of snow left within the clouds must be very small, and the cloud has the delicate appearance peculiar to cirrus; Hertz’s original diagram is quite covered by these systems of α, β, γ and δ lines, and the moisture lines. The lines show the density of the moist air at any stage of the process. The improved diagram by Neuhoff, published in 1900, is reprinted in the second volume of Abbe’s Mechanics of the Earth’s Atmosphere, and its arrangements help to solve many problems suggested by the recent progress of aerial research.
In von Bezold’s treatment of this subject only illustrative diagrams are published, because the accurate figures, drawn to scale; are necessarily too large and detailed. He presents graphically the exact explanation of the cooling by expansion, the loss of both mass and heat by the rainfall and snowfall, and the warmth of the remaining air when it descends as foehn winds in Switzerland and chinook winds in Montana. Even in the neighbourhood of a storm over low lands and the ocean, the warm moist air in front, after being carried up to the rain or snow stage, flows away on the upper west wind until a corresponding portion of the latter descends drier and warmer on the opposite side of the central low pressure. In order to have a convenient term expressive of the fact that two masses of air in different portions of the atmosphere having different pressures, temperatures and moistures, would, if brought to the same pressure, also necessarily attain the same temperature, von Bezold introduced the expression “potential temperature,” and devised a simple diagram by which the potential temperature may be determined for any mass of air whose present temperature, pressure and moisture are known. In an ascending mass of air, from the beginning of the condensation onwards, the potential temperature steadily increases by reason of the loss of moisture, but in a descending mass of air it remains constant at the maximum value attained by it at the highest point of its previous path. In general the potential temperatures of the upper strata of the atmosphere are higher than those of the lower. In general the so-called vertical temperature gradient is smaller than would correspond to the adiabatic rate for the dry stage. This latter gradient is 0·993° C. per hundred metres for the dry stage, but the actual atmospheric observations give about 0·6°. Apparently this difference represents primarily the latent heat evolved by the condensation of vapour as it is carried into the upper layers, but it also denotes in part the effect of the radiant heat directly retained in the atmosphere by the action of the dust and the surfaces of the clouds. Passing from simple changes due to ascent and descent, von Bezold next investigated the results of the mixture of different masses of air, having different temperatures and humidities, or different potential temperatures. The importance of such mixtures was exaggerated by Hutton, while that of thermodynamic processes was maintained by Espy, but the relative significance of the two was first clearly shown by Hann as far as it relates to the formation of rain, and further details have been considered by von Bezold. The practical tables contained in Professor Bigelow’s report on clouds, and those of Neuhoff as arranged for the use of those who follow up von Bezold’s train of thought, complete our methods of studying this subject.
A most important application of the views of von Bezold, Hertz and Helmholtz was published by Brillouin in his memoir of 1898. Just as we have learned that the motions of the atmosphere are not due either to the general distribution of heat or to local influences exclusively, but in part to each, and just as we have learned that the temperature of the air is not due either to radiation and absorption or to dynamic processes exclusively, but to both combined, so in the phenomena of rain and cloud the precipitation is not always due to the cooling by mixture, or to the cooling by expansion, or to radiation, but is in general a complex result of all. The effect of the evaporation of cloudy particles in the production of descending cold currents has always been understood in a general way, but was first brought to prominence by Espy in 1838, and perhaps equally forcibly by Faye in 1875. Helmholtz, in his memoirs on billows in the atmosphere, showed how contiguous currents may interact on each other and mix together at their boundary surface; but Brillouin explains how these mixtures produce cloud and rain—not heavy rains, of course, but light showers, and spits of snow and possibly hail. He says: “When the layers of clear or cloudy air are contiguous, but moving with very different velocities, their motion, relative to the earth because of the rotation of our globe, assumes a much more complicated character than that which obtains when the air has no horizontal but only a vertical motion. We know in a general manner what apparent auxiliary forces must be introduced in order to take into account this rotation, and numerous meteorologists have published important works on the subject since the first memoirs by Ferrel. But their points of view have been very different from mine. The subjects that I desire to study are the surfaces of discontinuity as to velocity, temperature and cloudiness in one special case only. Analytical methods permit us to resolve complex questions only for limited areas in longitude and for contiguous zones within which the movements are steady, but not necessarily uniform nor parallel. But it is evident that one can learn much as to the condition of permanence or destruction of annular zones having uniform and parallel movements. Thus simplified, the questions can be treated by elementary geometric methods, by means of which we at once rediscover and complete the results given by Helmholtz for zones of clear air and discover a whole series of new results for zones of cloudy air.” Among Brillouin’s results are the following theorems:—
A. If the atmosphere be divided into narrow zonal rings, each extending completely around the globe, thus covering a narrow zone of latitude, and if each is within itself in convective equilibrium so that the surfaces of equal pressure shall be surfaces of revolution around the axis of rotation, then within any such complete ring in convective equilibrium the angular velocity of any particle of the air will vary in inverse ratio of the square of its distance from the axis of rotation, or ar2 is constant; that is to say, the air will not move like a rotating solid, but will have a variable angular velocity, smaller far from the axis and greater near to it.
