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The Physics of Blown Sand and Desert Dunes
By Ralph Alger Bagnold
Dover Publications, Inc.Copyright © 1954 R. A. Bagnold
All rights reserved.
SAND AND DUST
1. THE ORDER OF SIZE OF SMALL PARTICLES. USE OF THE LOGARITHMIC SCALE
FIGURE 1 is intended to show approximately, in diagrammatic form, the relative sizes of material particles which are susceptible to the action of winds of normal speeds. They range from little pebbles, on the large side, through sand grains and rain and fog drops to specks of dust and the tiny water droplets which constitute clouds. Thence, smaller and smaller, to the minute particles that form thin smokes and hazes. The estimated diameter of large molecules of matter is included for completeness.
The diameter of the largest particle in the diagram is seen to be some fifty million times that of the smallest. For such a diagram the ordinary linear scale is clearly unsuitable. If a linear scale (one in which 10 is added to 10 in the ordinary way to give the position of the succeeding division 20) had been used, 99 per cent. of the horizontal space in the diagram would have been occupied by the pebbles and the larger sand grains ; and all the rest of the material, all the vast range of small particles from sand grains to hazes, would have been squashed up into the last single millimetre of the scale.
The linear scale, since it was first cut on the wall of an Egyptian temple, has come to be accepted by man almost as if it were the one unique scale with which Nature works and builds. Whereas it is nothing of the sort. Its sole value lies in giving due prominence to the differences and sums of quantities, when these are what we want to display. But Nature, if she has any preference, probably takes more interest in the ratios between quantities ; she is rarely concerned with size for the sake of size.
For many purposes a far more convenient scale is one in which equal divisions represent equal multiples—one which multiplies 10 by 10 to make 100 rather than one which adds 10 to 10 to make 20. This point is stressed, because throughout the subject with which we are dealing the relations between one quantity and another are very often found to be of the logarithmic type ; so that we have the choice, in representing relations of this kind diagrammatically, between dealing with awkward logarithmic curves on a linear scale (with the added disadvantage of being limited in the extent of the scale) or exhibiting simple straight-line relations on an unlimited ratio scale such as that of Fig. 1. I shall use both scales indiscriminately, according to which is most suitable to the occasion.
It will be seen from the diagram that of the whole size range of 50,000,000 to 1, sand occupies but a tiny belt between 1 mm. and mm., or a ratio of 50 to 1. That is, the size range of sand grains occupies but one-millionth of the whole range of size of small particles which are affected by the wind. The truth of this statement depends, of course, on how we define sand ; and that, in turn, from the nature of the subject, depends on the relative behaviour of small particles in a wind. It is to this question that we must first turn.
2. THE DEFINITION OF GRAIN SIZE
If an object of any size, shape, or material is allowed to fall from rest through any fluid, whether air, water, or oil, its velocity will increase, at first with the acceleration of gravity, but thereafter at a decreasing acceleration till it reaches a constant value known as the Terminal Velocity of Fall. The reason is that the net force on the object is the resultant of the pull of gravity acting downwards, and the resisting force of the fluid acting always in a direction opposite to that of the motion. As the velocity of the motion increases, so does the resistance against that motion, till eventually the two are equal. No net force any longer acts on the object, which therefore moves at a constant speed.
The downward force of gravity depends on the volume of the object and its density. The resisting force depends on the area of frontage exposed to the fluid, on the shape of the object, and on its speed through the fluid. Hence, since natural solid particles are of irregular and haphazard shape, the individuals, even of a collection of particles or grains chosen to be all of the same average size, will not have the same rate of fall.
The first task is to find a convenient way of specifying the average size and shape of the grains of a given sample so that the average rate of fall of these specified grains can be calculated. Now a great deal of experimental work has been done on the behaviour of spherical objects in air and other fluids, and the fluid resistance of a sphere of a given diameter and a given density can be accurately calculated. The most useful method, therefore, of specifying our sand grains is to replace them by a collection of imaginary spheres of the same material and of such a diameter that they will behave in air in the same way as the average sand grain of the sample. We then have a simple workable material of identical grains completely specified by diameter and density ; and, starting with any given initial conditions, we can calculate with confidence the subsequent paths of these ideal average grains through the air.
