Urban form strongly mediates the allometric scaling of airshed pollution concentrations

Many physical and socio-economic characteristics of urban areas scale with population [16], i.e. 'urban allometry' uncovers power-law relationships of the form:

$$\begin{eqnarray}&&{Y}_{j}={Y}_{j,0}{P}^{{\alpha }_{j}},\end{eqnarray} \tag{ 1 }$$

where coefficient, Y_j,o, and power-law exponent, α_j, describe the scaling of characteristic Y_j with some more commonly measured yardstick of 'scale' such as, in our case, population, P. Other yardstick measures, such as subsets of population [17, 18], GDP [19], or network fractal dimension [20], can be used if more appropriate to framing a particular study; we use population because it is a fundamental unit of social organization and to be consistent with prior work [21, 22]. The intention of allometric analysis is not to deduce causality, even regression-related Granger causality [23], but rather to search for simple emergent patterns in complex systems as an aid to understanding what is strictly contextual about instances of such systems and what holds more generally about them e.g. [24].

Sub-linear (α_j < 1) scaling relationships indicate 'economy (or parsimony) of scale', as seen, for example, with road length and the volume occupied by urban infrastructure (pipes, cables, etc) [21, 24]. Super-linear (α_j > 1) scaling relationships indicate 'increasing returns with scale' or 'scale-related excess', as is the case, for example, with incomes and the number of new inventions [25, 26]. Solid waste production from the world's current 27 megacities, and CO₂ emissions from urban clusters in the USA, show super-linear scaling [18, 27]. Linear power laws (α_j = 1 ± ε_j, where ε_j ≪ 1 is the error on the slope) include, for example, total employment and total housing of a city [16]; such linear relationships are neutral with respect to the distribution of population in one large city or several smaller urban settlements. When desirable traits scale super-linearly, or undesirable traits scale sub-linearly, the implication for policy-making is that organizing society into larger urban units can deliver improvements with respect to these traits. These scaling relationships prompt explanations in terms of a small set of basic principles that operate locally and describe the intrinsic socio-economic nature of urban living [16], in terms of parsimonious agent-based modeling [15], or via multiple regression model-finding of various kinds, e.g. [19, 28].

Here we extend the examination of the scaling behavior of urban settlements to include air pollution emissions and concentrations. When the scaling is robust, allometric scaling of pollution is a usefully emergent property of the complex urban system from which policy actions can be derived.

For evaluation of human exposure to air pollutants, the key metric is air pollutant concentration. In an urban area, air pollutant concentrations depend on imported background concentrations, local emission rates, atmospheric dilution and advection, chemical conversion, and local deposition, e.g. [29]. The likely scaling behavior of each of these influences on pollution concentrations is described below and in the SI, available online at stacks.iop.org/ERL/14/124078/mmedia. We begin with a focus on air pollutant emissions, which we would expect to be strongly related to population for a given level of economic development. Emissions inventories based on traffic flow, housing stock, and commercial activities are available for settlements of very different sizes across Great Britain (GB), which allows a detailed exploration of the scaling-law range across four orders of magnitude in population (10²–10⁶) and four orders of magnitude in urban area (1–10³ km²) (see SI).

Figure 1 shows very strong positive skew and changes in the variance around the power-law best fit for different values of population, similar to that seen in other analyses using large samples of urban areas; see, for example, figure 4 in [18]. To test the sensitivity of our results to sample size, the power-law relationship (i.e. the value of α_NOx) was re-calculated for sub-sets of the data, produced by removing more and more of the smaller settlements (figure 2).

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The power law exponent, α_NOx, for the relationship between NO_x emissions and population increases from 0.91 to 1.17 as smaller urban areas are progressively removed from the analysis. The peak value for α_NOx occurs when 20% (∼600) of the settlements from the original sample of 3030 remain. This corresponds to when settlements of less than 5000 inhabitants are excluded. The other pollutants behave similarly. When sampling the 20–140 largest urban areas in Britain, the scaling relationship is stable and close to unity (figure 2; minimum populations 200 000 and 50 000, respectively). Applying a categorization to the urban settlements—e.g. 'industrial cities', 'commercial cities', etc [19]—has been applied to help diagnose scaling relationships in some settings; we have not been able to find a suitable categorization.

