluxes and atmospheric temperature and humidity profiles were measured at 0930 PST (local time) on March 26 2005. The SCM is initialized with neutral virtual potential temperature and humidity profiles at the time when the CBL starts to develop (early morning). The initial values of temperature and humidity in the CBL are set to be the same as values at the top of the surface layer. Driven by the surface sensible and latent heat fluxes, the SCM determines corresponding atmospheric temperature and humidity profiles. Comparisons between predicted and measured vertical distributions of temperature and humidity in the mixed layer are in Fig. 3a, b. The RMSE values are 0.15 K for mean $\theta _\mathrm{v}$ and $1.07\times 10^{-4}\, \hbox {kg kg}^{-1}$ for mean $q$ in the mixed layer.

Fig. 3

Comparison of the SCM predictions and field measurements of a virtual potential temperature, and b specific humidity at 0930 PST on March 26 2005 at the Point Reyes site, California, USA, and c virtual potential temperature and d specific humidity at 1200 and 1500 AEST on Day 33 of the Wangara experiment in New South Wales, AUS

At the Wangara site, there were no direct measurements of surface sensible and latent heat fluxes. To drive the SCM, a simple “slab” model, reduced from the SLUCM without the presence of street canyons, was used to calculate these heat fluxes with the measured surface meteorological forcing, including net radiation, ground heat flux, near-surface air temperature/humidity, atmospheric pressure and wind speed. Figure 3c, d shows comparisons of the SCM predictions and measurements of $\theta _\mathrm{v}$ and $q$ at 1200 and 1500 AEST (local time). The RMSE values are 0.55 K and 0.48 K of mean $\theta _\mathrm{v}$ in the mixed layer, and $1.02\times 10^{-4}\, \hbox {kg\,kg}^{-1}$ and $9.23\times 10^{-5}\, \hbox {kg\,kg}^{-1}$ at 1200 and 1500 AEST respectively.

It is noteworthy that, up to date, the SCM has not been tested for urban areas due to the paucity of experimental measurements of ABL profiles over built terrains. In fact, ABL profile measurements in urban areas, especially in large cities, remain a challenge because of practical difficulties associated with logistics, instrument deployment and maintenance. Nevertheless, realistic atmospheric profiles in the urban CBL can be modelled numerically, given reliable boundary conditions for the surface layer (i.e. sensible and latent heat fluxes). Thus the coupled SLUCM–SCM framework developed herein is essential to quantify the transport of energy and moisture at the interface of the urban land–atmosphere system.

4 Case Study of the Coupled SLUCM–SCM

In this section, four cases are designed to assess the impact of urban LULC changes on the atmosphere using the coupled SLUCM–SCM framework, by changing (1) street canyon aspect ratio, (2) roof albedo, (3) roof vegetation fraction, and (4) aerodynamic roughness length. Given that the SLUCM predictions are more sensitive to roof properties (Loridan et al. 2010; Wang et al. 2011b), we focus more on the effect of roof characteristics than that of the street canyon. The coupled model is driven by meteorological forcing (air temperature, pressure, humidity, wind speed, and downwelling radiation) for a sunny day (June 13 2012) in Phoenix, Arizona, with sunrise at 0518 MST (local time) and sunset at 1939 MST. The list of input parameters required by the coupled SLUCM–SCM is presented Table 2, with calibrated values from Yang and Wang (2014). Surface temperatures for different urban canyon facets (roof, wall, and ground) were initialized based on the availability of field measurements from weather stations.

