Urban land–atmosphere interactions can be captured by numerical modeling framework with coupled land surface and atmospheric processes, while the model performance depends largely on accurate input parameters. In this study, we use an advanced stochastic approach to quantify parameter uncertainty and model sensitivity of a coupled numerical framework for urban land–atmosphere interactions. It is found that the development of urban boundary layer is highly sensitive to surface characteristics of built terrains. Changes of both urban land use and geometry impose significant impact on the overlying urban boundary layer dynamics through modification on bottom boundary conditions, i.e., by altering surface energy partitioning and surface aerodynamic resistance, respectively. Hydrothermal properties of conventional and green roofs have different impacts on atmospheric dynamics due to different surface energy partitioning mechanisms. Urban geometry (represented by the canyon aspect ratio), however, has a significant nonlinear impact on boundary layer structure and temperature. Besides, managing rooftop roughness provides an alternative option to change the boundary layer thermal state through modification of the vertical turbulent transport. The sensitivity analysis deepens our insight into the fundamental physics of urban land–atmosphere interactions and provides useful guidance for urban planning under challenges of changing climate and continuous global urbanization.

Land surface connects soil layers and the overlying atmosphere by transferring momentum, heat, and water through the interface. Thus landscape characteristics are critical in determining surface heat and moisture fluxes, which in turn regulate the atmospheric boundary layer dynamics in, e.g., mesoscale atmospheric modeling (McCumber and Pielke, 1981). Despite significant improvements of climate model predictability made in last decades, significant uncertainty still exists in model structures (i.e., mechanisms and equations), model parameters, numerical stability consideration, and scale transition (e.g., downscaling) (Hargreaves, 2010; Maslin and Austin, 2012). Statistical analyses on observational and numerical data sets have shown that land–atmosphere interaction is an importance source of uncertainty in climate predictability (Betts et al., 1996; Orlowsky and Seneviratne, 2010; Trier et al., 2011). Land–atmosphere interactions have significant impacts on climate both temporally (from seasonal to interannual) and spatially (from local to global) (Seneviratne and Stöckli, 2008). The predictive skill and robustness of regional and global climate models can be significantly improved with a better representation of land–atmosphere interactions, especially the soil moisture/temperature/precipitation interactions (Chen and Avissar, 1994; Chen and Dudhia, 2001; Phillips and Klein, 2014; Seneviratne et al., 2010).

Unprecedented rate of rapid urban expansion in last few decades has led to numerous environmental problems such as the urban heat island (UHI) effect, degradation of air quality, and increase of building energy consumption (Arnfield, 2003). Numerical weather and climate model uncertainties are further complicated due to the presence of complex built terrains. With a relatively small areal coverage, urban areas are manifested as hotspots of modified hydrothermal properties, altered flow fields, high surface heterogeneity, and anthropogenic heat and moisture sources (Arnfield, 2003; Flagg and Taylor, 2011; Wang et al., 2011b). Through land–atmosphere coupling, urban areas further impact hydroclimate in regional and even global scales via modified surface energy and water cycles. Thus sensitivity analysis is critical to quantify model uncertainties and improve model predictability, as the model performance is largely dependent on the accuracy of input parameters. With prescribed atmospheric forcing (i.e., air temperature, pressure, humidity, wind speed, and solar radiation) such as by measurements in the surface layer, the convective boundary layer (CBL) dynamics are largely dictated by boundary conditions at the bottom and the top of the CBL. In particular, previous studies have found that critical parameters for urban land surface modeling include the urban morphology, the hydrothermal properties of roofs, and the characteristics of the inversion layer (Loridan et al., 2010; Wang et al., 2011a; Wong et al., 2011; Ouwersloot and Vilà-Guerau de Arellano, 2013).

