Deciphering the sensitivity of urban canopy air temperature to anthropogenic heat flux with a forcing-feedback framework

We propose a forcing-feedback framework to diagnose the sensitivity of air temperature within the UCL, also called urban canopy air temperature hereafter, to anthropogenic heat flux based on the energy budget of air within the UCL (figure 1). This conceptualization of the UCL is consistent with the theoretical underpinning of nearly all single-layer urban canopy models (UCMs) in weather and climate modeling, including the CLMU to be used in this study (more details on the CLMU are presented later). Our starting point is that the UCL is our control volume (or system of interest) and is the direct recipient of anthropogenic heat flux (i.e. the forcing). At steady state, the energy budget of the air within the UCL can be written as

$$\begin{align}0 = {Q_{AH}} + R\end{align} \tag{ 1 }$$

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where ${Q_{AH}}$ is the anthropogenic heat flux and $R$ is the sum of heat fluxes other than the anthropogenic heat flux (we shall say more about $R$ later). When the anthropogenic heat flux is altered by a certain amount (indicated by ${{\Delta }}$), the energy balance of air within the UCL reaches a new equilibrium state

$$\begin{align}0 = {{\Delta }}{Q_{AH}} + {{\Delta }}R\end{align} \tag{ 2 }$$

where $\Delta {Q_{AH}}$ can be interpreted as the added anthropogenic heat flux compared with the scenario without anthropogenic heat flux and ${{\Delta }}R$ is the total change of other heat fluxes in response to ${{\Delta }}{Q_{AH}}$.

Changes in other heat fluxes (${{\Delta }}R$) are often related to changes in the urban canopy air temperature (${{\Delta }}{T_a}$). Denoting ${{\Delta }}R = {\lambda _{{\text{all}}}}{{\Delta }}{T_a}$, we can write the sensitivity of urban canopy air temperature to anthropogenic heat flux as

$$\begin{align}\frac{{\Delta {T_a}}}{{{{\Delta }}{Q_{AH}}}} = - \frac{1}{{{\lambda _{{\text{all}}}}}}\end{align} \tag{ 3 }$$

where ${\lambda _{{\text{all}}}}$ is the sensitivity parameter (called the total sensitivity parameter in order to distinguish it from other feedback parameters introduced later). The sensitivity $\Delta {T_a}/{Q_{AH}}$, which indicates how easily the urban canopy air temperature can be altered by a perturbation of anthropogenic heat flux, is thus equivalent to the negative reciprocal of the total sensitivity parameter (${\lambda _{{\text{all}}}}$). If the absolute value of ${\lambda _{{\text{all}}}}$ is larger, the urban canopy air warming per unit increase of anthropogenic heat flux is weaker. Therefore, to understand $\Delta {T_a}/{Q_{AH}}$, we need to examine ${\lambda _{{\text{all}}}}$ in the relation ${{\Delta }}R = {\lambda _{{\text{all}}}}{{\Delta }}{T_a}$.

The air within the UCL receives convective heat fluxes from various urban surfaces and the overlying atmosphere. The sum of these heat fluxes ($R$) received by the air within the UCL can thus be written as

$$\begin{align}R = \sum\limits_{i = 1}^n \rho {C_p}{w_i}{c_s}\left( {T_s^{\,i} - {T_a}} \right) + \rho {C_p}{c_a}\left( {{\theta _{{\text{atm}}}} - {T_a}} \right)\end{align} \tag{ 4 }$$

where $n$ is the number of urban surfaces (e.g. there are five urban surfaces in the CLMU that interact with the urban canopy air, including roof, previous ground, imperious ground, sun wall and shade wall), $i$ refers to the $i{\text{th}}$ urban surface, ${w_i}$ is the weight of the $i{\text{th}}$ surface based on the corresponding area fraction (converted to per unit area of urban canyon floor in the horizontal direction), $\rho$ is the air density (kg m⁻³), ${C_p}$ is the specific heat of air at constant pressure, assumed to be of a constant value of 1004.64 J kg⁻¹ K⁻¹, ${c_s}$ is the heat conductance between the air within the UCL and the urban surface (called the surface–canopy air heat conductance, m s⁻¹), ${c_a}$ is the heat conductance between the air within the UCL and the overlying atmosphere (called the atmosphere–canopy air heat conductance, m s⁻¹), ${T_s}$ is the urban surface temperature (K) and ${\theta _{{\text{atm}}}}$ is the atmospheric potential temperature (K). Here we have assumed that the heat conductances between the air within the UCL and different urban surfaces are identical, which is a common assumption made in the CLMU and many other single-layer UCMs. But this assumption can be relaxed by allowing ${c_s}$ to vary for different urban surfaces in future work.