Table 2 List of input parameters for the coupled SLUCM–SCM framework

4.1 Effects of Canyon-Aspect Ratio

The canyon-aspect ratio, defined as $h/w$, is a primary indicator of urban morphology in the SLUCM, and has a significant impact on the partitioning of radiative heat inside a street canyon (Theeuwes et al. 2014). A detailed formulation of the shortwave and longwave radiative fluxes in a street canyon, as functions of the canyon dimension, is presented in Appendix 1. Increasing $h/w$ values imply that the morphology of a built environment changes from sparse to dense building arrays or from shallow to deep street canyons. The variation of CBL height corresponding to $h/w = 0.25$, 2, and 8 is shown in Fig. 4a, where transition occurs at about 10.5 h after the formation of the CBL. According to Eq. 23, the evolution of CBL height is strongly related to the surface sensible heat flux as shown in Fig. 5, where a non-linear relation is found between $H_\mathrm{s}$ and $h/w$, i.e. larger $h/w$ values lead to larger $H_\mathrm{s}$ in early morning and late afternoon but smaller $H_\mathrm{s}$ in the middle of the day. We speculate that the main contributor to this non-linear effect is the evolution of surface temperatures and heat fluxes inside the street canyon, governed by two counteracting processes, viz. the shading effect of the direct shortwave radiation and the trapping effect of diffuse shortwave radiation and longwave radiation. Schematics of the shading and trapping effects for urban canyons with different building aspect ratios are presented in Fig. 6.

Fig. 4

Time evolution of the ABL height $z_\mathrm{h}$ with different land-surface characteristics: a aspect ratio $h/w = 0.25$, 2 and 8; b roof albedo $a_\mathrm{R,c} = 0.05$, 0.2 and 0.6; c roof vegetation fraction $f_\mathrm{veg} = 0$, 0.5 and 0.9; and d roof aerodynamic roughness $z_\mathrm{m,R} = 0.1$, 1 and 10 mm

Fig. 5

Time evolution of sensible heat flux in the surface layer, $H_\mathrm{s}$, for canyon aspect ratio $h/w = 0.25$, 2 and 8, respectively. The corresponding kinematic heat flux $\left( {\overline{{w}'{\theta }'}} \right) _\mathrm{s}$ values are also indicated plotted on the right (the bracket notation is used instead of overbar for the ensemble mean on the axis)

Fig. 6

Illustration of radiative trapping and shading effects in street canyons with different aspect ratios, at different time of the day: a early morning, b noon, and c late afternoon, with various angles of incidence of solar radiation. Note that the sketched canyon dimensions are not to scale and do not present the actual street canyons

To elaborate on the governing mechanisms, Fig. 7 conceptually illustrates the relationship between the change in canyon surface temperature $\Delta T_\mathrm{can}$, averaged over walls and ground, and building-aspect ratios and zenith angles. First, with the same magnitude of incoming solar radiation, canyon-aspect ratio is a significant factor in dictating $\Delta T_\mathrm{can}$. Canyons with larger $h/w$, i.e. taller buildings or narrower streets, tend to be warmer since more heat is trapped due to more reflections between canyon facets (trapping effect). In contrast, canyons with larger $h/w$ are cooler because of a larger shaded area (shading effect). The combined effect due to $h/w$ leads to a maximum $\Delta T_\mathrm{can}$ in the moderate$h/w$ range, as shown in Fig. 7a (Theeuwes et al. 2014). Secondly, the zenith angle of incoming solar radiation is also important for radiative trapping and shading effects. The zenith angle is directly related to, (i) the number of reflections experienced by a radiative ray inside the street canyon, and (ii) the magnitude of the incoming solar radiation depending on the time of the day. If zenith angle is close to 90 degrees, i.e. early morning or late afternoon, the trapping effect is significant due to multiple radiative reflections between canyon facets, while the shading effect is small due to low incoming solar radiation. In contrast, the street canyon exhibits a smaller trapping effect and a larger shading effect around noon when the zenith angle is close to zero. The resultant effect on $\Delta T_\mathrm{can}$ as a function of the zenith angle is shown in Fig. 7b. Overall, interactions between radiative trapping and shading effects lead to the non-linear effect of $h/w$ on the sensible heat flux arising from the urban canopy during daytime (Fig. 5), which is then manifest in the diurnal evolution of the CBL height (Fig. 4a).