The conventional approach to analyze model sensitivity is to change only one parameter at a time with all the other parameters fixed and compare the output results with the “control case” (i.e., results from original unchanged parameter sets). This approach, however, will result in high computational costs with large sets of parameters and potentially biased statistical correlations between uncertain parameters (viz. parameters subject to variability and lack of deterministic values in the analysis of interest). In contrast, statistical approaches handling the complete set of parameter uncertainty simultaneously in one simulation, e.g., those using Monte Carlo methods, are more suitable than the conventional sensitivity analysis (Wang et al., 2011a). For complex numerical frameworks involving multiple physics and large number of uncertain parameters, however, “curse of dimensionality” (i.e., a phenomenon that an algorithm works in low dimensions can break down in high dimensions) may arises in direct Monte Carlo simulations (MCS) (Bellman and Rand, 1957; Cherchi and Guevara, 2012). The curse of dimensionality necessitates more advanced Monte Carlo procedure, using importance sampling technique to improve computational efficiency, one example being the subset simulation model (Au and Beck, 2001). This model is based on Markov chain Monte Carlo (MCMC) procedure and particularly efficient for handling large dimensions of uncertain space and simulating small probability events (climate extremes, for example) with either short- or long-tail behavior. It has been used for risk, sensitivity, and extreme event analysis in a wide range of scientific applications including, e.g., seismic risk, fire safety, spacecraft thermal control, and climatic extremes (Au et al., 2007; Thunnissen et al., 2007; Wang et al., 2011a; Au and Wang, 2014; Yang and Wang, 2014b).

In this study, the subset simulation approach is adopted for sensitivity
analysis on urban land–atmosphere interactions. We will use the Phoenix
metropolitan as the prototype of cities in arid or semiarid regions. The
selection of this study area is primarily because Phoenix
has emerged as a hub of urban environmental study in last decades (Chow et
al., 2014) due to its undergoing rapid urban expansion, as well as the rich
portfolio of urban planning strategies adopted in this area. Located in the
northeast of the Sonoran Desert, Phoenix has a subtropical desert climate
with long hot summers with mean air temperature of 32

In addition, though the impact of landscape modification on urban environment has been extensively studied in last decades, most of the research focused on the thermal state in the urban canopy layer (from the ground to the tallest building height) or on the regional scale modeling of atmospheric dynamics with influence from synoptic scale (such as advection and cloud physics). In this study, the impact of urban land use changes will be assessed using a one-dimensional (1-D) numerical framework by coupling an urban land surface model with a single column atmospheric model (Song and Wang, 2015a), in order to single out the effect of direct land–atmosphere interactions primarily via turbulent transport in the vertical direction. Moreover, this new modeling framework enables us to look into changes of atmospheric dynamics due to landscape modification using physical planetary boundary layer parameterization, but not limited to the physics in the urban canopy layer (e.g., 2 m air temperature) prevailed in most previous study. The sensitivity analysis in this study will therefore allow us to ask fundamental questions: how effective is a certain urban mitigation approach in modifying the CBL structure and to what elevation? What alternative strategies do we have in urban landscape planning in addition to the popular options such as green/white roofs?

Schematics of coupled SLUCM–SCM framework: land surface processes are parameterized by a single layer urban canopy model; atmospheric processes under convective condition are parameterized by a single column model.

In this paper, urban land–atmosphere interactions are modeled using a 1-D stand-alone and scalable numerical framework (Song and Wang, 2015a), by coupling an advanced single layer urban canopy model (SLUCM) for urban land surface processes (Wang et al., 2011b; Wang et al., 2013) and a single column model (SCM) for boundary layer dynamics (Noh et al., 2003; Troen and Mahrt, 1986). To single out the direct impact of urban landscape modification, we test the sensitivity of the boundary layer only in the vertical direction without taking advection effect into consideration. The schematic of the coupled SLUCM–SCM framework is shown in Fig. 1, which captures three vertical layers. The lowest level is the surface layer, which is considered as the constant flux layer and consists of 10 % of the entire CBL with the built terrain located at the bottom. The middle level is a convective mixed layer, where distributions of temperature and humidity are determined by buoyant plumes arising from the surface layer and atmospheric turbulence. The top level is an entrainment zone with a temperature inversion, which inhibits upward mixing and confines subjacent air and pollution in the CBL. Temperature and humidity profiles in the entire vertical column are regulated by heat and moisture fluxes exchanged across the interfaces of two adjacent layers.

