In some ways cities can be thought of as living organisms because they consume energy and produce artifacts and waste. It is therefore not surprising that there are biological metaphors associated with the “scaling” phenomenon illustrated above (as already noted in Chapter 1). For example, as indicated above, the surface area of an object is proportional to (i.e., scales as) the square of its size (linear dimension), and its volume scales as the cube of its size. More specifically, many if not most physiological properties of biological organisms scale with body mass M with an exponent β that is a multiple of ¼ (this is a generalization of Kleiber’s law, mentioned above). Thus the metabolic rate per unit mass R ∝ M−1/4 decreases with body size. This means that larger organisms consume less energy per unit time and per unit mass. Inversely, life spans and maturation times scale as M1/4. Nevertheless, there are differences: larger organisms live by slower biological clocks, by virtue of the inverse quarter power law referred to above, whereas the pace of city life increases with size, as we shall see below. As the authors of the article point out, scaling has proved to be a valuable tool for revealing underlying dynamics and structure for many problems encountered in science and technology. In what follows, we discuss some implications of the “urban growth equation” derived in that paper.
We are interested in deriving and subsequently solving an equation for the population growth rate, dN/dt, in terms of resources for maintenance and growth per unit time mentioned above (y) and two other parameters, R and E. R is defined as the amount of resource(s) per unit time required to maintain an individual; E is the amount required to add another individual to the population. Then for ΔN individuals added in time Δt the resource allocation rate equation may be written, in the appropriate limit as
The left-hand side of this equation is a measure of the available resources, balanced on the right by consumption terms due to maintenance and population growth, respectively. Substituting for y from equation (18.1) and rearranging we find that
This is a Bernoulli equation; the substitution v = N1−β reduces this to a linear equation with integrating factor
The resulting solution of equation (18.3) for β ≠ 1 is
where N0 = N(0).
Exercise: Derive equation (18.4).
There are three basic forms of solution depending on whether β > 1, β = 1 or β < 1. The simplest case, β = 1, has the solution (using equations 18.1 and 18.2)
This represents, of course, simple exponential growth or decay depending on the sign of the exponent. For β < 1 the growth curve is sigmoidal (sometimes called Gompertz-like); as can be seen from equation (18.4), there is a horizontal asymptote—the carrying capacity (see Chapter 15)—found by taking the limit as t → ∞. Its value is . This is quite a significant result, for it means that there is an upper limit to the size of the population: it will eventually stop growing (in practical terms, after a very long but finite time). As we saw in Chapter 15, this is very similar to biological systems in which the competition between finite resources and population growth eventually results in saturation of the population (even in the simplest models). The urban equivalent of “finite resources” in biology or ecology is referred to in the article as “economies of scale.” A typical graph of N(t) is shown in Figure 18.1 for a simple choice of parameters in equation (18.4), expressing in units of, say, 100,000. The initial value N0 = 1, with β = 0.5, and the carrying capacity N∞ = 25 are chosen here for illustrative purposes.
Figure 18.1. Typical solution N(t) based on equation (18.4).
The case of β > 1 is even more interesting. The solution can be rearranged in the form
If , then the term inside the curly brackets will become zero for some time tc; it corresponds to unbounded (or explosive) growth at finite time (and therefore the growth is faster than exponential). In fact, this condition on N0 is readily obtained from equation (18.3). We have previously encountered this kind of growth, albeit in a less sophisticated form, in Chapter 15.
Since equation (18.6) is of the form N(t) = [A − BeCt]−D for positive constants A, B, C, and D, it follows that
Nothing that , we see from the above equation that, provided the quantity is small “enough” compared with 1, then
A graph of N(t) in this case showing the (theoretically) unbounded growth is shown in Figure 18.2. Of course, for a finite city size (or population) and finite resources, such a singularity cannot occur. However, the authors point out that left unchecked, this lack of sustainability triggers a “phase transition” to the point where N(t) collapses; according to equation (18.3) this will occur when N exceeds the critical value (this was the value of N∞ for β < 1). At this point N′(t) < 0 and an inexorable population decline follows (see Figure 18.3) unless, as the authors point out, major qualitative changes occur which effectively “reset” the initial conditions and parameters of the governing solution defined by equation (18.4). In order to maintain growth, then, a new cycle must be initiated wherein β > 1 as before, and at the “new” time t = 0 the parameters N0, R, and y0 are such that, as before, . While in principle this process can be repeated continually, leading to multiple cycles (and pushing potential collapse into the future), there is an inherent problem. Note from equation (18.8) that the time between cycles tc is inversely proportional to Clearly this decreases with each new cycle, since N0 increases accordingly. Therefore major innovations and adaptations must occur at an accelerated rate; in other words, the pace of life increases with city size.
Figure 18.2. Typical super-exponential growth with finite-time singularity tc (vertical line).
Figure 18.3. Population collapse when growth cannot be maintained.