Fig. 7

Conceptual sketches of the effects of radiative trapping and shading on changes of canyon surface temperatures as a function of a aspect ratio, and b time of day

The effect of $h/w$ on the spatial distribution of $\theta _\mathrm{v}$ in the mixed layer is presented in Fig. 8. To illustrate the temporal variation of the $\theta _\mathrm{v}$ profiles, we select six times for comparison, viz. early morning (0700 MST), late morning (1000 MST), noon (1200 MST), early afternoon (1400 MST), late afternoon (1600 MST) and dusk (1900 MST). In the morning, when the trapping effect is dominant, the canyon with $h/w = 8$ exhibits largest $\theta _\mathrm{v}$, while the differences among $h/w$ cases decrease with time. This can be interpreted as the trapping effect diminishes and the shading effect becomes dominant due to the change of zenith angle. This trend continues in the afternoon until the excess temperature arising from dense urban areas (large $h/w$) is completely offset by the shading effect. Eventually, close to dusk, the trapping effect overtakes the shading effect again when the solar angle of incidence decreases. But the combined effect of complex radiative interactions results in the street canyon with moderate $h/w = 2$ has the highest $\theta _\mathrm{v}$ in the mixed layer, consistent with the speculation illustrated in Fig. 7a.

4.2 Effects of Surface Albedo

Roofs with high albedo (customarily referred to as “cool roofs” in building communities) have attracted extensive interest as a popular strategy to mitigate the canopy UHI (Akbari et al. 2012; Jacobson and Hoeve 2012). However, the impact of surface albedo on the ABL over an urban area is less frequently discussed. Here we simulated three cases with different albedos of conventional (paved) roofs, viz. $a_\mathrm{R,c} = 0.05$, 0.2 and 0.6, to investigate the effect of cool roofs on the CBL temperature. Physically, higher albedo leads to lower roof surface temperature, which reduces sensible heat fluxes above the roof and the urban surface, and reduces the CBL height in turn, as shown in Fig. 4b. In Fig. 9, it is clear that higher $a_\mathrm{R,c}$ values also result in lower $\theta _\mathrm{v}$ in the mixed layer. Note that the simulation results in this case are based on meteorological conditions for a clear day in a desert city without clouds and aerosols in the ABL. In the presence of clouds and aerosols, the transport of shortwave and longwave radiation in the ABL, atmospheric stability and the land-surface energy partitioning are all modified (Stull 1988). Such complexities are not incorporated in the current numerical framework.

Fig. 8

Model predictions of virtual potential temperature profiles in the ABL at 0700, 1000, 1200, 1400, 1600 and 1900 MST for canyon aspect ratios $h/w = 0.25$, 2 and 8

Fig. 9

Model predictions of virtual potential temperature profiles in the ABL at 0700, 1000, 1200, 1400, 1600 and 1900 MST for roof albedo $a_\mathrm{R,c} = 0.05$, 0.2 and 0.6

4.3 Effects of Vegetation Fraction

“Green” urban infrastructures, e.g. green roofs, urban lawns, urban agriculture, etc., are gaining increasing popularity as an effective means of reducing adverse urban environmental effects such as UHI mitigation, stormwater management and the preservation of ecological diversity (Dvorak and Volder 2010; Sailor et al. 2012; Sun et al. 2013a). Previous studies using the SLUCM have found that the vegetation fraction in built environments is critical in determining the vertical transport of heat and moisture within urban canopies (Wang et al. 2011b; Yang and Wang 2014). Using the coupled SLUCM–SCM framework, we choose vegetated roof fractions $f_\mathrm{veg} = 0, 0.5, 0.9$ to investigate the effect of green roofs on the CBL. The growth of ABL height is presented in Fig. 4c, while the distributions of $\theta _\mathrm{v}$ and $q$ in the mixed layer are shown in Figs. 10 and 11, for different $f_\mathrm{veg}$. In general, larger $f_\mathrm{veg}$ values lead to greater latent heat and smaller sensible heat fluxes. Consequently, an increase in green roof fraction in urban canopies results in a significant cooling effect, which can effectively “penetrate” throughout the entire CBL.