To obtain the diameter of a sphere which will be equivalent to a given sand grain, the mean dimensions of the grain are found by passing it through a series of sieves each having a known size of aperture which differs but slightly from that of the next in the series. The mean dimension is taken to be that midway between the size of the aperture of the sieve through which the grain will just pass and of that of the next sieve which will retain it. This mean diameter is then multiplied by a suitable shape-factor. For desert sand this factor can be taken as 0.75.
The value of this factor can be found very simply by experiment. A small shower of mixed sand of all grain sizes is allowed to fall from a known height on to a slowly rotating disc covered with sticky paper, from a hopper which is made to open by a trigger operated by a cam on the disc. By this means the beginning of the fall is made to correspond in time with the passage of a fixed zero radius on the disc below. The exact time of fall can then be measured by the angle between the zero radius and the radius passing through any required grain (which is held in position by the sticky paper). The continuous curve in Fig. 2 gives the calculated time of all of quartz spheres of various diameters given by the scale on the left. The dots show the measured times of fall of actual sand grains whose measured diameters have been multiplied by a shape-factor of 0.75 to reduce them to their corresponding equivalent diameters, as explained above. The figure shows fairly well the degree to which sand grains of the same general size may be expected to vary as regards their wind resistance. It also shows the degree of closeness with which one can calculate the wind resistance and the rate of fall of a real grain of a given size.
The above is but a very brief reference to the subject of grain size and rate of fall. In contrast to our imperfect knowledge of the general effect of the presence of numerous particles on the motion of a fluid, a really formidable amount of detailed work has been devoted both to the rate of fall of individual particles through a fluid at rest, and to the measurement of the size of small particles. An excellent bibliography of original papers on the subject is given in a paper by Heywood ; and the whole subject is treated of at length by Krumbein and Pettijohn.
3. THE SUSPENSION OF SMALL PARTICLES. THE DISTINCTION BETWEEN SAND AND DUST
Returning to Fig. 1; in the lower part will be found given the rates of fall of the various kinds of particles shown in the upper part, the velocities being measured downwards. Owing to the enormous range of velocities, a logarithmic scale is again necessary. It will be observed that the smallest particles, those constituting the thin smokes and hazes, do not fall at all. This is because they are so susceptible to displacement by collisions with molecules of air that the very feeble downward pull of gravity is counter-balanced by the dispersive tendency of the jostling they receive from the air molecules.
The velocity of the wind is never constant. The short-period variation of speed, or the gustiness, is due to the internal movements of the air. As these movements, or eddies, circulate haphazard about axes in all directions, there are internal air currents which move upwards and downwards as well as those which move forwards, backwards, and sideways relatively to the general direction of the wind. Close to the ground the upward and downward components of the eddy velocity have been found to be less than the components in other directions. Though the ratio of the upward eddy velocity to the mean velocity of the wind is very variable, an average figure of for this ratio is probably not far out. If, therefore, there are solid particles in the air whose terminal velocities of fall are less than of the mean velocity of the wind, some of these particles may be carried upwards and may remain for a time in partial suspension. On the other hand, larger particles with greater terminal velocities will remain on or near the ground.
Now it will be shown later that when sand is being driven by the wind, the grains rarely rise higher than 1 metre above the ground, and that the average height is much less—of the order of 10 cm. And it will also appear that the wind velocity, as measured at this height, which is just strong enough to set the grains on the ground in motion, is in the neighbourhood of 5 metres/sec., or 11 miles per hour. Taking as a rough estimate the value of 1/5 × 5 = 1 metre/sec. as the maximum upward velocity of the internal movement of this wind, we see from Fig. 1 that sand grains having this velocity of fall have a diameter of about 0-2 mm. We might therefore expect that at somewhere about this size there should be a noticeable change in the character of the distribution of small loose particles existing on the Earth's surface.
This is found to be the case. When samples of natural sand are analysed by sifting, it is found that, in general, grains of one diameter predominate, and that the weights of sand of diameters both larger and smaller fall off rapidly as the diameter departs from the ' peak ' value. And in the finest wind-blown sands the predominant diameter is never less than 0.08 mm. Usual values, depending on the locality, lie between 0.3 and 0-15 mm.
We can thus define the lower limit of size of sand grains, without reference to their shape or material, as that at which the terminal velocity of fall becomes less than the upward eddy currents within the average surfaces wind. Particles of smaller size tend to be carried up into the air and to be scattered as dust.