At this point the reader may wonder yet again whether this scenario has any basis in reality. Citing several reputable sources, the authors note that “these predicted accelerating cycles . . . are consistent with observations for the population of cities . . . , waves of technological change, and the world population.” And to some degree these ideas can be quantified: the famous Scottish scientist Lord Kelvin is reputed to have said that “I have no satisfaction in formulas unless I feel their arithmetical magnitude,” so let’s try to do just that. In equation (18.8) the ratio E/y0 appears. This may be interpreted as the time required for an average individual to reach “productive maturity,” or more simply, the time needed to “create” a new individual. Bettencourt et al. [32] express this as E/y0 ~ (20 yr)T, T being a number of order unity. With β = 1.1 and the initial population measured in millions (N0 = 106n), equation (18.8) reduces to the simple form
For a large city tc will typically be a few decades, but clearly this timescale decreases with increasing city size. This transition to successive cycles of super-exponential growth accompanied by a reduction of tc has, it seems, been a common pattern in urban development [29] as well as for the world population [33]. As an example of this urban pattern, Batty and Longley [1] consider the population growth of the New York metropolitan area from 1790 to the present day. It can be decomposed into successive periods of super-exponential growth, and the period of faster growth in the 1960s was followed by the decline of the 1970s as individuals left the city under “the perception of spiraling increases in costs, crime and congestion.”
But super-exponential growth, you ask? Are we back at the Doomsday equation once more? We can do no better than to quote again from Johansen and Sornette [29]:
It is estimated that 2000 years ago the population of the world was approximately 300 million and for a long time the world population did not grow significantly, since periods of growth were followed by periods of decline. It took more than 1600 years for the world population to double to 600 million and since then the growth has accelerated. It reached 1 billion in 1804 (204 years later), 2 billion in 1927 (123 years later), 3 billion in 1960 (33 years later), 4 billion in 1974 (14 years later), 5 billion in 1987 (13 years later) and 6 billion in 1999 (12 years later). This rapidly accelerating growth has raised sincere worries about its sustainability as
well as concerns that we humans as a result might cause severe and irreversible damage to eco-systems, global weather systems etc.
The interested reader is recommended to consult this rather technical paper for further details of the suggested “finite-time singularity” and its possible manifestations and consequences for humanity. Related, but much earlier papers (including von Foerster et al. [28]) are also well worth reading; the reader should also consult the additional references for details.
Chapter 19
AIR POLLUTION IN THE CITY
I spent several years as a student living in London, but fortunately I never had to experience something that plagued the city in the first half of the twentieth century: smog ( = smoke + fog). The last major occurrence of London smog was in 1952, and while estimates vary, it is thought that as many as 12,000 people died in the weeks and months following the outbreak. Basically, smog is caused by the chemical reaction of sunlight with chemicals in the atmosphere.
But more generally, what is air pollution? Essentially, it is the presence of substances in the atmosphere that can adversely affect the quality of human, animal, and plant life, and the environment. Of course, this is rather vague, and the definition itself is rather “fluid,” changing somewhat as more is known about pollutants. Furthermore, it is usually the addition of such substances resulting directly or indirectly from human activity that is of most concern. The biggest such contributions are from the burning of fossil fuels, including coal, oil, and gas in cars, trucks, factories, and homes. However, natural sources such as forest fires and volcanic eruptions can cause local and even global havoc, as was seen in April 2010 when the Icelandic volcano Eyjafjallajökull erupted.
Air pollution is designated primary or secondary. The former results from pollution that is introduced directly into the air, such as smoke and car exhausts. Secondary pollution forms in the air as a result of sunlight-induced chemical reactions changing the nature of the primary pollutants.
X = Vt: PARTICLES IN THE CITY
Let’s examine the behavior of small particles like aerosols or tiny cloud droplets as they fall slowly through air (or indeed, sediment as it settles down to the bottom of a lake). Both are well described by Stokes’ law, stated below, provided their speed of descent V is small enough that no turbulence is generated in their wake. Not surprisingly, the upper limit for aerosol sizes is determined by sedimentation—unless the particles can stay aloft for reasonable periods of time (days or longer), they will contribute little to the lack of long-term visibility.
Consider for simplicity a spherical particle of radius R and density ρ falling through the air. The downward force acting on the particle is its weight,
g being the gravitational acceleration. (Although forces are vector quantities, all the ones acting on this particle are directed upward or downward, so the vector notation will be dispensed with here.) There is an upward buoyancy force, but since the density of the air is so small compared with that of the particle, it will also be neglected here. The other force acting to resist the downward fall of the particle is the drag force F; as the particle initially accelerates downward because of gravity, this resistive force will also increase until the two are in balance, assuming the time of fall is long enough to permit this, as it will usually be in this context. Then there is no net force acting on the particle, and it proceeds to fall with a constant speed, the terminal speed. (The correct term is terminal velocity, but again, we are not concerned with vector notation here.) Naturally, the magnitude of the terminal speed will determine how long the particle how long the particle remains in the air. Very small particles (< 0.1μ) are continually buffeted by a molecular process known as Brownian motion, and remain suspended indefinitely.
