DIFFUSION AS A RESULT OF ATMOSPHERIC TURBULENCE Watching the drift of smoke from a bonfire or factory will convince the observer that wind, instead of having a steady streaming motion, is characteristically turbulent as described in Chapter III. According to Brunt (1934), large numbers of small-scale eddies, whose periods are of the order of I second, arc usually present in the turbulent boundary layer, and at least two-thirds of the eddying energy is associated with eddies of less than 5 seconds. The action of these very numerous eddies of varying size on the very numerous spores produced from plant sources, makes some regularity in the dispersal pattern possible.
The study of eddy diffusion has proved difficult, but it provides the most promising approach to the elucidation of dispersal. Before describing the methods in detail, a few general notions—familiar to physicists, but mostly unfamiliar to biologists—must be introduced.
We are attempting to discover laws governing spore diffusion in the atmosphere. In nature this is often a complex process, as there are obstacles preventing the free flow of air. We therefore use a device familiar to physicists—making a simplified model in the hope that, if we can under stand the process of diffusion under simple conditions, we shall be able to attack the more complex situations found in nature. The assumptions we have to make for a simplified model are as follows.
(i) The field. Diffusion is assumed to be taking place in three dimen sions in the atmosphere over a plane surface which is of indefinite extent, free from topographical irregularities—not necessarily `smooth', but, if aerodynamically rough, then uniformly so.
(ii) Co-ordinates. To describe movement over the plane surface we need a system of co-ordinates. Their origin, `0', is conveniently taken to be the point of liberation of the pollen or spores. The 'x'-axis is horizontal and positive in the down-wind direction, and the 'y'-axis is also hori zontal but at right-angles to the direction of the wind. Lengths above and below the origin are measured on the vertical 'z'-axis.
(iii) Sources. Particles are liberated from a source. The simplest form of source is a `point source', and this may either liberate a number `Q of spores at a single instant (an `instantaneous point source'), or it may be a `continuous point source' emitting Qspores per second.
Instead of a point source we may have a `line source'. For simplicity we assume that the line is horizontal, and is emitting Qspores per centi metre of its (effectively) infinite length. The line source in turn may be instantaneous or continuous. Furthermore, we may have an `area source' (emitting Q spores per square centimetre), or a `volume source'. Real sources in the field that correspond approximately to these ideal sources would be a single plant (point), a hedge (line), a ground crop (area), and an orchard or forest stand (volume). The dimensions of the source must be treated as relative to their distance: thus a field would be regarded as effectively a point source when considered from distances many times its own width.
All these sources may be instantaneous or continuous. The cloud from an instantaneous point source is a puff or spherical cloud, whereas the conical cloud arising from a continuous point source is familiar in the smoke plume from a chimney. A continuous point source can be viewed as made up of a succession of overlapping instantaneous emissions.
(iv) Standard deviation. Suppose that a `puff' of spores has been liber ated at an instant from a point source into a wind and has become subject to the action of atmospheric eddies which move individual spores apart at random. After a short time the particles composing the cloud will show a scatter around their origin (Fig. 6). At any instant such a cloud has two characteristics which we could compute if we had all the data: (t) the mean position of the particles, i.e. the centre of the cloud, which can be expressed as a point on the system of x, y, and z ordinates; and (2) the standard deviation, a, of the particles from their mean position. When the cloud has travelled farther down-wind it will have a new mean position, and during the time the cloud has been travelling it will have been further diluted by eddies, its particles will have got farther apart, and consequently their standard deviation will have become larger.
The next problem is to find a relation between the standard deviation and the distance travelled. How does a grow as x grows ? This is a prob lem that has excited the interest of many workers since the First World War, who were attempting to predict the concentration of gas clouds, smoke screens, smoke trails, and crop pathogens.
The pioneer in the subject was the Austrian meteorologist, Wilhelm Schmidt (1918, 1925), who put forward a theory similar to those being developed almost simultaneously in Britain by G. I. Taylor and L. F. Richardson. Schmidt supposed that, with a given state of turbulence of the air, diffusion of particles proceeds like the diffusion of heat in a solid, but with an atmospheric turbulence coefficient Alp replacing the coeffi cient of thermal conductivity. He showed that for these conditions His work is now mainly of historical interest, but we should note one interesting feature: according to Schmidt the standard deviation squared is proportional to the time during which diffusion has been taking place, so that on his theory the standard deviation will not be constant at a given distance, but will depend on the time taken to reach that distance, i.e. upon the speed of the wind.