At the bottom of numerical framework, the urban canopy layer is
parameterized by a SLUCM, which is also adopted in the latest version of
Weather Research and Forecast (WRF) model (v3.7.1) (Yang et al., 2015). This
new SLUCM features enhanced urban hydrological processes coupled with the
urban energy balance model, which enables a more realistic representation of
the transport of energy and water over built terrains. The energy balance
equation for the urban canopy layer is given by

The turbulent sensible and latent heat fluxes arising from the urban area
(

The turbulent fluxes from street canyon are aggregated over walls and
the ground:

To resolve the overlying atmospheric boundary layer, a modified version of
the Yonsei University (YSU) boundary layer scheme commonly used in the WRF
model (Hong et al., 2006; Noh et al., 2003) was applied by incorporating an
analytical prognostic formula (Ouwersloot and Vilà-Guerau de Arellano,
2013) rather than a diagnostic formula related with Richardson number (Hong
et al., 2006) for determining the boundary layer height. In the mixed layer,
the governing equation for mean profiles of virtual potential temperature
and specific humidity due to boundary layer turbulence in SCM is given by
(Troen and Mahrt, 1986)

The upper boundary condition is at the height of CBL (

The turbulent kinematic heat and moisture fluxes at the upper boundary of
mixed layer (Hong et al., 2006; Kim et al., 2006) are

The kinematic turbulent heat and moisture flux in the mixed layer with the
account of non-local mixing and entrainment effect can be parameterized as
(Noh et al., 2003)

Comparison of simulated and measured atmospheric profiles of
virtual potential temperature

To evaluate the coupled SLUCM–SCM framework outlined in Sect. 2.1,
experiment data of temperature and humidity profiles were obtained from
NOAA/ESRL radiosonde database (

In urban climate modeling, the capability of assessing critical responses of atmospheric processes to urban land use/land cover change is of paramount significance for assessment of climatic extremes. The SLUCM–SCM framework coupling urban land surface processes and CBL dynamics involves a large number of input parameters, which leads to high dimensionality of input space for the following statistical analysis. Hence we adopt subset simulation (Au and Beck, 2001; Au and Wang, 2014) for subsequent sensitivity study, which is efficient in simulating rare (very small probability) events and robust for high dimensionality. Instead of simulating rare events as in direct MCS method with expensive computational cost, subset simulation breaks down extreme events with small exceedance probability into a sequence of more frequent events by introducing intermediate exceedance events. The targeted small exceedance probability is then expressed as a product of larger conditional probabilities of each intermediate event. In addition, MCMC technique is adopted based on effective accept/reject rules in subset simulations to improve computational efficiency.

Schematic of subset simulation procedure:

As illustrated in Fig. 3, the sampling technique employed in the subset simulation proceeds as follows: in level 0 (initial state), the
unconditional samples of uncertain parameters follow a prescribed
probability distribution function (PDF) (Fig. 3a). Conditional samples in
level 1 are defined using a given intermediate conditional probability

Comparison of the coefficient of variation of exceedance probability in subset simulation and direct MCS.

To evaluate the statistical quality of subset simulation, we computed the
coefficient of variation (c.o.v., defined as the ratio of the standard
deviation to the mean) using a typical statistical average of 30 independent
runs. The resulted c.o.v. of subset simulation as a function of exceedance
probability is shown in Fig. 4, where c.o.v. of direct MCS is also shown for
comparison. Estimate of c.o.v. of direct MCS can be analytically formulated
as [(1

The diurnal surface atmospheric forcing of 14 June 2012 (a clear
day) in Phoenix, AZ:

Input parameters of the coupled SLUCM–SCM numerical framework.