Fig. 10

Model predictions of virtual potential temperature profiles in the ABL at 0700, 1000, 1200, 1400, 1600 and 1900 MST for roof vegetation fraction $f_\mathrm{veg} = 0$, 0.5 and 0.9

Fig. 11

Model predictions of specific humidity profiles in the ABL at 0700, 1000, 1200, 1400, 1600 and 1900 MST for roof vegetation fraction $f_\mathrm{veg} = 0$, 0.5 and 0.9

4.4 Effects of Aerodynamic Roughness Length

The turbulent transport of momentum, heat and moisture in the ABL is sensitive to the surface aerodynamic roughness (Mascart et al. 1995). As shown in Eqs. 7–9, the aerodynamic resistance $r_\mathrm{a}$ directly affects the magnitude of sensible and latent heat fluxes. It is well-known that $r_\mathrm{a}$ decreases non-linearly with the surface roughness length (e.g. Stull 1988). In this case study, we modify the values of roof roughness length for momentum transfer, viz. $z_\mathrm{m,R} = 0.1, 1$, and 10 mm, and investigate the same set of ABL responses (i.e. $z_\mathrm{h}$ and mixed-layer $\theta _\mathrm{v}$ and $q$). It is expected that roofs with larger $z_\mathrm{m,R}$ induce lower $r_\mathrm{a}$, which in turn leads to larger $H_\mathrm{s}$ and ${LE}_\mathrm{s}$ due to the enhanced vertical turbulent transfer of heat and moisture. The enhanced sensible and latent heat fluxes from the surface layer then lead to higher $z_\mathrm{h}$ and higher $\theta _\mathrm{v}$ and $q$ in the mixed layer, as shown in Figs. 4d and 12 and 13. In the context of urban planning, particularly for UHI mitigation, results of this case study also demonstrate that altering roof roughness lengths is effective in regulating the transport of heat and moisture from built terrains to the overlying CBL, without fundamental changes to the urban morphology.

Fig. 12

Model predictions of virtual potential temperature profiles in the ABL at 0700, 1000, 1200, 1400, 1600 and 1900 MST for roof aerodynamic roughness length $z_\mathrm{m,R} = 0.1$, 1 and 10 mm

5 Concluding Remarks

A new SLUCM–SCM framework is developed for modelling the urban land–atmosphere interactions, with the SLUCM enabling the realistic representation of urban surface hydrologic processes including evapotranspiration, infiltration, irrigation, and sub-surface soil moisture. We test the model against field measurements of net radiation and sensible and latent heat fluxes in urban canopy layers, as well as vertical distributions of temperature and humidity and the evolution of ABL height. The coupled model is found to be robust and captures the vertical transport of heat and moisture from the surface layer to the overlying CBL via land–atmosphere interactions.

We then apply the numerical framework to study the impact of urban landscape characteristics, including morphology, albedo, vegetation fraction and aerodynamic roughness on the growth of the ABL and the distributions of temperature and humidity in the mixed layer under convective conditions. Results of case studies show that changes in land-surface properties (hydrothermal or geometric) have a significant impact on the evolution of the overlying boundary layer. In particular, the urban morphology, represented by the canyon-aspect ratio $h/w$, imposes non-linear effects on the ABL responses ($z_\mathrm{h}$ growth and $\theta _\mathrm{v}$ distribution in the mixed layer), through rather complex interactions of the opposing radiative trapping and shading effects co-evolving throughout the daytime. It is also found that widely-used urban planning strategies especially for surface UHI mitigation, such as cool and green roofs and modification of the vertical turbulent transfer through enhanced aerodynamic conductance, are effective in influencing the transport of momentum, heat and moisture in the urban boundary layer.

