Now let’s think “laterally” for a moment; the right-hand side of equation (19.9) looks suspiciously like the equation of the normal distribution. (I have a now-defunct ten Deutschmark bill with the equation and graph of that distribution on it.) This has the classic “bell-curve” shape. In statistics, the random variable X is said to be normally distributed with mean μ and standard deviation σ if its probability distribution is given by
The total area between the X-axis and the graph of the function f(X) is equal to one (see Figure 19.1). If we compare equation (19.11) for f with that for C(x, y) above in equation (19.9), it is clear that and Carrying this equivalence over to the solution (19.10) for C(x, y, z), we see that the plume concentration is distributed normally in the crosswise direction (y) but also vertically (z), because now and The standard deviation σ is a measure of how tightly the concentration sits about the mean value μ (zero here); so the plume width in the x- and z-directions is here proportional to , the square root of the distance downstream. It is also proportional to the square root of the diffusion coefficient and inversely proportional to the square root of the wind speed U. It is known that turbulence can increase D (which in this context is often referred to as the eddy diffusivity).
In light of these comments, let us summarize the predictions of the above model based on the solution (19.10) for C(x, y, z) now dropping the subscript on K2:
Figure 19.1. Normal distribution f(X) given by equation (19.11) for μ = 4 and σ = 4.
I. The downwind concentration is directly proportional to K(D, U), the source emission rate.
II. The more turbulent the atmosphere, the wider the lateral spread of the plume after any given time.
III. The maximum concentration at any location is found at ground level on the line y = 0, and is inversely proportional to distance x downwind from the source.
A more general analysis of the particle emission problem shows that K ∝ U−1, so that the maximum concentration will be smaller for higher wind speeds. But even a less sophisticated model such as ours can be adapted to emission from an elevated source. A common example is the plume of smoke from a smokestack. Because the particles have a longer time to diffuse before they reach the ground, we expect that the maximum concentration will depend on the height H of the stack. Let’s investigate this. The simplest and most obvious modification to the model is to replace z by z − H in equation (19.10) for C(x, y, z). However; there is now a lower boundary for the particles—the ground—and the associated problem of what boundary condition to impose on the problem. At this point the reader may object: in the previous model the source is on the ground at “height” z = 0; so why worry about the effect of ground in this model? In that case the maximum concentration was always at ground level, so the effect was implicit in the model. As we shall see, a simple modification is to assume that particles are partially reflected from the ground when they diffuse and settle downward. Perfect reflection is unlikely, of course; the ground is not a mirror and the particles are not perfectly elastic—indeed, recalling the difference that clay or grass courts can make in tennis, the problem is rather more complex than we can investigate in depth here. Nevertheless, some insight into the modeling process and the conclusions drawn from it can be illustrated by assuming that there is an “image” smokestack emitting pollutants at a rate αK(0 ≤ α ≤ 1) from z = −H. Of course, this is just a mathematical artifact (and a common one in this kind of problem). We are only interested in what happens for 0 z ≥ 0. To this end, the suggested form for the concentration distribution is
The divisor 1 + α in this expression is required to ensure that C reduces to the case above when H = 0. Again, we are interested in the concentration at ground level, which simplifies to
For any given x > 0 this will have its maximum value when y = 0, so it will be of interest to determine the location of the maximum of the function
Using the first derivative test it is straightforward to show that the maximum ground-level concentration of
at x = xm = UH2/4D. Using this very simple model we have been able to reproduce a result first derived in 1936 [34], namely that the maximum ground-level concentration from a plume released from a height H is inversely proportional to H2.
Exercise: Verify this result.
Summarizing, therefore, the model predicts that
I. Cmax is inversely proportional to the square of the plume-release height H.
II. Cmax is inversely proportional to the wind speed U.
III. xm is directly proportional to the square of the plume-release height H.
IV. xm is directly proportional to the wind speed U.
V. Cmax and xm are both independent of the ground “reflection coefficient” α.
The model developed here is very simplistic, so it is encouraging to note that the first two predictions are the same as those from more sophisticated models. Physically, they make sense, since the plume is diffusing in both the y- and z-directions as the wind carries it downstream, and the pollutants are spread over a wider area (which has dimensions of (length)2). Furthermore, a stronger wind will stretch out the plume more per unit time, diluting it all the more as it does so.
As for the remaining predictions, III and IV follow naturally for the same reasons as I and II. In reality, the effect of ground reflection must play a role, though perhaps only a minor one compared to that of H and U. One mechanism neglected here is that of buoyancy; very often the effluents released (or, indeed, smoke from forest fires) is warmer than the surrounding air, and it continues to rise for a time after it is released. But to keep things relatively simple, that effect has not been included here.