One might hypothesize that the NO_x emissions scaling with population is entirely the result of road-length scaling with population in urban settlements. Comparing the population-scalings of NO_x emissions and road length (figure 2), it is evident that the scalings start to converge for samples containing settlements with populations above about 5000. For any pair of scalings against population, the scaling of one with respect to the other is given by the ratio of the population scalings (see figures 2 and S3 in SI). When two scalings with respect to population converge to a common value, the scaling between the two parameters approaches unity (i.e. α ∼ 1, a linear scaling). The derived relationship between NO_x emissions and road length diminishes (from ∼1.3 to 1.1, see caption to figures 2 and S3 in SI) when small settlements (populations < 5000) are removed. The NO_x emissions scaling directly against road length stabilizes at ∼1.10, i.e. a 10% increasing pollution 'scale-related excess'. From this, we can deduce that it is a substantial change in source character or the relative 'under-use' of roads in small urban settlements that is the larger cause of the super-linear relationship between NO_x emissions and road length for the whole sample of 3030 settlements, but that a substantial element due to congestion in larger settlements remains. The relationship between population and road length for the largest cities (rightmost point in figure 2) is similar to that found previously for cities in the USA (α_L = 0.85) [21].

The health effects of air pollution depend on exposure to pollution, and atmospheric pollutant concentrations are a much closer surrogate of exposure than emissions. To explore scaling relationships for pollutant concentrations and satellite-derived column densities, we examine the budget for an air pollutant in a well-mixed urban airshed, e.g. [29]

$$\begin{eqnarray}&&\displaystyle \frac{dC}{dt}=\displaystyle \frac{{Y}_{E}}{A.h}-\displaystyle \frac{{V}_{d}.C}{h}-\displaystyle \frac{\left(C-{C}_{0}\right).u}{x}+p-L\left(C\right).\end{eqnarray} \tag{ 2 }$$

As discussed in more detail in the SI, C is the well-mixed airshed concentration of the pollutant, t is time, Y_E is the total city-wide emission rate of the pollutant per unit time, A is the area of the settlement (of length x), h is the atmospheric mixing height, V_d is the deposition velocity for the pollutant, C₀ is the upwind concentration of the pollutant, u is the average wind speed across the urban area and through the height of the well-mixed boundary layer, p is the photochemical production rate of the pollutant, and L(C) is its chemical loss rate, which is a function of the abundance, C. We use this budget (equation (2)) to investigate the factors driving scaling properties of urban-airshed air quality in GB, not to model ground-level pollutant concentrations, a task for which many other more suitable but more complex models are available.

The steady-state pollutant concentration, C_ss, in the airshed is:

$$\begin{eqnarray}&&{C}_{ss}=\left(\displaystyle \frac{{Y}_{E}}{A.h}+p+\displaystyle \frac{{C}_{0}.u}{x}\right){\left(\displaystyle \frac{{V}_{d}}{h}+\displaystyle \frac{u}{x}+k^{\prime} \right)}^{-1},\end{eqnarray} \tag{ 3 }$$

where k' is a pseudo first-order rate coefficient describing chemical loss, and so k'[C] is a simple proxy for L(C). The scaling behavior of equation (3) depends on the relative sizes of the terms in each parenthesis (see SI for a discussion of the influence of each term on allometric scaling). The simplest informative situation is where deposition is negligible, the pollutant is not formed or lost by chemical reaction in the atmosphere over the timescales of interest, and the upwind concentration is very much smaller than the concentration in the urban airshed. This simplification is appropriate for NO_X and ultrafine aerosol (PM_0.1), but not for PM_2.5, for which primary emissions of particles [34] and transport of regional secondary aerosol, both contribute significantly to the ground-level urban concentrations [35]. The steady-state relationship then reduces to

$$\begin{eqnarray}&&{C}_{ss}=\displaystyle \frac{{Y}_{E}.x}{A.h.u}\approx \displaystyle \frac{{Y}_{E}}{{A}^{\displaystyle \frac{1}{2}}.h.u}.\end{eqnarray} \tag{ 4 }$$