The upper limit of sand size is that at which a grain resting on the surface ceases to be movable either by the direct pressures of the wind or by the impact of other moving grains.
Any substance consisting of solid non-cohesive particles which lie within these limits of size may be, classed as 'sand '. Such substances all possess one peculiar characteristic : alone of all artificial or natural solids they have the power of self-accumulation —of utilizing the energy of the wind to collect their scattered components together into definite heaps, leaving the intervening country free of grains. They can do this in the open, unsheltered by wind-breaks other than those of their own making ; and the heaps, or dunes, can retain their identity and can move about from place to place.
4. THE COMPOSITION OF NATURAL SAND. THE PREVALENCE OF QUARTZ
The relative prevalence of any material on the Earth's surface depends upon (1) the abundance or otherwise of the source from which it is derived, (2) the rate at which it is formed, and (3) the rate at which it is destroyed or transformed into something else. Consider, for instance, the case of dry snow which, according to the definition just given, is but a special form of 'sand', and which may display all its characteristics. Its source, water, is abundant; and it is formed rapidly and in great quantity in suitable climates. Yet, as dry snow, it exists in very limited amount. This is because, being unstable, it is so soon changed into firn-ice, water, or vapour. In addition, its tiny crystals, being brittle and easily cracked by impact, tend to be reduced to powder or dust when driven along the surface.
As rocks are degraded by the action of water and weather into smaller and smaller particles, fragments which are either soft, brittle or easily soluble pass rapidly down the scale of size. These have but a short life as sand grains, and do not contribute much material to the existing sand of the Earth's surface.
Thus, in order that a substance may be present as sand in large quantities, it must satisfy the following requirements ; its mother substance must be, or have been, plentiful; it must be one which resists the action of chemical weathering, of solution, and of abrasion, and it must be tough enough to resist fracture by the impact of other grains during transport.
Of all natural substances, crystalline silica (quartz) complies best with these requirements; and it is of quartz that the bulk of sand grains are composed.
Quartz is not the only material, however, which occurs in the form of sand. Sands which are predominantly quartz often contain grains of other materials, and under special conditions, near a plentiful source of supply, sands are found which are composed entirely of other substances—e.g. of broken sea shells, magnetic iron-ore, flint, &c. But as regards their behaviour in a fluid, whether air or water, neither the composition of the grains, nor their shape, are found to have any considerable effect on the character of the accumulations produced. Grain size is far more important: for though the weight of a grain of a given size may vary with the material in a ratio of two to one, this variation in weight will be offset by a change of only the cube root of two, or 1.26, in the size.
Since quartz so greatly predominates in the sands found on the Earth's surface, the experimental work on which much of this book is based has been done with quartz sand. But the results in nearly all cases can be applied without change to other sands.
5. THE ORIGIN AND FORMATION OF SAND GRAINS
It is generally accepted by geologists that the bulk of the quartz sand grains found in the earth's crust, whether occurring free on the surface or in sandstones and other sedimentary rocks laid down long ago, have originated from the disintegration of quartz-bearing rock followed by some process of mechanical abrasion. But apart from the fact that certain quartz-rich rocks—e.g. granites—contain ready-made quartz particles of suitable size, there seems to be no unanimity as to what process is mainly responsible for having reduced the grains to their present size and shape.
Of the possible processes we have dry weathering on the one hand, and the mechanical action of water and frost on the other. In the case of a relatively insoluble and chemically inert substance such as quartz, the only kind of dry weathering available appears to be temperature splitting. But though there is plenty of evidence that the violent temperature changes experienced by surface rocks in desert regions can and do cause rocks and large stones to split, there is little or no evidence that this process extends to fragments smaller than 1 cm. in diameter. For the stresses set up depend on the temperature differences within the body of the material, and the material does its best to reduce these differences by the conduction of heat from the hotter to the colder parts. Though the surface may be exposed to rapid and violent changes of air temperature, the maximum possible rate of change is not unlimited. Hence the smaller the body the more easily can its internal adjustment of temperature keep pace with the externally applied changes. In the case of such small particles as sand grains it is extremely unlikely that any temperature splitting could ever take place in Nature.
Excerpted from The Physics of Blown Sand and Desert Dunes by Ralph Alger Bagnold. Copyright © 1954 R. A. Bagnold. Excerpted by permission of Dover Publications, Inc..
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