A simple argument can be used to determine the dependence of the drag force on the particle size and speed V. It is reasonable to assume that F ∝ η, η being the (dynamic) viscosity coefficient of the air, that is, F = Kη, where K is a (dimensional) constant. If we equate the dimensions of both sides of this equation using [M], [L], and [T] for mass, length, and time, respectively, then
Therefore the dimensions of K are [L]2 [T]−1 and this is accomplished, in particular, by the combination RV. The magnitude of K cannot be determined by dimensional arguments, but in 1851 George Gabriel Stokes carried out more detailed calculations and found that for small Reynolds numbers R (where R = VRρa/η < 1, ρa being the density of the air),
(The reader may recall that earlier in this book (Chapter 3) a more generic form of the Reynolds number was used, namely R = ul/, where u, l, and represented a typical speed, length scale, and (kinematic) viscosity, respectively. To avoid confusion here with the radius R, we use the alternative notation Re for the Reynolds number. Also, since this chapter is a little more technical and mathematically precise, the latter form for the Reynolds number is more appropriate here (note also that = η/ρa).)
To determine the dependence of the terminal speed Vt on the particle size, we just equate the two opposing forces and solve equation (19.2) for this speed to find that
Clearly, the terminal speed increases quite rapidly with particle size. Let us apply this result to calculate the terminal speed of a particle of radius 10 microns (10−5 m) in still air at 5°C at an altitude of 1000 m; g ≈ 9.8 m/s2, ρ ≈ 2 × 103 kg/m3 and η ≈ 1.8 × 105 Ns/m2. Therefore
or about 2 cm/s. The Reynolds number for this particle in air is, from the definition above with ρa ≈ 1.1 kg/m3,
so the use of Stokes’ law is readily validated. Note that a particle ten times smaller (R = 1μ) falls one hundred times more slowly, at about 2 × 10−4 m/s. In Chapter 20 these ideas of sedimentation rates will be discussed again, but in the context of reduced visibility caused by particulate scattering of light.
X = C(x, t): POLLUTION IN THE CITY
How do the suspended particles spread in the air? In the absence of wind and other air currents the simplest approach to this question is to consider the one-dimensional diffusion equation derived in Appendix 10:
For simplicity we have assumed that the diffusion coefficient D is constant (although there are certainly situations where this is not the case). This type of argument can be readily generalized to the case of two or three dimensions and geometries other than Cartesian, and we will just introduce them as needed from this point on. This equation describes a trend to a uniform distribution of the pollutant concentration C(x, t) over time. To see this, suppose that for a particular interval of time, C(x, t) has a local minimum; then the right-hand side of equation (19.4) is positive, and C will increase in time. Correspondingly, if C has a local maximum, C will decrease in time.
There is another mechanism that must be included in any realistic discussion of pollution: wind. We shall consider the effects of a wind with constant speed U in the x-direction only (even when a higher-dimensional diffusion equation is used). Again, as shown in Appendix 10, equation (19.4) can be generalized to become
In most cases of interest, this new wind “advection” term far outweighs the effects of the diffusion term. A more practical variant of the problem for our purposes is represented by the equation
What, then, does this equation signify? In this context, it describes the temporal and spatial behavior of the concentration of pollutant particles as they diffuse in the y-z plane perpendicular to the wind direction (x) wafting them downstream. A further simplification is often justified, namely, to consider a steady-state situation. Frequently the source of pollutants emits them at a constant rate, and has a plume whose average shape doesn’t change much in time, unless the wind direction changes or some new weather pattern otherwise modifies it significantly. If these do not occur, then we may set the left-hand side of equation (19.6) to zero, resulting in an impressive-sounding (but less impressive-looking) time-independent advection-diffusion equation! Here it is:
We proceed to justify a solution of this equation in a nonrigorous way as follows.
Exercise: Show by direct substitution
that the equation
possesses a solution for x > 0
The source is a “point” located at x = 0, y = 0, but do not be concerned about the apparent (and real) singular behavior of this solution there; textbooks on partial differential equations discuss this type of problem and its resolution (with time t replacing x). K1 is a constant that depends on the rate of pollutant emitted in the plane x = 0, and possibly on D and U (see below).
Because of the symmetry in y and z possessed by equation (19.7), and based on the solution (19.9), we expect the solution of (19.7) to be of the form
This is also readily verified by direct substitution (exercise!). Generally, K2 will also depend on the constants D and U. (We have simplified things greatly here; generally D will be different in the horizontal y-direction, perpendicular to the wind, from that in the vertical z-direction.) Note that, for a given distance x downstream, the maximum concentration is C(x, 0, 0) = K2(D, U)/x.
X and the City: Modeling Aspects of Urban Life Page 15