A Mathematical Model for the Seasonality of Urban Heat Islands

By Lakshmi Chandrasekaran 

Cities generally exhibit higher air and surface temperatures than surrounding rural areas due to differences in their surface energy balance [5]. This phenomenon, which is known as the urban heat island (UHI) effect, poses a major threat to human health. Studies show that more than half of the world’s population now resides in cities, where warming can increase morbidity and mortality — especially during heat waves [5, 10]. For example, many parts of Western Europe suffered from sweltering heat during the summer of 2019. The heat wave compounded the misery of passengers in a broken-down Eurostar train from Brussels to London, necessitated the deployment of municipal misters in Vienna, and prompted Parisians to cool off in the Trocadero fountain near the Eiffel Tower to escape the unrelenting heat.

Researchers have extensively studied UHIs all over the globe [5]. Several studies suggest that anthropogenic heating—from sources that include carbon and pollutant emissions, the building sector, and human metabolism—can increase the intensity of the UHI effect [12]. In addition, research indicates that anthropogenic activity (especially in the 20th century) has significantly contributed to climate change, resulting in higher temperatures and longer, more severe heat waves [7]. Climate change will have the greatest impact on urban areas, which already suffer from the heat island effect and will bear the brunt of these heat events. According to Brice Tréméac, director of the Laboratory of Cold, Energy and Thermic Systems in Paris, more air conditioning is not a solution. “By cooling off the inside and warming the outside, we are feeding a disastrous vicious circle,” he told The New York Times in 2019.

A group of international scientists in the U.K., U.S., Singapore, and Switzerland has been attempting to tackle this challenge, ultimately producing a mathematical model that describes the variation of UHI intensity across cities and seasons. Gabriele Manoli, Simone Fatichi, Elie Bou-Zeid, and Gabriel Katul use just two equations—a hydrologic budget and a radiative-energy balance—to describe the complex interplay between seasonal rainfall, solar radiation, and vegetation conditions [4]. These factors regulate the intensity, timing, and geographical variations of the surface urban heat island (SUHI) effect, which measures the land surface temperature difference between an urban area and the surrounding rural region. 

The study was motivated by unexplained remote sensing observations of seasonal “hysteretic patterns” in SUHIs [13]. This hysteresis between the intensity of SUHIs \((\Delta T_s)\) and the background land surface temperature \((T_s)\) was previously identified by a comprehensive statistical analysis of European cities and confirmed via numerical simulations for Greater London [11]. These findings observed that the hysteretic behavior is clockwise (see Figure 1c) and hysteretic patterns are distinctive—depending on the climate (temperate versus dry)—with \(\Delta T_s\) either increasing or decreasing with \(T_s\) (see Figure 1). 

Figure 1. 1a. Conceptual representation of a daytime surface urban heat island (SUHI). 1b. Conceptual representation of an input-output system that exhibits hysteresis. 1c. Hysteretic behavior of \(\Delta T_s\) observed in Paris, France (circles) and Madrid, Spain (squares). Data are digitized from [12] and represent daytime values retrieved at 13:30 local time. 1d. Time series of input \(X\) (gray line) and output \(Y_1\) (black solid line) and \(Y_2\) (black dashed line) associated with the phase shift \(\Delta_{\phi1}=30 \:\: \rm{d}\) and \(\Delta_{\phi2}= -150 \:\: \rm{d}\), respectively. 1e. The resulting hysteretic curves. The input-output system is described by equations 1 and 2 with parameters \(\omega=\frac{2\pi}{365}\), \(\mu_X=0\), \(A_X=10\), \(\phi_X=-100\), \(\mu_{{Y}_{1}}=2\), \(A_{{Y}_{1}}=5\), \(\mu_{{Y}_{2}}=-5\), and \(A_{{Y}_{2}} =5\). Figure courtesy of [4].