Fig. 13

Model predictions of specific humidity profiles in the ABL at 0700, 1000, 1200, 1400, 1600 and 1900 MST for roof aerodynamic roughness length $z_\mathrm{m,R} = 0.1$, 1 and 10 mm

Preliminary results from applications of the new SLUCM–SCM framework in the current study necessitate a few important questions for future research on urban land–atmosphere interactions. First, model uncertainties inherent in the parameter space, as well as parametrization schemes, need to be carefully quantified. With a relatively large number of input parameters, model uncertainty and sensitivity analysis necessarily requires computationally efficient numerical procedures, e.g. using advanced stochastic methods. It is also noteworthy that in the offline (stand-alone) setting, the current numerical framework has no predictive skill in the absence of projections of future atmospheric forcing. Thus it is of critical importance to run the model in an online setting, e.g. by incorporating the framework into numerical weather prediction models such as the WRF model. This way, the proposed numerical framework can be driven by meteorological forcing from numerical predictions at regional scales and become predictive for scenarios of city-scale land–atmosphere interactions downscaled from future climatic projections. With land-surface processes represented by the latest SLUCM, online simulations using the coupled framework should help to provide important guidelines for future development of cities with sustainable urban planning, e.g. UHI mitigation and adaptation strategies.

Appendix 1: Calculation of Net Radiation in a Street Canyon

The net shortwave and longwave radiative fluxes for walls and ground inside a street canyon can be computed using a two-reflection model (Kusaka et al. 2001; Wang et al. 2013) as,

$$\begin{aligned} S_\mathrm{W}&= (1-a_\mathrm{W})\left[ {\begin{array}{l} S_\mathrm{D} \frac{l_\mathrm{shadow} }{2h}+S_\mathrm{Q} F_{\mathrm{W}\rightarrow \mathrm{S}} +S_\mathrm{D} \left( {\frac{w-l_\mathrm{shadow} }{w}} \right) a_\mathrm{G} F_{\mathrm{W}\rightarrow \mathrm{G}} \\ +S_\mathrm{Q} F_{\mathrm{W}\rightarrow \mathrm{G}} +S_\mathrm{D} \frac{l_\mathrm{shadow} }{2h}a_\mathrm{W} F_{\mathrm{W}\rightarrow \mathrm{W}} +S_\mathrm{Q} a_\mathrm{W} F_{\mathrm{W}\rightarrow \mathrm{S}} F_{\mathrm{W}\rightarrow \mathrm{W}} \\ \end{array}} \right] , \end{aligned}$$

(34)

$$\begin{aligned} S_\mathrm{G}&= \left( {1-a_\mathrm{G} } \right) \left[ S_\mathrm{D} \left( {\frac{w-l_\mathrm{shadow} }{w}} \right) +S_\mathrm{Q} F_{\mathrm{G}\rightarrow \mathrm{S}} +S_\mathrm{D} \frac{l_\mathrm{shadow} }{2h}a_\mathrm{W} F_{\mathrm{G}\rightarrow \mathrm{W}}\right. \nonumber \\&\left. +\,S_\mathrm{Q} a_\mathrm{W} F_{\mathrm{W}\rightarrow \mathrm{S}} F_{\mathrm{G}\rightarrow \mathrm{W}} \right] , \end{aligned}$$

(35)