X = C(x, t): A DISTRIBUTED SOURCE
We have regarded the source of effluent to be on the ground or at the top of a smokestack. In each case the source is in effect a point source. This is because of the nature of the solution at x = 0, referred to above. But consider a long line of slowly moving bumper-to-bumper traffic along a straight stretch of road. This can be considered a distributed source of particulates (current emission regulations notwithstanding)—a line source. Of course, the average speed of the traffic, the length of the road, and the wind strength and direction will affect the concentration of particles (such as hydrocarbons from the tailpipes) at any point on or off the road. To build a simple model of pollutant dispersal for a line of traffic of length L, we now use the emission rate per unit length, namely K/L. We will again neglect buoyancy and regard the line source as being placed at ground level along the y-axis from −L/2 to L/2 (though it is not necessary to specify this in what follows). We shall utilize the earlier models by considering only a cross-wind U in the x-direction as before; thus the pollutant is blown directly from the road into the neighboring land or cityscape. For a long traffic line L (strictly, an infinitely long line) there can be no variation of C in the y-direction because the source is uniform along that line. Therefore we approximate the finite-L case by requiring that the particles diffuse only in the vertical direction (again, the effect of wind dominates any diffusion in the x-direction). The governing equation now simplifies to
Again, dropping the subscript, this time on K1, the solution, based on equation (19.9), is now
Note that at ground level (z = 0) the concentration varies more slowly downwind, as x−1/2, compared with x−1 for a point source. Clearly the concentration for any x > 0 is maximized at ground level, but this result is a means to an end. In many situations, the source will be better approximated, not by a point or a line, but by an area composed of multiple sources in an urban region. These can effectively combine because of wind and diffusion in such a way as to render the individual sources unidentifiable. Since we are interested in the ground-level concentration, we set z = 0 in the above equation and imagine for simplicity a rectangular source by integrating the result with respect to x. Therefore the accumulated concentration CA(x) has the following dependence on x:
More realistic models indicate a greater dependence on x
than this, especially if the atmospheric conditions preclude the particles from unlimited diffusion vertically. There is evidence to suggest that the rate at which pollutants are emitted and the region affected by pollution both increase faster than the population does. Modeling this would be a very substantial exercise for the reader!
Chapter 20
LIGHT IN THE CITY
With such particles suspended in the atmosphere for sometimes days or weeks at a time, smog presents a danger to health, but in London it was also known as a “pea souper” because one could not see one’s hand in front of one’s face! In fact, as a result of the Great London Smog of 1952 (caused by the smoke from millions of chimneys combined with the mists and fogs of the Thames valley), the Clean Air Act of 1956 was enacted. With this in mind we now turn to the topic of how air pollution may affect visibility.
X = Is: VISIBILITY IN THE CITY
We start with an apparent non sequitur by asking the following question. Have you ever been in an auditorium of some kind, or a church, in which your view of a speaker is blocked by a pillar, but you can still hear what is being said? I’m pretty sure you must have experienced this. Why can your ears receive auditory signals, but your eyes cannot receive direct visual ones (excluding Superman of course)? The reason for this is related to the wavelengths of the sound and light waves, being ≈1m and ≈5 × 10−7 m, respectively. The latter, in effect, “scatter” more like particles while the former are able to diffract (“bend”) around an obstacle comparable in size to their wavelength. By the same token, therefore, we would expect that light waves can diffract around appropriately smaller obstacles, and indeed this is the case, as evidenced by softly colored rings of light around the moon (coronae) as thin cloud scuds past its face. Another diffraction-induced meteorological phenomenon is the green, purple-red, or blue iridescence occasionally visible in clouds. But it is the collective phenomenon known as scattering that we wish to discuss in some detail, in order to better appreciate the character of air pollution and its effect on the light that reaches our eyes.
So: when light is deflected in some manner from its direction of travel, it is said to be scattered. There are several mechanisms that contribute to the scattering of light by particles in the atmosphere: reflection, refraction, and diffraction being the most common, though they are not necessarily mutually exclusive effects. The size of the particles determines which mechanism is the predominant one.
Visibility is reduced to some extent by the absorption of light, but scattering by particles and droplets is the primary source of this reduction. We perceive distant objects by contrast with their environment, and this contrast is reduced by the scattering of light from particles and droplets in the line of sight. Thus visibility is reduced. Depending on their size, particles may settle out of the air in due course; this sedimentation process had already been mentioned in the previous chapter.
Although the particles will be irregular in shape, we can define an effective radius as the average of that of (i) the largest sphere that can be inscribed in the particle and (ii) the smallest sphere that contains the particle. If this radius exceeds about 10 microns (10−5 m), they settle out in several hours. As we saw, when an object falls in air, it is subject to at least two separate forces; its weight (acting downward) and air resistance or drag (acting upward). A third force is that of buoyancy (also acting upward), but since the air density is negligible compared with that of the particle, this can be neglected. The particle weight is proportional to its mass and hence to the cube of its radius. As we saw in the previous chapter, the air resistance is only proportional to its radius, so the weight dominates the drag by a factor that increases as the square of the radius; hence larger particles fall faster, at least initially. As the speed increases, so does the air resistance. If the particle falls from a sufficient height, these competing forces eventually balance each other, and the net force is zero. At this point the particle falls with a constant speed, the terminal speed. On the other hand, if the particles are smaller than 10 microns in size, they can remain suspended in the air for several days, buffeted by air currents.