A standard model of hysteresis is provided by an input-output system that transforms a sinusoidal input \(X(t)\) into a delayed output \(Y(t)\), i.e.,

\[X(t)=\mu_X + A_X \cdot\sin [\omega\cdot(t+ \phi_X)]\tag1\]

\[Y(t)=\mu_Y + A_Y \cdot\sin [\omega\cdot(t+ \phi_Y+ \Delta \phi)].\tag2\]

Here, \(t\) is time, \(\omega\) is frequency, \(\Delta\phi\) is the input/output time lag, \(\mu_X\) and \(\mu_Y\) are the mean values, \(A_X\) and \(A_Y\) are the amplitudes, and \(\phi_X\) and \(\phi_Y\) are the respective phase shifts of the input and output signals. One can normalize the variables to

\[X_n(t)=(X(t)- \mu_X)/A_X \:\:\: \textrm{and} \:\:\: Y_n(t)=(Y(t)-\mu_Y)/A_Y,\]

where \(n\) represents a normalized version of \(X\) and \(Y\). Assuming that \(\omega=1\) and \(\phi_X=0\), equations \((1)\) and \((2)\), without loss of generality, result in

\[Y_n=X_n \cos(\Delta \phi)+ \cos [\arcsin (X_n)] \sin (\Delta \phi).\tag 3\]

One can express this phenomenon, which is known as “rate-dependent hysteresis,” by converting equation \((3)\) into a first-order linear non-homogenous ordinary differential equation:

\[\frac{1}{\csc (\Delta\phi)}\frac{dY_n(t)}{dt}-Y_n(t)\cos(\Delta\phi)=-X_n(t). \tag 4\]

When the forcing term \(X_n(t)=0\), it renders the equation homogenous. Then \(Y_n(t)\) does not exhibit loops and exponentially decays to its zero equilibrium value, thus evidencing the rate dependency of hysteresis.

In their current work, Manoli and his colleagues focus on five European cities—Paris, Madrid, London, Milan, and Nicosia—that are characterized by a range of climatic conditions and hysteretic behaviors, as demonstrated in previous studies [11, 13]. Their analysis found that the hysteretic behavior in cities with relatively wet summers, like Paris and London, is concave up and characterized by strong surface urban heat intensities in summertime (with \(\Delta T_s>0\)). This is due to high rates of evaporative cooling in the surrounding rural areas (see Figure 2a). The impermeable heat-absorbing surfaces in urban areas limit evaporative cooling and make the cities a few degrees warmer, which can significantly impact local energy consumption, climate adaptation policy, and public health.

Figure 2. 2a. Observed/simulated seasonality of \(\Delta T_s\) in Paris (wet climate). 2b. Observed/simulated seasonality of \(\Delta T_s\) in Madrid (seasonally dry climate). Data are digitized from reference [12]. Given that simulated quantities represent monthly averages while data are observations at 13:30 local time [12], both observations and model results in 1a and 1b are rescaled using their respective annual averages of surface temperature \(\mu T_S\) and SUHI intensity \(\mu\Delta T_s\).

However, what happens in locations that experience dry summers due to scarce rainfall, such as Phoenix, Ariz., or Madrid? These regions exhibit a concave-down hysteresis (see Figure 2b); \(\Delta T_s<0\) during summer and autumn and is lower than the annual mean. With less rainfall and vegetation to spur evaporative cooling, the countryside heats up and the city may experience an “oasis effect.” This means the cities can be a degree or two cooler than the surrounding rural areas, even in blisteringly hot conditions. 

The authors hypothesized that time lag between urban and rural dynamics, specifically of energy and water fluxes, may control the observed hysteresis of SUHIs. To explain this behavior, they employed a well-established stochastic soil moisture balance equation that computes the seasonality of relative soil moisture [2, 6, 8]. This probabilistic approach integrates information on daily rainfall stochasticity and seasonality to describe the average seasonal cycle of water fluxes [2]. The corresponding equation is as follows:

\[nZ_r \frac{ds(t)}{dt}=R(t)-ET(s(t),t)-LQ(s(t),t). \tag5\]

Here, \(s\) denotes soil saturation at daily time scales. The interplay between rainfall \((R)\), evapotranspiration \((ET)\), and runoff \((LQ)\) describe the dynamics of this saturation. The term \(n\) represents soil porosity and \(Z_r\) is the rooting depth.

