Scaling intra-urban climate fluctuations

TL;DR

Urban climate variability within cities exhibits sensitivity to urban form, yet boundary-based scaling introduces biases. The authors assemble high-resolution data for $T$ and $PM_{2.5}$ across $142$ cities and link intra-urban fluctuations to a logarithmic dependence on urban features $x$ via $\Delta y = \alpha + \beta \ln x$, revealing a universal location–scale scaling form. After rescaling by city-specific means and variances, marginal and joint PDFs of $T$ and $PM_{2.5}$ collapse onto a common function $G$, approximately Gaussian, with clustering into $K=3$ groups improving collapse without altering $G$. A stochastic radial-decay extension, $\Delta y(r) = y_A \exp(-r^2/(2\lambda_y^2)) + \mathcal{N}(0,\sigma_{r,city}^2)$, reconciles traditional decay models with observed Gaussian statistics, linking urban morphology to intra-urban climate variability and enabling downscaling and planning applications.

Abstract

Urban-induced changes in local microclimate, such as urban heat islands and air pollution, are known to vary with city size, leading to distinctive relations between average climate variables and city-scale quantities (e.g., total population). However, these approaches suffer from biases related to the choice of city boundaries and they neglect intra-urban variations of urban characteristics. Here, we use high-resolution data of urban temperatures, air quality, population, and street networks from 142 cities worldwide and show that their marginal and joint probability distributions collapse onto a set of general scaling functions. Using a logarithmic relation between urban spatial features and climate variables, we find that average street network properties are sufficient to characterize the entire variability of the temperature and air pollution fields observed within and across cities. These findings provide a unified statistical framework for characterizing intra-urban climate variability, with important implications for climate modeling and urban planning.

Figures (16)

• Figure 1: Conceptual framework for analyzing the covariation of urban structure and climate. (a) In this study we consider intra-urban variations of climate variables $y(s)$ (i.e., temperature $T$ and particulate matter concentrations $PM$) and their relation to urban features $x(s)$ (i.e., street intersections $n$ and population counts $p$), where $s$ is the spatial coordinate. Urban-rural differences $\Delta y = y-\langle y_{0}\rangle$ are calculated considering background climatic conditions $\langle y_{0}\rangle$. (b) Example of a regular square grid used to aggregate all datasets, allowing both climate variables and urban features to be compared across cities. (c) PDFs of urban features $\rho(x)$ and urban climate perturbations $\rho(\Delta y)$, where each curve represents one city. Previous work Hendrick2024 showed that the PDFs of urban features collapse onto a single curve following a finite-size scaling form $F$ (inset). (d) Methodology employed in this study: first we investigate the relation between $x$ and $\Delta y$ and then we rescale their marginal and joint PDFs to obtain a data collapse - indicating the existence of general scaling functions ($G$ and $J$, see main text).
• Figure 2: Scaling intra-urban temperature ($T$) variations and covariations with street network intersections ($n$). (a) Distinct PDFs of temperature observed across the analyzed cities, where each curve represents one city. (b) After rescaling using the empirical climate data according to equations \ref{['eq:ansatz']}-\ref{['eq:y_rescaled']}, these PDFs collapse onto a common scaling function $G$. A Gaussian fit (dashed black line) indicates that $G$ is well approximated by a normal distribution. The inset shows the residual $R(\phi)$. (c) Global distribution of cities and their grouping into three clusters. (d) Collapse of the marginal PDFs after rescaling using the street network information according to equations \ref{['eq:ansatz_rescaled_kmeans']}-\ref{['eq:y_rescaled_kmeans']}. As for panel b, the Gaussian fit (dashed black line) and residual $R(\phi)$ (inset) are also shown. (e) Joint PDFs for temperature and street network intersections for all cities analyzed. Colors refer to the different clusters illustrated in panel c. (f) After both variables are rescaled using equation \ref{['eq:ansatz2']}, the data collapse follows a common joint scaling function. The inset shows the residual $R(\phi,\delta)$.
• Figure 3: Scaling intra-urban air quality ($PM$) variations and covariations with street network intersections ($n$). (a) Distinct PDFs of particulate matter concentrations observed across the analyzed cities, where each curve represents one city. (b) After rescaling using the empirical climate data according to equations \ref{['eq:ansatz']}-\ref{['eq:y_rescaled']}, these PDFs collapse onto a common scaling function $G$. A Gaussian fit (dashed black line) indicates that $G$ is well approximated by a normal distribution. The inset shows the residual $R(\phi)$. (c) Global distribution of cities and their grouping into three clusters. (d) Collapse of the marginal PDFs after rescaling using the street network information according to equations \ref{['eq:ansatz_rescaled_kmeans']}-\ref{['eq:y_rescaled_kmeans']}. As for panel b, the Gaussian fit (dashed black line) and residual $R(\phi)$ (inset) are also shown. (e) Joint PDFs for temperature and street network intersections for all cities analysed. Colors refer to the different clusters illustrated in panel c. (f) After both variables are rescaled using equation \ref{['eq:ansatz2']}, the data collapse follows a common joint scaling function. The inset shows the residual $R(\phi,\delta)$.
• Figure 4: Stochastic radial decay model of climate variables. (a) Probability density functions of $\Delta y$ from numerical simulations: when $\sigma_{r,city}=0.0$, the deterministic decay model is recovered, while for $\sigma_{r,city}=0.3$ an approximately Gaussian PDF shape emerges, using a peak value $y_A = 2.5$, decay rate $\lambda_y = 5$ km, and city radius $R = 25$ km. (b) Illustration of the radial decay of climate variables (i.e., urban–rural temperature or pollution differences, $\Delta y$) together with the deterministic decay model fit for Bucharest. (c) Distribution of empirical noise strengths ($\sigma_{r,city}$) of climate variables across all cities. (d) Scatter plots of the street network decay rate ($\lambda_n$) versus both the decay rate of temperature ($\lambda_T$) and particulate matter ($\lambda_{PM}$), along with their corresponding Pearson correlation values $C$ ($p < 0.001$).
• Figure S.1: Logarithmic relationship between climate variables and urban characteristics for urban-rural differences of (a) particulate matter concentration and (b) temperature.