With equation (4), ${\lambda _{{\text{all}}}}$ can be written as the sum of the direct effect and feedbacks. Using the chain rule on equation (4) yields

$$\begin{align}{\lambda _{{\text{all}}}} & = \frac{{\Delta R}}{{\Delta {T_a}}} = \frac{{\partial R}}{{\partial {T_a}}} + \sum\limits_{i = 1}^n \frac{{\partial R}}{{\partial T_s^{\,i}}}\frac{{\Delta T_s^{\,i}}}{{\Delta {T_a}}} + \frac{{\partial R}}{{\partial {c_a}}}\frac{{\Delta {c_a}}}{{\Delta {T_a}}} \nonumber\\ & \quad + \frac{{\partial R}}{{\partial {c_s}}}\frac{{\Delta {c_s}}}{{\Delta {T_a}}}+ \frac{{\partial R}}{{\partial {\theta _{{\text{atm}}}}}}\frac{{\Delta {\theta _{{\text{atm}}}}}}{{\Delta {T_a}}}\nonumber\\ & = {\lambda _0} + {\lambda _1} + {\lambda _2} + {\lambda _3} + {\lambda _4}\end{align} \tag{ 5 }$$

where the partial and total derivatives are denoted by $\partial$ and $\Delta$, respectively. In this equation, ${\lambda _0}$ is the baseline sensitivity parameter, representing the direct effect of anthropogenic heat flux on urban canopy air temperature, with everything else (e.g. surface temperature, atmosphere–canopy air heat conductance, etc) held the same. Other $\lambda$ parameters represent different feedback processes: ${\lambda _1}$ refers to the strength of feedback from changes in surface temperatures; ${\lambda _2}$ is the feedback parameter for changes in atmosphere–canopy air heat conductance; ${\lambda _3}$ is the feedback parameter for changes in surface–canopy air heat conductance; ${\lambda _4}$ is the parameter for atmospheric feedback. A positive (or negative) feedback means that the process leads to an amplification (or dampening) of the direct effect of anthropogenic heat flux on urban canopy air temperature.

Combining equations (4) and (5), the baseline sensitivity parameter and feedback parameters can be derived as

$$\begin{align}{\lambda _0} = \frac{{\partial R}}{{\partial {T_a}}} = - \rho {C_p}\left( {\mathop \sum \limits_{i = 1}^n {w_i}{c_s} + {c_a}} \right)\end{align} \tag{ 6 }$$

$$\begin{align}{\lambda _1} = \mathop \sum \limits_{i = 1}^n \frac{{\partial R}}{{\partial T_s^{\,i}}}\frac{{\Delta T_s^{\,i}}}{{\Delta {T_a}}} = \mathop \sum \limits_{i = 1}^n \rho {C_p}{w_i}{c_s}\frac{{\Delta T_s^{\,i}}}{{\Delta {T_a}}}\end{align} \tag{ 7 }$$

$$\begin{align}{\lambda _2} = \frac{{\partial R}}{{\partial {c_a}}}\frac{{\Delta {c_a}}}{{\Delta {T_a}}} = \rho {C_p}\left( {{\theta _{{\text{atm}}}} - {T_a}} \right)\frac{{\Delta {c_a}}}{{\Delta {T_a}}}\end{align} \tag{ 8 }$$

$$\begin{align}{\lambda _3} = \frac{{\partial R}}{{\partial {c_s}}}\frac{{\Delta {c_s}}}{{\Delta {T_a}}} = \left( {\mathop \sum \limits_{i = 1}^n \rho {C_p}{w_i}\left( {T_s^{\,i} - {T_a}} \right)} \right)\frac{{\Delta {c_s}}}{{\Delta {T_a}}}\end{align} \tag{ 9 }$$

$$\begin{align}{\lambda _4} = \frac{{\partial R}}{{\partial {\theta _{{\text{atm}}}}}}\frac{{\Delta {\theta _{{\text{atm}}}}}}{{\Delta {T_a}}} = \rho {C_p}{c_a}\frac{{\Delta {\theta _{{\text{atm}}}}}}{{\Delta {T_a}}}.\end{align} \tag{ 10 }$$