Next, the researchers used a novel coarse-grained SUHI model to represent the changes in surface temperature as an impact of urbanization [5]. This model estimates urban-induced changes in surface temperature \((\Delta T_s)\) that account for urban-to-rural changes in surface albedo (a unitless quantity that indicates a surface’s ability to reflect solar energy), emissivity, evapotranspiration, convection efficiency, and anthropogenic heat. Based on the derivation in Manoli’s previous work [5], one can express the urban-to-rural surface temperature difference \(\Delta T_s\) as

\[\Delta T_s(t)= \frac{1}{f_s(t)-\frac{\eta}{a_T}f_a(t)} \Delta S(t), \tag6\]

where \(f_s\) and \(f_a\) are energy redistribution factors associated with surface and air temperature respectively. The differential energy forcing due to urban-induced changes in surface energy balance is given by \(\Delta S\), and the parameters \(\eta\) and \(a_T\) account for the coupling between air and surface temperatures. \(\Delta T_s \) represents mean daily values of SUHI intensity as daytime and nighttime conditions are smoothed over on monthly time scales, neglecting heat storage effects [5].

Despite its relative simplicity, the model captures the major features of the observed seasonality of water and energy fluxes at the land surface, as well as the hysteretic behavior of \(\Delta T_s\). Unlike previous models that attribute hysteretic behavior only to incoming solar radiation [11], this new model links rainfall variability to urban-rural differences in evapotranspiration and albedo, thus demonstrating that soil moisture plays a key role in the seasonality of SUHIs in seasonally dry climates.

In wet climates, the impact of soil water availability is negligible due to its limited variability; the hysteretic loop is mainly caused by the time lag between radiation and temperature, which produces the concave-up curve. Conversely, the shape of hysteresis in dry climates is driven by the lag between radiation and rainfall, as well as the induced vegetation water stress in the natural surroundings. These factors give rise to the concave-down loop. 

Katul noted that these seasonal patterns of warming and cooling have significant implications for heat mitigation strategies, as urban green spaces can reduce heat island intensity during the summer. Meanwhile, the seasonality of solar radiation mitigates any potentially negative effects of albedo management during winter.

Experts in the field are employing other numerical approaches to tackle UHIs. Diego Rybski and his colleagues at the Potsdam Institute for Climate Impact Research use climate simulations to describe the UHI effect in several generated cities under constant background climate but with variable area, volume, and density. They found that the UHI intensity (defined in terms of air rather than surface temperature) is associated with both building density (i.e., more compact morphologies experience stronger SUHI effects) and an amplifying effect that urban sites have on each other [3]. The researchers believe that their findings could be useful for urban development.

A key takeaway of the study by Manoli and his collaborators is that climate-vegetation characteristics influence not only the mean intensity of urban warming, but also its seasonality. These findings are in line with observations in parts of Europe [13], India [9], China [14], and the U.S. [1], thus demonstrating that the model captures the features of different cities despite its simplicity.

However, the authors did note a caveat: their study focuses on remotely-sensed surface temperatures and considers city-scale values over monthly time scales, so its findings are not sufficient to guide site-specific urban planning solutions. Doing so requires knowledge of local microclimatic conditions—such as air temperature, humidity, and wind speed—at a much more granular scale (from buildings to blocks at sub-hourly to interannual time scales). Furthermore, thermal comfort depends on the overall climatic conditions experienced by city dwellers, not just the UHI effect. As a result, the authors note that the SUHI intensity is a necessary but insufficient metric to characterize heat stress. 

Nevertheless, this coarse-grained model provides a new approach for modelling UHI intensity and complements more detailed and city-specific tools. Manoli believes that one could use the interplay between urban properties and climate in terms of city-scale and monthly averages to develop general guidelines and frameworks for heat mitigation strategies in cities where extensive data and detailed simulations are unavailable. Given these findings, the team believes that rising temperatures and shifting rainfall patterns may impact the seasonality of UHIs in coming decades.

With world energy consumption projected to rise 1.6 times from 2010 to 2040, anthropogenic emissions due to air conditioning, industry, transportation, and other sources will continue to rise. Unique and novel strategies, such as the aforementioned study, could pave the way to a better understanding of the interactions between cities and their environments. This is key to the creation of innovative solutions to cool our warming planet.

Lakshmi Chandrasekaran received her Ph.D. in mathematical sciences from the New Jersey Institute of Technology. She earned her masters in science journalism with a health and science writing concentration from Northwestern University's Medill School of Journalism. She is a freelance science writer whose work has appeared in PLUS math magazine, HELIX - Science in Society magazine at Northwestern University, and The Munich Eye, an online newspaper based in Munich, Germany.