Substituting population scaling for each variable into equation (4) suggests a heuristic for the overall scaling behavior of airshed average concentrations:

$$\begin{eqnarray}&&{\alpha }_{C}={\alpha }_{E}-\displaystyle \frac{1}{2}{\alpha }_{A}-{\alpha }_{h}-{\alpha }_{u},\end{eqnarray} \tag{ 5 }$$

where α_k is the power-law exponent against population for each of the parameters, k, in equation (4). One can think of this as a transformation of the pollution budget in physical space (equation (2) and more detailed continuity equations) into an equivalent budget in socio-economic 'space', in which population (or some other scale) provides the measuring yardstick. Equation (5) implies that the scaling of urban airshed concentration against population depends, therefore, on the relative magnitudes of the α parameters for the scaling of emissions, area, mixing height, and wind speed against population. Since the last three of these scalings depend crucially on urban form we can combine them into an urban form factor: α_UF = α_A/2 + α_h + α_u.

We have shown above how emissions scale with population: focusing on the largest 146 GB settlements (populations > 50 000), α_NOx ∼1 (1.02 ± 0.03) for NO_x emissions (figure 3(A), top-left panel). The equivalent power law relating settlement area to population in GB has α_A ∼ 0.9 (0.88 ± 0.01; figure 3(A), center panel; note that −0.5 log(A) has been plotted, making the slope of the graph = −0.44). The power law generating α_A relates the area of an urban settlement to its population; α_A < 1 implies that urban area grows more slowly than in direct proportion to population (i.e. is a measure of the average extent of urban densification in the sample of settlements); while α_A < 1 is a measure of the urban 'sprawliness' of the settlements). Although increased roughness and the urban heat island effect can cause mixing height, h, and wind speed, u, to change with settlement size [36], the changes are opposite in sign and approximately equal in magnitude (see SI). Hence, equation (5) simplifies to α_C = α_E – α_A/2. So, the emergent pattern of pollution concentration as a function of urban population is the balance of emissions scaling and urban form scaling, with the former being about twice as important as the latter.

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Substituting our results for how emissions and settlement area scale with population in GB gives a strongly sub-linear (α_C = 0.55 ± 0.03), but still positive, scaling of pollutant concentrations with population (figure 3(A), top-right panel). This scaling is marginally larger than that of city-scale column-integrated airshed NO₂, derived from satellite observations globally (α_C = 0.48 ± 0.03) [22], as discussed further in the SI. The strongly sub-linear concentration scaling against population demonstrates that, in terms of the average behavior of the British settlements we examined, people living in large urban areas are much less impacted by urban airshed pollutant concentrations than would be expected if the emissions scaled linearly with population. This is not to say that some cities are more impacted by air pollution than others, either because of their deviation from the average picture or because of the existence of pollution 'hot spots'. We discuss both of these issues in turn, below, having first discussed the implications of the principal drivers for the general scaling behavior.

Equation (5) implies that there are three primary drivers that can aid understanding of urban airshed pollutant concentrations across a sample of urban settlements: (i) the scaling of emissions with population as a result of socio-economic patterns, only some of which are expressed spatially; (ii) the scaling of settlement area with population, usually referred to as 'urban density' or its converse, 'sprawl'; and (iii) the scaling of meteorological factors with population as a result of socio-economic patterns expressed in those elements of urban form that affect aerodynamic roughness (e.g. heterogeneity in building height) and the surface energy balance. Because of their tendency to cancel, the scaling of meteorological factors is only effective insomuch as the magnitude of the scaling of mixing height, α_h, becomes different from that for wind speed, α_u, conditions that are not simple to envisage.

