X and the City: Modeling Aspects of Urban Life (54 page)

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
/
y
₀
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
/
y
₀
~ (20 yr)
T
,
T
being a number of order unity. With
β
= 1.1 and the initial population measured in millions (
N
₀
= 10
⁶
n
), equation (18.8) reduces to the simple form

For a large city
t
_c
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
t
_c
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
]:

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.

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.

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 (< 0.1
μ
) 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
=
VRρ
_a
/η
< 1,
ρ
_a
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
/
ρ
_a
).)