In this section, we apply subset simulation to analyze the sensitivity of the coupled SLUCM–SCM to different input parameters. The meteorological forcing in the surface layer was prescribed using field measurements of an eddy covariance tower on a clear day (14 June 2012) provided by the Central Arizona–Phoenix Long-Term Ecological Research (CAP LTER) project (Chow et al., 2014). The inputs of diurnal air temperature, relative humidity, and downwelling shortwave and longwave radiation are plotted in Fig. 5, with the daytime from 06:00 to 19:30 (local time) for the development of CBL. With the prescribed meteorological forcing, the surface sensible and latent heat fluxes are predicted by the SLUCM, which then in turn drive the SCM to estimate temperature and humidity profiles in the mixed layer. The input parameters of SLUCM–SCM (including surface dimensional and hydrothermal parameters for the SLUCM and atmospheric parameters for the SCM) are presented in Table 1. Note that the initial soil water content for green roofs in the SLUCM is set as 90 % saturated for the subsequent 13.5 h of simulation after the beginning of CBL development such that the evaporative power of green roofs is not constrained by soil water availability. Among the model inputs, 15 parameters are selected for subsequent sensitivity analysis, including 6 surface thermal parameters, 3 surface hydrological parameters, 4 surface dimensional parameters, and 2 atmospheric parameters, as listed in Table 2. In addition, PDFs of these parameters are determined based on previous studies (Ouwersloot and Vilà-Guerau de Arellano, 2013; Wang et al., 2011a; Yang and Wang, 2014b) and local conditions in our study area. Care must be taken here that this particular selection of uncertain parameter space is by no means exhaustive or unique and is subject to the limitation of parameterization used in the numerical framework, and the subsequent analysis can, at best, represent only the model physics. Since the initial parameter distribution by direct MCS are pivotal to the statistical sampling efficiency of subset simulations, PDFs for uncertain parameters are carefully selected to constitute a physically realistic parameter space. In addition, it was found that normal (Gaussian) distribution is more realistic for thermal and hydrological parameters with the expected value in a physical range having higher probability, while the distributions of dimensional (geometric) parameters are subject to engineering design and is therefore more uniform (Wang et al., 2011a). The two atmospheric parameters at the top of CBL (i.e., entrainment rate and lapse rate) are also set as uniform distribution to achieve same probability for different top boundary conditions according to Ouwersloot and Vilà-Guerau de Arellano (2013).

Summary of statistics of uncertain parameters used in the sensitivity study.

Three atmospheric variables, i.e., the critical CBL height (

Plots of exceedance probabilities versus various model responses averaged
over 30 simulations are presented in Fig. 6. The variations of critical
model outputs with three different green roof fractions indicate the
sensitivity of roof greening degrees on CBL dynamics. In Fig. 6a and b,
we monitored CBL height and virtual potential temperature of mixed layer
under three conditions of green roof fractions (i.e.,

Estimates of exceedance probabilities for model outputs of
critical

It is also noteworthy that there exist log concavities for the exceedance
probabilities of both critical

Histogram of conditional samples at different conditional levels
for

In general, for an uncertain parameter, the deviation between the
distribution of MCMC-generated conditional samples (in levels 1, 2, and 3)
and the initial prescribed distribution sampled using direct MCS (level 0)
indicates the significance of parameter sensitivity with respect to the
corresponding model output. Figure 7 shows the comparison between
conditional distribution (histograms) and initial distribution (dashed line)
for two sample parameters, i.e., heat capacity of green roof

To better quantify the parameter sensitivity, a percentage sensitivity index
(PSI) (Wang et al., 2011a) is adopted here to measure the model sensitivity
to an uncertain parameter

PSI values for model outputs of critical

PSI values of all uncertain parameters for three different monitored
outputs, i.e.,

The UHI effect has attracted significant effort are even heated debate from urban climate researchers and city planners. UHI is characterized by elevated temperature in built environments compared to surrounding rural areas (Oke, 1982). Major contributors of UHI include (a) excess storage of thermal energy due to radiative trapping by street canyon and thermal properties of pavement materials, (b) reduced vegetation cover and evaporative cooling, and (c) the release of anthropogenic heat, moisture, and greenhouse gases (Santamouris, 2014; Sun et al., 2013a). Correspondingly, there are several popular UHI mitigation strategies, including (1) changing canyon geometry (characterized by aspect ratio and roughness lengths) to alter the energy distribution through radiative shading and trapping; (2) changing thermal properties, such as installing cool roofs or cool pavements to reflect more solar radiation by increasing surface albedo; (3) adding green spaces, such as green roofs to increase evapotranspiration in urban area. We will discuss the effects of these UHI mitigation strategies on the overlying atmosphere based on the sensitivity study and its implication to urban planning.