This, then, is a qualitative summary of the hydrodynamic aspects of sedimentation discussed earlier. By contrast, the optical aspects are more complicated because of the range of particle sizes compared with the wavelengths of visible light (approximately 0.4–0.7μ) A convenient measure of relative size is the radius-to-wavelength ratio R/λ. When this ratio is at least about ten, the particles are considered to be large, and it is convenient to regard light in terms of rays. This is the domain of geometrical optics, and as illustrated in Figure 20.1, the three processes mentioned above can occur. Light rays can be partially reflected from the surface of the particles, refracted on passing through the interior, or diffracted (“bent”) around the edges. All three mechanisms are exhibited in the phenomenon of the rainbow (see Appendix 11); light is refracted and reflected by raindrops to produce this beautiful colored arc in the sky. Less familiar is the third important mechanism—diffraction—a consequence of the wavelike properties of light. This is responsible for some of the more subtle rainbow features—pale fringes below the top of the bow and, as already noted, iridescence in clouds near the sun.
Figure 20.1. Light incident on a large particle may undergo some or all of the indicated processes: (i) reflection at the surface; (ii) refraction into and out of the particle; (iii) internal reflection; and (iv) edge diffraction. Redrawn from Williamson (1973).
For large particles, the amount of light “scattered” by diffraction is as much as that by the other two mechanisms. Some of the refracted light may be absorbed by the particles; if so, this will affect the color of the outgoing radiation. An extreme example of this is black smoke—in this case most of the incident radiation is absorbed. When little or no absorption occurs, large particles scatter light pretty much in the forward direction, so the observer looking toward the light source—the sun, usually—will see a general whitish color. When the particles are not large, the “light ray” approach of geometrical optics is inadequate to describe the scattering processes; the wave nature of light, as mentioned above, must be taken into account. Particles for which R/λ ≈ 1 scatter light in a wider “band” away from the incident direction (resulting in the sky appearing hazy), but smaller particles (for which R/λ 1) are better able to scatter light multidirectionally. If the aerosols are smaller than about 0.1 μ, the light is scattered much more uniformly in all directions; as much backward as forward and not much less off to the sides. Furthermore, the amount of scattering is very sensitive to the wavelength of the incident light; as we will see below the degree of scattering is ∝ λ−4. This means that the light of shorter wavelength, such as blue or violet, is scattered much more than the longer wavelength red light. Only when we look in the direction of the setting sun, for example, do we see the red light predominating—most of the blue has been scattered out of the line of sight. Think for a moment of cigarette smoke curling upward from an ashtray; typically it is bluish in color—a consequence of the smoke particles being smaller than the wavelengths of light. It is the blue light that is scattered more, and this is what we see.* This is an example of Rayleigh scattering, the same phenomenon that makes the sky blue. Rayleigh scattering arises because of wavelength-dependent molecular scattering
To understand the phenomenon of scattering from a more analytic point of view, we need to recall some basic physics. An electromagnetic wave has, not surprisingly, both an electric and a magnetic field that are functions of time and space as it propagates. The direction of propagation and the directions of these fields form a mutually orthogonal triad (Figure 20.2). And when an electromagnetic field encounters an electron bound to a molecule, the electron is accelerated by the electric field of the wave. It’s a type of “chicken and egg” situation, because an accelerated electron will also radiate electromagnetic energy in the form of waves in all directions (to some extent), and this is the scattered radiation that we have b
een discussing. Consider Figure 20.3, which illustrates such a situation for a small particle with R/λ 1 as a snapshot in time.
Figure 20.2. Orthogonal triad formed by the direction of the electric and magnetic fields and the direction of propagation for an electromagnetic wave.
Figure 20.3. A spherical particle of radius R λ experiences a nearly constant electric field E. (The wavelength of the incident light is λ.) Redrawn from Williamson (1973).
As shown in the figures, the wave propagates with speed c in the x-direction, with the electric field in the z-direction (it is said to be polarized in that direction; this is a qualification we shall address below). This field varies periodically with frequency, = c/λ (where λ is the wavelength), and its fluctuations affect the electrons it encounters. To a much lesser extent, the more massive nuclei are also affected, but this will be ignored here. We shall denote the incident electric field by
where x0 is the location of this particular electron on the x-axis, E0 is the amplitude of the wave, and ω = 2π is called the angular frequency of the wave. As the wave passes, electrons will be accelerated back and forth in the z-direction, which in turn will radiate electromagnetic waves—this radiation is the scattered light. An important consequence of our assumption that the particle is small is this: since R λ the electric field is almost uniform throughout the particle, so every electron (with charge e) experiences close to the same force (eE) accelerating it, proportional to its displacement s from its former position of equilibrium in the absence of the wave. The force will always be such as to move the electron back toward that position, so it can be incorporated in Newton’s second law of motion as follows:
X and the City: Modeling Aspects of Urban Life Page 16