Equations (1)–(10) constitute our forcing-feedback framework for diagnosing the sensitivity of urban canopy air temperature to anthropogenic heat flux. The aim of the proposed forcing-feedback framework is not to predict $\Delta {T_a}/\Delta {Q_{AH}}$ but to provide a diagnostic tool for quantifying the strengths of direct effects and feedback processes. In this study, the inputs for this framework are the simulated results from the CLMU. However, this framework is not limited to the CLMU and can be applied to diagnosing outputs from other UCMs.

The CLMU model represents urban parameterization within the Community Land Model (CLM), which is the land component of the Community Earth System Model (CESM) (Danabasoglu et al 2020). In this study, the most recent released version of the CLM (CLM5) within the framework of CESM version 2 (CESM2) is used. Within each land grid cell, CLM5 can have multiple land units including vegetated, crop, urban, glacier and lakes. For each urban land unit, three urban categories (tall building district, high density and medium density) are allowed. In CLMU, the urban canyon system consists of five surfaces: roofs, sunlit and shaded walls, impervious and pervious floors. The energy and water fluxes from each urban surface interact with the canopy air (see figure 1). A more detailed description of CLMU including the main urban parameters can be found elsewhere (Oleson et al 2010, Oleson and Feddema 2020). The CLMU input data are supplied by a global dataset (Jackson et al 2010). The model has been widely used to study urban energy and water fluxes, as well as surface and air temperatures (Oleson et al 2008a, 2008b, Grimmond et al 2011, Demuzere et al 2013, Karsisto et al 2016, Oleson and Feddema 2020). In this study, we use an improved CLMU that includes parameterizations of urban heat mitigation strategies (e.g. cool roofs and green roofs) that have been proposed and validated in our previous work (Wang et al 2020, 2021), although these new features are not used in this study.

We run CLM5 in an offline mode (i.e. forced by meteorological data) at a 1/8 degree spatial resolution over the CONUS and at an hourly time step (Wang and Li 2022). The hourly meteorological forcing data are from the North America Land Data Assimilation System phase II (NLDAS2) dataset (Xia et al 2012). The model is first spun up for 84 years by recycling the 1979–1999 NLDAS2 forcing four times. Four sets of numerical experiments are then conducted from 1979 to 1999 using the same initial condition obtained from the spin-up run (table S1). These four numerical experiments are designed to quantify how the urban canopy air temperature (figure 1) responds to a prescribed increase anthropogenic heat flux. In the control (CTL) experiment, no anthropogenic heat flux is added to the urban canopy air heat budget, and the simulated canopy air temperature is denoted as ${T_{a,0}}$. In the first sensitivity experiment (AH1), we add 1 ${\text{W }}{{\text{m}}^{ - 2}}$ of anthropogenic heat flux into the urban canopy air heat budget at each time step and compute a new canopy air temperature (hereafter ${T_{a,1}}$). Therefore, the difference between ${T_{a,1}}$ and ${T_{a,0}}$ (which is numerically equivalent to $\Delta {T_a}/\Delta {Q_{AH}}$ given that the added anthropogenic heat flux is 1 ${\text{W }}{{\text{m}}^{ - 2}}$) is the total impact of 1 ${\text{W }}{{\text{m}}^{ - 2}}$ of anthropogenic heat flux, which includes both the direct effect and the feedbacks. In another two sensitivity experiments AH10 and AH100, the added anthropogenic heat flux is 10 ${\text{W }}{{\text{m}}^{ - 2}}$ and 100 ${\text{W }}{{\text{m}}^{ - 2}}$, respectively. We denote the simulated urban canopy air temperatures in these two experiments as ${T_{a,10}}$ and ${T_{a,100}}$, respectively. The sensitivity $\Delta {T_a}/\Delta {Q_{AH}}$ is thus calculated as $\left( {{T_{a,10}} - {T_{a,0}}} \right)/10$ and $\left( {{T_{a,100}} - {T_{a,0}}} \right)/100$, respectively (see table S1). These two sensitivity experiments (AH10 and AH100) are designed to quantify whether the sensitivity $\Delta {T_a}/\Delta {Q_{AH}}$ is influenced by the magnitude of anthropogenic heat flux due to nonlinearity in the feedback processes. We choose these values (1, 10, 100 ${\text{W }}{{\text{m}}^{ - 2}}$) to cover a wide but reasonable range of anthropogenic heat fluxes.