Building geometry and density in an urban area have a significant impact on
the partitioning and redistribution of solar energy in the surface layer,
which in turn modulate the energy transport processes in the overlying
atmosphere. The canyon aspect ratio

Illustration of the nonlinear effect of aspect ratio

As shown in Fig. 8, CBL states (

It is also noteworthy in Fig. 8 that thermal properties of conventional
roofs and those of green roofs have opposite correlation to different CBL
dynamics, which can be explained by plausible mechanisms governing surface
energy balance. For a conventional roof, larger heat capacity implies that
more thermal energy is needed to heat the roof, while higher thermal
conductivity implies that less time is needed for heat dissipation, both
leading to lower roof surface temperature (Wang et al., 2011b). Lower roof
surface temperature will then reduce the sensible heat (given other
conditions invariant), causing lower CBL height and lower temperature in the
mixed layer, as shown in Fig. 8a and b. In Fig. 8c, it is shown that to
increase

Due to their ability to modify energy and water budgets in the urban surface
layer, city planners are increasingly using green roofs as an effective
strategy to mitigate UHI effect (Sailor et al., 2012; Susca et al., 2011;
Wang et al., 2016). In our study, four sets of green roof parameters are
studied: (1) thermal parameters, i.e.,

Threshold values at different conditional levels as functions of
green roof fractions for critical

In contrast, CBL dynamics are very sensitive to green roof width and areal
fractions, as they determine the area of green roof in a built environment,
which in turn strongly influence the soil water availability for
evaporation. It is shown that larger green roof width

Roughness lengths of momentum and heat transfer are important land surface
characteristics that regulate the aerodynamic resistance related to
turbulent transport of mass, momentum and energy in the surface layer
(Grimmond and Oke, 1999). Specifically, aerodynamic resistance is a function
of roughness length based on MOST (Mascart et al., 1995; Wang et al., 2013).
In this study, we set the roughness lengths of momentum at the roof level as
uncertain parameters for both conventional and green roofs. The roughness
lengths of heat transfer follow a simple parameterization that

In addition to urban landscape characteristics, the coupled SLUCM–SCM
numerical framework also involves physical parameterizations at the top of
CBL, i.e., in the inversion layer. The uncertainties of two atmospheric
parameters, namely the entrainment rate

In this study, we use an advanced Monte Carlo method to quantify the sensitivity of atmospheric boundary layer dynamics to urban land surface characteristics based on a coupled urban land–atmosphere model. Results show that in general the CBL dynamics over a built terrain are largely dictated by the urban geometry, roughness lengths, and hydrothermal properties of landscape materials. In particular, the urban geometry, represented by canyon aspect ratio, introduces a nonlinear impact on the CBL height and temperature. This is inherited from the nonlinear impact on bottom conditions of the CBL, viz. surface energy processes with two counteracting mechanisms of radiative trapping and shading in the street canyon. Specifically, rooftop planning strategies strongly modulate CBL dynamics. Besides, changing roughness lengths or thermal properties on rooftops (e.g., by planting different species of vegetation for green roofs or using porous pavement materials for conventional roofs) can also be effective means to reduce urban environmental temperatures in both the surface layer and the CBL.

In addition, we would like to reiterate here that results of sensitivity analysis in this study are based on the model physics of the stand-alone coupled SLUCM–SCM numerical framework; the actual urban land–atmosphere interactions involve more complicated physical processes in transferring momentum, heat, and moisture in the soil–land–atmosphere continuum. Nevertheless, as various research groups worldwide have extensively tested the numerical framework, either separately or in integrated platforms (e.g., WRF), we are confident that this physically based model captures the basic physics of urban land–atmosphere interactions. Results of sensitivity study of the numerical framework thus shed new light on the impact of urban land surface characteristics on the overlying atmosphere and provide useful guidelines for urban planning under future expansion and emergent climatic patterns.

Temperature and humidity profiles at the Phoenix Radiosonde site are obtained from NOAA/ESRL radiosonde database, and available at

This work is supported by the US National Science Foundation (NSF) under grant no. CBET-1435881 and CBET-1444758. The authors thank Melissa Wagner for the help in retrieving the radiosonde data in Phoenix and the two anonymous reviewers for their constructive feedback in improving the quality of this paper. The help of the Handling Editor, Stefan Buehler, is gratefully acknowledged.Edited by: S. Buehler