$$\begin{aligned} L_\mathrm{W}&= \varepsilon _\mathrm{W} \left( {F_{\mathrm{W}\rightarrow \mathrm{S}} L^{\downarrow }+\varepsilon _\mathrm{G} F_{\mathrm{W}\rightarrow \mathrm{G}} \sigma T_\mathrm{G}^4 +\varepsilon _\mathrm{W} F_{\mathrm{W}\rightarrow \mathrm{W}} \sigma T_\mathrm{W}^4 -\sigma T_\mathrm{W}^4 } \right) \nonumber \\&+\,\varepsilon _\mathrm{W} \left( {1-\varepsilon _\mathrm{G} } \right) L^{\downarrow }F_{\mathrm{G}\rightarrow \mathrm{S}} F_{\mathrm{W}\rightarrow \mathrm{G}} \nonumber \\&+\,2\left( {1-\varepsilon _\mathrm{G} } \right) \varepsilon _\mathrm{W} \sigma T_\mathrm{W}^4 F_{\mathrm{G}\rightarrow \mathrm{W}} F_{\mathrm{W}\rightarrow \mathrm{G}} +\varepsilon _\mathrm{W} \left( {1-\varepsilon _\mathrm{W} } \right) L^{\downarrow }F_{\mathrm{W}\rightarrow \mathrm{S}} F_{\mathrm{W}\rightarrow \mathrm{W}} \nonumber \\&+\,\left( {1-\varepsilon _\mathrm{W} } \right) \varepsilon _\mathrm{G} \sigma T_\mathrm{G}^4 F_{\mathrm{W}\rightarrow \mathrm{G}} F_{\mathrm{W}\rightarrow \mathrm{W}} +\varepsilon _\mathrm{W} \varepsilon _\mathrm{W} \left( {1-\varepsilon _\mathrm{W} } \right) \sigma T_\mathrm{W}^4 F_{\mathrm{W}\rightarrow \mathrm{W}} F_{\mathrm{W}\rightarrow \mathrm{W}}, \end{aligned}$$

(36)

$$\begin{aligned} L_\mathrm{G}&= \varepsilon _\mathrm{G} \left( {F_{\mathrm{G}\rightarrow \mathrm{S}} L^{\downarrow }+2\varepsilon _\mathrm{W} F_{\mathrm{G}\rightarrow \mathrm{W}} \sigma T_\mathrm{W}^4 -\sigma T_\mathrm{G}^4 } \right) +2\varepsilon _\mathrm{G} \left( {1-\varepsilon _\mathrm{W} } \right) F_{\mathrm{W}\rightarrow \mathrm{S}} F_{\mathrm{G}\rightarrow \mathrm{W}} L^{\downarrow } \nonumber \\&+\left( {1-\varepsilon _\mathrm{W} } \right) \varepsilon _\mathrm{G} F_{\mathrm{G}\rightarrow \mathrm{W}} F_{\mathrm{W}\rightarrow \mathrm{G}} \sigma T_\mathrm{G}^4 +2\varepsilon _\mathrm{G} \varepsilon _\mathrm{W} \left( {1-\varepsilon _\mathrm{W} } \right) F_{\mathrm{W}\rightarrow \mathrm{W}} F_{\mathrm{G}\rightarrow \mathrm{W}} \sigma T_\mathrm{W}^4, \end{aligned}$$

(37)

where $S_\mathrm{W}$ and $S_\mathrm{G}$ are the net shortwave radiative fluxes for wall and ground respectively, $L_\mathrm{W}$ and $L_\mathrm{G}$ are the net longwave radiative fluxes for wall and ground respectively, $S_\mathrm{D}$ and $S_\mathrm{Q}$ are the direct and diffuse solar radiative fluxes, $a$ is the albedo (solar reflectivity), $F_{i\rightarrow j}$ are the view factors for radiation emitted from a generic surface $i$ and received by surface $j$, and $l_\mathrm{shadow}$ is the normalized shadow length. The shadow length is estimated by (Kusaka et al. 2001),

$$\begin{aligned} l_\mathrm{shadow} =\left\{ {\begin{array}{l@{\quad }l} h\tan \theta _z \sin \theta _\mathrm{n}, &{} l_\mathrm{shadow} < w \\ w,&{} l_\mathrm{shadow} \ge w \\ \end{array}} \right. , \end{aligned}$$

(38)

where $\theta _{z}$ is the solar zenith angle, $\theta _{n}$ is the difference between the solar azimuth angle and canyon orientation. All view factors for radiative exchange between canyon facets are directly related to the aspect ratio $h/w$ (Wang 2010).