We should emphasize that the added anthropogenic heat flux in all our experiments is prescribed, not computed by the BEM in CLMU (Oleson et al 2011, Demuzere et al 2013). We prescribe the added anthropogenic heat flux because we are mostly interested in the sensitivity of urban canopy air temperature to anthropogenic heat flux, not what processes generate the anthropogenic heat flux. Moreover, when the BEM in CLMU is used, the generated anthropogenic heat flux is added to the pervious and impervious surface energy budgets, which seems unphysical and is avoided in our study. Another way of interpreting our results is that they represent the sensitivity of urban canopy air temperature to anthropogenic heat fluxes from non-building (e.g. transportation) sectors with magnitudes of 1, 10 and 100 ${\text{W }}{{\text{m}}^{ - 2}}$.

For all simulations, we output the hourly urban canopy air temperature, the temperatures of different urban surfaces (i.e. roofs, walls and canyon floors), the atmospheric potential temperature, the surface–canopy air heat conductance (${c_s}$), as well as the atmosphere–canopy air heat conductance (${c_a}$). Note that the outputted ${c_s}$ and ${c_a}$ are computed internally via their parameterizations in CLMU. These hourly outputs are then used in the forcing-feedback framework described in section 2.1. Specifically, we compute the hourly sensitivity parameters based on equations (6)–(9). Given that we do not have atmospheric feedbacks in our simulations, ${\lambda _4} = 0$. With the sensitivity parameters calculated using equations (6)–(9), the total sensitivity $\Delta {T_a}/\Delta {Q_{AH}}$ can be diagnosed using equations (3) and (5). The diagnosed $\Delta {T_a}/\Delta {Q_{AH}}{\text{ }}$ is then compared with the directly computed $\Delta {T_a}/\Delta {Q_{AH}}$ mentioned above (e.g. for AH1 the directly computed $\Delta {T_a}/\Delta {Q_{AH}}$ is simply ${T_{a,1}} - {T_{a,0}}$). We average the hourly results over 20 years from 1980 to 1999.

Before we move to the results section, it is informative to briefly discuss the physics behind ${c_a}$ and ${c_s}$ and their parameterizations in CLMU, as they are key parameters in the forcing-feedback framework (see, e.g., equation (6)). Physically, ${c_a}$ (${c_s}$) represents the efficiency of convective heat transfer between the overlying atmosphere (the urban surfaces) and the canopy air. Given that the flow within the UCL is turbulent, both ${c_a}$ and ${c_s}$ are strongly affected by shear and buoyancy, the two main sources of turbulence kinetic energy. However, ${c_a}$ and ${c_s}$ are fundamentally different because they represent the convective heat transfer efficiencies across different levels. In terms of their parameterizations in CLMU, ${c_a}$ is parameterized through the classic Monin–Obukhov similarity theory (Oleson et al 2008a). Hence, ${c_a}$ is strongly affected by atmospheric stratification. However, ${c_s}$ is parameterized as a function of wind speed alone in the urban canyon (Oleson et al 2008a) and is thus much less affected by atmospheric stratification than ${c_a}$.

We first present the sensitivity $\Delta {T_a}/\Delta {Q_{AH}}$ simulated by CLMU in summer (June–August, or JJA) and winter (December–February, or DJF) seasons (figure 2). The results shown here have been averaged over 20 years (1980–1999) and are based on the AH1 experiment where the added anthropogenic heat flux is 1 ${\text{W }}{{\text{m}}^{ - 2}}$. The effect of increasing the magnitude of anthropogenic heat flux will be discussed in section 3.4. In summer, the median value of $\Delta {T_a}/\Delta {Q_{AH}}$ is around 0.01 ${\text{K }}{\left( {{\text{W }}{{\text{m}}^{ - {\text{2}}}}} \right)^{ - 1}}$, broadly comparable with previous studies presented in table 1. Here the median values are shown to minimize the influence of outliers. In winter, $\Delta {T_a}/\Delta {Q_{AH}}$ becomes stronger, with the median value increased by about 20%. In some cities in the southwestern USA (e.g. Los Angeles and Phoenix), the winter values of $\Delta {T_a}/\Delta {Q_{AH}}$ even reach 0.03 ${\text{K }}{\left( {{\text{W }}{{\text{m}}^{ - {\text{2}}}}} \right)^{ - 1}}$.