Retaining the deposition parameter in the simplification of steady-state equation (3) produces

$$\begin{eqnarray}&&{C}_{ss}=\displaystyle \frac{{Y}_{E}}{{A}^{\displaystyle \frac{1}{2}}\left({V}_{d}{A}^{\displaystyle \frac{1}{2}}+u.h\right)}\end{eqnarray} \tag{ 6 }$$

in which the scaling will depend on the relative size of the terms in parentheses in the denominator. The deposition term in this boundary-layer budgeting approach is much less than 10% of the u.h term for all but the largest (A ≈ 100 km²) conurbations (see SI). Even for these large urban areas, deposition will have a minor influence on the steady-state airshed concentration and on the scaling of those concentrations with population. The effect of expanding urban green infrastructure at the airshed scale will likely be much stronger through its effect on urban form, especially total urban area (equation (4), above) [37]. The effects of vegetation on surface roughness and, hence, the mean wind speed, u, and the effects of roughness and local heat balance on the mixing height, h [36, 38], will be more influential on urban airshed-average pollution concentrations than the effect of deposition on vegetation (see [39] and similar approaches which focus solely or mainly on the deposition term). However, at the local scale, the effect of green infrastructure on pollutant concentrations is a context-dependent balance of fumigation and deposition effects with potentially large impact [10, 38, 40, 41].

Calculating the residual between the point for each settlement and the scaling best-fit line (see SI) provides a measure of the degree to which each settlement is doing relatively 'better' or 'worse' than expected, based on the overall allometry [24]. Figure 4 shows residuals for the 146 largest GB settlements, ranked by their NO_x emission residual (red line), calculated such that a positive residual represents a better than expected (i.e. lower) emission than predicted by the allometric relationship with population.

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Superimposed on figure 4 are the combined residuals for emissions and settlement area scaling against population (blue line). Where the blue line is more positive than the red line in figure 4, the relevant settlement benefits more than expected from its urban form, making settlements that are doing better than expected on the basis of emissions (the left-hand side of figure 4) do even better, and ameliorating the 'worse than expected' performance for settlements that are doing worse than expected on the basis of emissions (the right-hand side of figure 4). Where the red line is less positive than the blue line, the situation is reversed: urban form makes the performance of individual settlements worse than expected on the basis of emissions alone. The top 20 and bottom 20 settlements whose concentration ranking is most changed by the urban form factor, compared to an emissions-only ranking, are listed in table S2 of the SI. The city whose rank is most improved by urban form is the post-world-war-2 'garden city', Milton Keynes. Of the largest British cities, Birmingham, Manchester, Glasgow, and Sheffield are in the top 20 'most improved by urban form', whereas London is largely unchanged.

Not all pollution is well-mixed throughout the urban airshed. Pockets of high concentration exist, particularly in poorly ventilated street canyons through which high volumes of traffic flow [10, 38, 41, 42]. We expect the likelihood of the occurrence of pollution hotspots, H, to be

$$\begin{eqnarray}&&H=\displaystyle \frac{{Y}_{E}}{{Y}_{L}}F,\end{eqnarray} \tag{ 7 }$$

where F is the fraction of roads liable to poor ventilation (fumigation), and Y_E/Y_L is the average emission per unit road length in the urban area.

We have shown above that road length is a stronger and stronger linear predictor for NO_x emissions as GB urban areas increase in size, but that a ∼10% pollution scale-related excess persists, presumably as a result of increasing congestion in the largest settlements. The scaling of F with population will depend on changes in urban form [42]. Some changes in urban form with scale—e.g. the emergence of an increasingly dense and increasingly high-rise central business district—will generate a larger fraction of poorly ventilated streets. Strict self-similarity (i.e. 'tiling' of urban form) would suggest that, neglecting any wind speed scaling as discussed in the SI, F scales linearly with population (i.e. we expect α_F ≥ 1). Overall, then, we deduce that, as cities grow, local traffic-related air pollution problems become exacerbated super-linearly (i.e. the scaling of H with population, α_H ≥ 1.1). Engineered solutions to urban air pollution hotspots could focus, therefore, on changes to urban form that reduce the fraction of poorly ventilated urban areas until traffic emissions are sufficiently reduced. Since the same canyon-like urban forms trap heat as well as pollutants [36, 43, 44], and so contribute to the urban heat island, any changes in urban form made to ameliorate air pollution hotspots will have the co-benefit of relieving urban heating (itself a phenomenon with undesirable human health impacts [45]).