The role of city size and urban form in the surface urban heat island

Urban climate is determined by a variety of factors, whose knowledge can help to attenuate heat stress in the context of ongoing urbanization and climate change. We study the influence of city size and urban form on the Urban Heat Island (UHI) phenomenon in Europe and find a complex interplay between UHI intensity and city size, fractality, and anisometry. Due to correlations among these urban factors, interactions in the multi-linear regression need to be taken into account. We find that among the largest 5,000 cities, the UHI intensity increases with the logarithm of the city size and with the fractal dimension, but decreases with the logarithm of the anisometry. Typically, the size has the strongest influence, followed by the compactness, and the smallest is the influence of the degree to which the cities stretch. Accordingly, from the point of view of UHI alleviation, small, disperse, and stretched cities are preferable. However, such recommendations need to be balanced against e.g. positive agglomeration effects of large cities. Therefore, trade-offs must be made regarding local and global aims.


Multi-linear regression
The result obtained from a step-wise multi-linear regression based on 5,000 city clusters, without normalizing the dependent variables -ln S C , ln A, D f .
Analogous to Eq. (3) in the main text.
The relation based on normalized dependent variables Analogous to Eq. (4) in the main text.
Since the nighttime surface UHI is generally weaker than that during daytime (which can be seen from the value range of the Y-axis), the parameters obtained from the multi-linear regression are also correspondingly smaller. However, these results are consistent with the findings in the main text, i.e. city size exerts the strongest influence (0.17 ln S * C ), followed by fractality (0.09 D * f ), and anisometry (0.01 ln A * ).
2 Correlations between ln S C , D f and ln A

Linking heat transfer coefficient, area, and fractal Energy balance of urban surfaces
The energy balance of urban surface can be written as Regardless of the anthropogenic heat release, surface temperature (T surface ) is mainly determined by the sensible heat flux Q H , as Q H = h∆T = h(T surface − T air ), where h is the convection transfer coefficient. To simplify the problem, we idealize the urban surface as a flat horizontal isotropic plate, without taking into account the surface roughness. The convection heat transfer coefficient h -more precisely, how h is related to object size -is crucial to study the scale effect of the surface temperature 1 .

Convection heat transfer coefficient h
The convection heat transfer coefficient h can be expressed by using the dimensionless Nusselt number Nu (the ratio of convective to conductive heat transfer), The Prandtl number is defined as the ratio of momentum diffusivity to thermal diffusivity, The Reynolds number is defined as the ratio of inertial forces to viscous forces According to 3,4 , the perimeter P of an object is given by the area (S) raised to the power of D f /2, i.e. P ∼ S D f /2 , where D f is the fractal dimension of the perimeter. Thus, the characteristic length is L = S/P ∼ S 1− D f /2 . Equation (7) can be rewritten as To illustrate the influence of surface area S and fractal dimension D f , we consider a constant influx of solar radiation. For a fixed fractal dimension, the convection transfer coefficient h decreases with increasing surface area S, resulting in a higher surface temperature. Analogously, for a fixed surface area S, the convection transfer coefficient h decreases with increasing fractal dimension D f , resulting in a higher surface temperature.
It is worth mentioning that the fractal dimension D f of the perimeter, also called as envelope fractal dimension, is different from the fractal dimension calculated in this study. The latter one is also known as box fractal dimension. We estimated both the envelope and box fractal dimensions by applying the box counting method to the urban outline (envelope) and urban area, respectively 4 . As shown in Fig. S3, the box fractal dimension is always larger than the envelope fractal dimension.
Analyzing our data, we obtained empirically P ∼ S aD f , with a ≈ 0.43. Since the term 1 − a D f in Equation (8) is still positive, the conclusions are not affected. ρ =0.65 Figure S3. The box fractal dimension D f versus the envelope fractal dimension D f . It can be seen that the box fractal dimension is always larger than the envelope fractal dimension.