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To understand the directly computed $\Delta {T_a}/\Delta {Q_{AH}}$ from CLMU simulation results, we employ the forcing-feedback framework described in section 2.1. The total sensitivity diagnosed from this framework (i.e. using equations (5)–(9)) matches very well with the directly computed $\Delta {T_a}/\Delta {Q_{AH}}$ (figure 2), with spatial correlation coefficients larger than 0.99. These results give us confidence to use the forcing-feedback framework to interpret $\Delta {T_a}/\Delta {Q_{AH}}$.

Based on the forcing-feedback framework, the total sensitivity parameter (${\lambda _{{\text{all}}}}$) can be decomposed into the sum of the baseline sensitivity parameter (${\lambda _0}$) and the feedback parameters (${\lambda _1},{\text{ }}{\lambda _2}$ and ${\lambda _3}$). We find that the magnitude of ${\lambda _{{\text{all}}}}$ is almost identical to the magnitude of ${\lambda _0}$ (the spatial median value is −122 ${\text{W }}{{\text{m}}^{ - {\text{2}}}}{\text{ }}{{\text{K}}^{ - 1}}$ for both ${\lambda _{{\text{all}}}}$ and ${\lambda _0})$ in summer (figure 3). This is because the sum of the three feedback parameters (${\lambda _1} + {\lambda _2} + {\lambda _3}$) is very small, with the positive feedbacks and negative feedbacks nearly canceling each other. The positive feedback is mainly from changes in surface temperature (${\lambda _1}$, with a median value of 24 ${\text{ }}{{\text{m}}^{ - 2}}$ ${{\text{K}}^{ - 1}}$). This is expected as increases in surface temperature due to the added anthropogenic heat flux can in turn amplify the urban canopy air warming. On the other hand, the negative feedback is mainly the result of changes in atmosphere–canopy air heat conductance (${\lambda _2}$, with a median value of −25 ${\text{ }}{{\text{m}}^{ - 2}}$ ${{\text{K}}^{ - 1}}$). As the anthropogenic heat flux is added, the atmospheric stratification is altered (i.e. relatively more unstable), resulting in increased atmosphere–canopy air heat conductance (${c_a}$). This in turn leads to an increase in heat transfer into the overlying atmosphere and a dampening of the urban canopy air warming signal. The feedback from changes in surface–canopy air heat conductance (${\lambda _3}$, with a median value of 1 ${\text{ }}{{\text{m}}^{ - 2}}$ ${{\text{K}}^{ - 1}}$) is much weaker than the other two feedback processes. This can be explained by the parameterization of surface–canopy air heat conductance (${c_s}$) in CLMU, which is only dependent on the wind speed in the UCL and is thus a much weaker function of atmospheric stratification than the atmosphere–canopy air heat conductance (${c_a}$).

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In winter (figure 3), the negative feedback from atmosphere–canopy air heat conductance (${\lambda _2}$) decreases in magnitude by 11 ${\text{W }}{{\text{m}}^{ - 2}}$ ${{\text{K}}^{ - 1}}$ (in terms of median value) when compared with its summer counterpart (see also figure S1 for a comparison between summer and winter results). Namely, ${\lambda _2}$ becomes less negative, implying that the negative feedback from atmosphere–canopy air heat conductance (${c_a}$) is weakened. Unlike the reduced magnitude of ${\lambda _2}$ in winter, the winter–summer differences in ${\lambda _1}$ and ${\lambda _3}$ are much smaller (about 1 ${\text{W }}{{\text{m}}^{ - 2}}$ ${{\text{K}}^{ - 1}}$ in terms of median values) and almost negligible. As a result, the sum of feedbacks (${\lambda _1} + {\lambda _2} + {\lambda _3}$) becomes positive in winter (compared with nearly zero in summer). The absolute value of the total sensitivity parameter (${\lambda _{{\text{all}}}}$) therefore decreases, which further leads to an increase in $\Delta {T_a}/\Delta {Q_{AH}}$. The weakened negative feedback from atmosphere–canopy air heat conductance (${c_a}$) explains why stronger canopy air warming is observed in winter than in summer with the same amount of anthropogenic heat flux (figure 2), a typical result in the literature.