Schmidt also assumed that the particles in the diffusing cloud are brought to ground-level by their fall under gravity, and he used measured values of terminal velocity to fix dispersal limits for various organisms (cf. Table XXVII).
Sutton (1932) recognized that diffusion in the atmosphere differs from molecular diffusion of heat in a solid in one important respect. Diffusion in a solid is constant (depending on the mean free path of the molecules) however long the diffusion has been going on. Diffusion in the atmosphere is much more complex, because atmospheric eddies are of a vast range of sizes, varying from a centimetre or so up to eddies that we recognize as fluctuations in wind direction, and even to cyclones and anti cyclones. Sutton realized that the size of eddy effective at a given moment in diluting a cloud is of the same order as the size of the cloud itself at that moment. Thus a r-cm. eddy would not effectively dilute a cloud I-metre in diameter, and a i,000-metre eddy would merely carry a I-metre cloud around bodily without diluting it. The eddy that dilutes a 1-metre cloud is itself of the order of 1 metre. This led Sutton to an equation for the standard deviation which is fundamentally different from that of Schmidt: 2 = where t = time; u = wind-speed; `C' is a new coefficient of diffusion with dimensions (L)t; and `in' is a number varying between 1•24 in extremely stable, non-turbulent wind, and under conditions of extreme turbulence. The value for normal overcast conditions with a steady wind is na = Because wind-speed multiplied by time equals distance we can write Sutton's formula: = °. This suggestion that is a function of the distance, x, is not unreasonable, because the surface roughnesses which generate eddies are spread out along the distance travelled by the cloud. It is moreover a tempting theory, because we need not know the wind speed under which dispersal takes place.
Values of C decrease with height because conditions at great heights are unfavourable for the formation of eddies. Values for in appear to increase with longer sampling periods, and Sutton suggests that in itself is a function of time. In making continuous observations over a long period on the density of a cloud, he suggests that the random element may become smoothed out, so that, over a sufficiently long period, in = 2. These possibilities should be borne in mind when the density formulae described below arc applied to some biological data where the sampling period is very long.
In practice it is found that near ground-level, diffusion takes place faster on the x- and y-axes than on the vertical z-axis. Turbulence is then said to be `non-isotropic', and C has to be represented by its com ponents: and C.
The number in is an indicator of the degree of turbulence of the air and is, as a first approximation, independent of the mean wind-velocity. It is primarily affected only by those factors which tend to damp out or increase turbulence, such as the vertical temperature gradient and the roughness of the ground. For conditions of spore dispersal tests it seems appropriate to assume values of = o 5-1•o (metre)*, = (metre), and m = 1.75-2•o.
Expressions for the concentration of particles in a cloud emitted from various types of source were deduced by Sutton (1932), and are analogous to heat-conduction equations, as follows: (i) An instantaneous point source, such as a puff of Qgrams of smoke, or a number Qof spores emitted at an instant of time. Here the concen tration in the cloud is given by where `r' = distance from the centre of the puff or cloud.
(ii) A continuous point source, such as a factory chimney emitting Q particles per second. Here, to obtain an integral that can be handled conveniently, the assumption is made that the spread of the cloud laterally and vertically is small compared with its spread down-wind. When emission has continued long enough for the distribution to reach a steady state, the concentration is given approximately by The cross-wind concentration shows a `normal' distribution of particles. On the axis of the cloud (y = z = 0) the concentration is given by the simpler expression and because, according to the theory, in cannot exceed a•o, the fall-off in concentration on the axis of a point-source cloud cannot be more rapid than the inverse square, no matter how turbulent the wind may be.
(iii) A continuous line source at right-angles to the mean direction of the wind, emitting Qparticles per second per centimetre, and assuming the line to be of infinite length Values obtained by Sutton suggest that, as a rough and ready rule, a finite line source behaves as a line of infinite length for distances of travel of the cloud up to 4 times the actual length of the line. For points on the xOy plane Sutton's statistical method does not exhaust the possible approaches to the problem of atmospheric diffusion, and attempts to find a still more useful model continue (see H. L. Green & Lane, 1957). From the theory of T. von Korman, Calder (1952) developed an equation which is said to give better predictions than Sutton's theory up to distances of loo metres, but not for greater distances. It is also difficult to apply Calder's equations except to point sources. Another theory of dispersion, based on fluctuations in wind direction, is outlined by Sheldon & Hewson (1958), and a recent theory by Clarenburg (196o) will have to be taken into account.