The canopy layer urban heat island (UHI) effect, as manifested by elevated near-surface air temperatures in urban areas, exposes urban dwellers to additional heat stress in many cities, specially during heat waves. We simulate the urban climate of various generated cities under the same weather conditions. For mono-centric cities, we propose a linear combination of logarithmic city area and logarithmic gross building volume, which also captures the influence of building density. By studying various city shapes, we generalise and propose a reduced form to estimate UHI intensities based only on the structure of urban sites, as well as their relative distances. We conclude that in addition to the size, the UHI intensity of a city is directly related to the density and an amplifying effect that urban sites have on each other. Our approach can serve as a UHI rule of thumb for the comparison of urban development scenarios.
We propose an upgraded gravitational model which provides population counts beyond the binary (urban/non-urban) city simulations. Numerically studying the model output, we find that the radial population density gradients follow power-laws where the exponent is related to the preset gravity exponent γ. Similarly, the urban fraction decays exponentially, again determined by γ. The population density gradient can be related to radial fractality and it turns out that the typical exponents imply that cities are basically zero-dimensional. Increasing the gravity exponent leads to extreme compactness and the loss of radial symmetry. We study the shape of the major central cluster by means of another three fractal dimensions and find that overall its fractality is dominated by the size and the influence of γ is minor. The fundamental allometry, between population and area of the major central cluster, is related to the gravity exponent but restricted to the case of higher densities in large cities. We argue that cities are shaped by power-law proximity. We complement the numerical analysis by economics arguments employing travel costs as well as housing rent determined by supply and demand. Our work contributes to the understanding of gravitational effects, radial gradients, and urban morphology. The model allows to generate and investigate city structures under laboratory conditions.
More than 30 years ago, Diffusion-Limited Aggregation (DLA) has been studied as mechanism to generate structures sharing similarities with real-world cities. Recently, a stochastic gravitation model (SGM) has been proposed for the same purpose but representing a completely different mechanism. Both approaches have advantages and disadvantages, while e.g. the dendrites emerging via DLA are visually very different from real-world cities, the SGM does not preserve undeveloped locations close to the city center. Here we propose a unification of both mechanisms, i.e. a particle moves randomly and turns into an urban site with a probability that depends on the proximity to already developed sites. We study the cluster size distributions of the structures generated by both models and find that SGM generates more balanced distributions. We also propose a way to assess to which extent the largest cluster is a primate city and find that in both models, beyond certain parameter value, the size of the largest cluster becomes inconsistent with being drawn from the same distribution of remaining clusters.Various models have been proposed to generate structures that share similarities with real-world cities. The correlated percolation model combines an exponential radial gradient with correlated random numbers (Makse, Havlin, and Stanley 1995). However, the largest city is too big and consequently omitted when Zipf's law for cities, i.e. a power-law size distribution with specific exponent, is validated. The model by Schweitzer & Steinbrink (1998) is based on two processes, the emergence of new clusters and cluster growth. In this model a similar problem is encountered -due to coagulation the largest cluster dominates the growth process. The authors avoid this by excluding the largest cluster from growth.
Urban growth can take different forms, such as infill, expansion and leapfrog development. Here we focus on leapfrogging, which is characterised as new urban development bypassing vacant land. Analysing a sample of 100 global locations, we study the probability that land cover is converted from non-urban to urban as a function of the minimum distance to existing urban cells. The probability decreases with the distance but in many of the considered real-world samples it increases again just before the maximum possible distance. Comparing these empirical findings with numerical ones from a gravitational model, we discover that the characteristic increase can be found in both. Our results indicate that the conversion probability as a function of the distance to urban land cover includes three urban growth domains. (i) Expansion of existing settlements, (ii) discontinuous development of coincidental new settlements rather close to existing ones and (iii) leapfrogging of new settlements far away from existing ones. We conclude that gravitational effects can explain discontinuous development but leapfrogging can be attributed to a scarcity of developable land at long distances to settlements.
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