Figure 2 shows the strong spatial variabilities in $\Delta {T_a}/\Delta {Q_{AH}}$. To understand these spatial variabilities, we first note that the spatial pattern of the baseline sensitivity parameter ${\lambda _0}$ is very close to that of ${\lambda _{{\text{all}}}}$, with spatial correlation coefficients of 0.77 and 0.95 in summer and winter, respectively. Therefore, the spatial variability of ${\lambda _0}$ largely determines the spatial variability of $\Delta {T_a}/\Delta {Q_{AH}}$. From equation (6), ${\lambda _0}$ is proportional to the sum of atmosphere–canopy air heat conductance (${c_a}$) and surface–canopy air heat conductance (${c_s}$). We find that ${c_s}$ is less than 20% of ${c_a}$ and shows little spatial variability (not shown). As a result, one would expect that the spatial variability of ${\lambda _0}$ is mainly controlled by the spatial variability of ${c_a}$.

This is indeed the case. We find that the spatial correlation coefficients between ${\lambda _0}$ and ${c_a}$ are very strong (−0.87 and −0.98 in summer and winter, respectively). The negative correlations are understandable since physically the atmosphere–canopy air heat conductance (${c_a}$) indicates how strongly the air within the UCL communicates with the overlying atmosphere in terms of convective heat transfer. In places with larger (smaller) ${c_a}$, it is easier (more difficult) to transfer heat from the UCL to the overlying atmosphere, and thus the canopy air warming signal is weaker (stronger) with the same amount of anthropogenic heat flux.

We further analyze the diurnal variation of $\Delta {T_a}/\Delta {Q_{AH}}$. To do this, we select four metropolitan cities that have widely different climates and geographical locations (San Francisco, Boston, Chicago and Houston), instead of presenting averaged results over the CONUS.

In summer (figure 4(a)), all four cities experience a higher $\Delta {T_a}/\Delta {Q_{AH}}$ in the early morning than at other times. The morning peak of $\Delta {T_a}/\Delta AH$ is around 0.038 ${\text{K }}{\left( {{\text{W }}{{\text{m}}^{ - {\text{2}}}}} \right)^{ - 1}}$ in Houston, followed by San Francisco, Boston and Chicago. In the afternoon, the sensitivity in all cities is close to 0.01 ${\text{K }}{\left( {{\text{W }}{{\text{m}}^{ - {\text{2}}}}} \right)^{ - 1}}$, which is consistent with the findings of Kikegawa et al (2014) that also suggested a summer afternoon sensitivity of 0.01 ${\text{K }}{\left( {{\text{W }}{{\text{m}}^{ - {\text{2}}}}} \right)^{ - 1}}$. In contrast, there exist large differences in the diurnal variation of $\Delta {T_a}/\Delta {Q_{AH}}$ in winter (figure 4(b)). The sensitivity $\Delta {T_a}/\Delta {Q_{AH}}$ in Boston and Chicago is around 0.01 ${\text{K }}{\left( {{\text{W }}{{\text{m}}^{ - {\text{2}}}}} \right)^{ - 1}}$ throughout the day with small diurnal variations, while the diurnal variations of $\Delta {T_a}/\Delta {Q_{AH}}$ in Houston and San Francisco are strong, with much larger nighttime values than daytime values. San Francisco has the largest sensitivity in winter among the four cities, with a peak value of 0.036 ${\text{K }}{\left( {{\text{W }}{{\text{m}}^{ - {\text{2}}}}} \right)^{ - 1}}$.

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According to the forcing-feedback framework, the diurnal variations of $\Delta {T_a}/\Delta {Q_{AH}}$ are linked to the diurnal variations of feedback parameters, including the baseline sensitivity parameter (${\lambda _0}$). As shown in figures 4(c)–(l), ${\lambda _0}$ and, to a lesser extent, ${\lambda _2}$ exhibit diurnal variations that resemble those of ${\lambda _{{\text{all}}}}$, implying that the diurnal variations of $\Delta {T_a}/\Delta {Q_{AH}}$ are controlled by processes encoded in ${\lambda _0}$ (equation (6)) and, to a lesser extent, ${\lambda _2}$ (equation (8)). Close inspection of equations (6) and (8) indicates that a common process in equations (6) and (8) is the atmosphere–canopy air heat conductance (${c_a}$), suggesting that the diurnal variation of ${c_a}$ (and $\Delta {c_a}$) are the key to understanding the diurnal variation of $\Delta {T_a}/\Delta {Q_{AH}}$.