B. The surfaces of equal pressure are more concave towards the centre than is the surface of the globe itself, and they are tangent to the latter only along the parallel where calms prevail.
C. A heavy gaseous atmosphere resting upon a rotating frictionless globe divides itself into concentric rings whose angular movements increase as we pass from the polar region towards the equatorial ring; the central globe rotates more rapidly than the equatorial atmospheric ring.
D. The surface of separation between two contiguous concentric rings must be such that the atmospheric pressure shall have the same value as one approaches this surface from either direction, and the surface of separation is stable if the differences of pressure in different parts of this surface are directed towards the surface of equilibrium. As the distribution of pressure along a line parallel to the axis of rotation is independent of the velocity of rotation, the ordinary condition of stability, viz. that the gas of which the lower ring is composed shall be denser than that above, will hold good for this line. In general, any inclination of the surface of separation to the horizon amounting to 10° must be associated with very small differences of density and large differences of velocity; in practice the inclinations are far less than 10°.
E. If the surfaces of equal pressure or isobars are nearly horizontal, as in ordinary cases, the calculations are comparatively easy to make. Let the inclination of the isobaric surface ascending towards the pole be φ; let h1 be a distance counted along the axis of the earth, and H1 the distance measured in the direction of the attraction of gravity; then the angle of inclination of the isobaric surface is given by the equation
where λ is the complement of the angle between the direction of gravity and the line drawn to the poles, or the axis of rotation of the earth. The surface of separation is that over which the pressure is the same in two contiguous masses or zones, and is identical with a vertical plane only when the densities and velocities in the two layers have certain specific relations to each other. It can never lie between the isobaric surfaces that Brillouin designates as 1 and 2. In order that the equilibrium may be stable, it is necessary that when ascending in the atmosphere along a line parallel to the polar axis one should traverse layers of diminishing density. In the midst of any zone there cannot exist another zone of limited altitude; it must extend upwards indefinitely. Whenever there is any zone of limited altitude it must necessarily have, near its highest or lowest point, an edge by which it is attached to the surface of separation of two other neighbouring zones. In other words, the surfaces of separation of the three zones, of which one is limited and the other two are indefinite, must all run together at a common point or edge, very much as in the problem of the equilibrium of thin films.
F. When the contiguous zones are cloudless the mixtures take place under the following conditions: Starting from the stable conditions, the cloudless mixture ascends on the polar side when the west wind which prevails on the equatorial side of the surface of separation is warmer, but descends between the pole and the equatorial side of the horizon when the west wind which prevails on the equatorial side of the surface of separation is colder. The mixtures of cloudless air rapidly occupy the whole height of the two layers that are mixing. When they form along a surface that becomes unstable the whirlwind that is thus engendered is sensibly cylindrical at first, but finally becomes extremely conical; This whirlwind may be limited as to height when the two contiguous masses that are mixing are surmounted by a third clear or cloudy layer which intersects the other two and whose lower surface is stable. (Brillouin suggests that possibly this corresponds to the formation of water-spouts and tornadoes.)
G. When the contiguous zones are cloudy and the mixtures produce decided condensations, and sometimes even precipitation, the study of these must follow closely in the train of thought followed out by von Bezold. When the contiguous winds are feeble, but the temperatures are very different and the zones are near the equator, then the position of the mixture can be inverted by condensation, since the influence of difference of pressure becomes predominant. At the equator, whatever may be the difference of temperature, a mixture that is accompanied by condensation always rises if the surface of separation is stable. The condensation increases by the expansion, each zone of mixture being an outburst of ascending cumuli. At the equator, whatever may be the difference of