The ${c_a}$ is controlled by shear- and buoyancy-generated turbulence and thus is strongly affected by atmospheric stratification. In winter, the air within the UCL experiences more stable conditions at night, and hence ${c_a}$ is smaller, ${\lambda _0}$ is less negative (figure 4(f)) and $\Delta {T_a}/\Delta {Q_{AH}}$ is larger (figure 4(b)) than their daytime counterparts, assuming that the shear is the same between daytime and nighttime. In summer, the accumulation of stable stratification throughout the night reduces ${c_a}$ (leading to a less negative ${\lambda _0}$, figure 4(e)) and increases $\Delta {T_a}/\Delta {Q_{AH}}$ (figure 4(a)). After sunrise, the stratification transitions from stable to unstable, which increases ${c_a}$, causes a more negative ${\lambda _0}$ and reduces $\Delta {T_a}/\Delta {Q_{AH}}$. These two processes yield a morning peak of $\Delta {T_a}/\Delta {Q_{AH}}$, as observed in figure 4(a). Shear also plays an important role. For example, the stronger winds in Boston and Chicago in winter are likely to cause larger shear, leading to a larger ${c_a}$ and smaller $\Delta {T_a}/\Delta {Q_{AH}}$, when compared with Houston and San Francisco (figure 4(b)).

The above results are from the AH1 experiment, which adds 1 ${\text{W }}{{\text{m}}^{ - 2}}$ of anthropogenic heat flux into the UCL. We also conduct experiments to investigate how the urban canopy air temperature responds to different amounts of anthropogenic heat flux. The aim of these experiments is to test whether any of the feedbacks scale nonlinearly with $\Delta {Q_{AH}}$, thereby creating nonlinear responses of $\Delta {T_a}$ to $\Delta {Q_{AH}}$. Note that the baseline sensitivity parameter (${\lambda _0}$, see equation (6)) does not change with the magnitude of anthropogenic heat flux. Thus, any nonlinear response must stem from the feedback processes.

Figure 5 presents the relative changes in $\Delta {T_a}/\Delta {Q_{AH}}$ and feedback parameters (${\lambda _1}$, ${\lambda _2}$, ${\lambda _3}$) by comparing AH10 and AH100 with AH1 (i.e. the results of AH10 and AH100 minus the results of AH1 and then normalized by the results of AH1). The relative changes in $\Delta {T_a}/\Delta {Q_{AH}}$ are all negative, implying that the sensitivity becomes smaller as the magnitude of anthropogenic heat flux increases. The relative changes between AH100 and AH1 in terms of $\Delta {T_a}/\Delta {Q_{AH}}{\text{ }}$ have median values of −27% and −35% in summer and winter, respectively. This suggests that $\Delta {T_a}{\text{ }}$ does respond nonlinearly to $\Delta {Q_{AH}}$. Here we should stress that this result does not mean that changes in urban canopy air temperature $\Delta {T_a}$ become smaller as the magnitude of anthropogenic heat flux increases. It is rather $\Delta {T_a}/\Delta {Q_{AH}}$ that reduces as the magnitude of the anthropogenic heat flux increases.

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The relative changes in feedback parameters suggest that the nonlinear response of urban canopy air warming to the addition of anthropogenic heat flux is mostly due to decreases in ${\lambda _2}$ (i.e. ${\lambda _2}$ becomes more negative) as $\Delta {Q_{AH}}$ increases (figure 5). For example, the differences between AH100 and AH1 in terms of ${\lambda _2}{\text{ }}$ give median values of −13% and −28% in summer and winter, respectively. As alluded to earlier in section 3.1, ${\lambda _2}$ is associated with changes in the atmosphere–canopy air heat conductance ($\Delta {c_a}$). These results imply that with a larger $\Delta {Q_{AH}}$, the increase in ${c_a}$ is stronger, leading to a more negative ${\lambda _2}$ and a weaker $\Delta {T_a}/\Delta {Q_{AH}}$. Therefore, the nonlinear response of $\Delta {T_a}$ to $\Delta {Q_{AH}}$ is traced to the role of ${c_a}.$