expressed by the following formula for the pressures that prevail at sea-level:—
G＝[(2n sin φ + cos iv/r )v sec i ] / [83,000,000].
A popular exposition of this and other results of Ferrel’s work is given by Archibald in Nature (May 4, 1882), and still better in Ferrel’s Treatise on the Winds (New York, 1889, and later editions).
The charts of mean annual pressure, temperature and wind above referred to show certain broad features that embrace the whole system of atmospheric circulation, viz. the low pressures at the equator and the poles, the high pressures under the tropics, the trade winds below and the anti-trades above, with comparative calms under the belts of equatorial low pressure and tropical high pressure. The first effort of the mathematician was to explain how these mean average conditions depend upon each other, and to devise a system of general circulation of the wind consistent with the pressures, resistances and densities. But, as we have already said, such a system may be very far from that presented by the real atmosphere, and little by little we are being led to a different view of the question of the general circulation. The earlier students of storms generally accepted one of two views as to the cause of whirlwinds. They were either (1) formed mechanically between two principal currents of air flowing past each other, the so-called polar and equatorial currents; or (2) they were due to the ascent of buoyant air while the heavier air flowed in beneath, the whirling motion being communicated by the influence of the rotation of the earth, or by the greater resistances on one side than on the other. In order to explain why hurricanes and typhoons exist continuously for many days, or even weeks, it is necessary that there should be a source of energy to maintain a continued buoyancy and rising current at the centre, and this was supposed to be fully provided for by Espy’s proof of the liberation of latent heat consequent on the formation of cloud and rain. To this latter consideration Abbe in 1871 added the important influence of the sun’s heat intercepted at the upper surface of the cloud. At this stage of the investigation the whirlwind is but an incident in the general circulation of the atmosphere, but further consideration shows that it ought rather to be regarded as an essential portion of that circulation, and that when temperature gradients and density gradients exceed a certain limit the formation of great whirlwinds is inevitable. Therefore an atmosphere containing several whirlwinds is just as truly a system of general circulation in the one case as an atmosphere without a whirlwind is in the other. The formation of rain, the evolution of latent heat, and even the absorption of heat at the upper surface of the cloud really constitute a normal general circulation in this special case. We may therefore consider a system of vortices, which is a system of discontinuous motions, as the most natural solution of the equations of motion—but the mathematical treatment of this form of motion has not yet been sufficiently well developed, for the discontinuity relates not only to the motion but to the thermal conditions and the interchange of vapour and water.
In 1890 Professor Hann published a careful analysis of the actual temperature conditions prevailing over an extensive area of high pressure in Europe, and showed that the temperatures of the upper strata in both high and low areas, namely, in anti-cyclones and cyclones are often directly contrary to those supposed to prevail by Espy and Ferrel. This study necessitated a more careful examination into the radiation of heat from the dust and moisture of the atmosphere, and Professor Abbe seems to have shown that in areas of high pressure and clear weather a very slow descending movement throughout each horizontal layer gives time for a radiation of heat that explains the anomalies of temperature, but the dynamic phenomena still remained unexplained. On the other hand, von Helmholtz in several memoirs of 1888–1891 showed that waves or billows may be formed in the atmosphere of great extent at the dividing surface between upper and lower strata moving in different directions and with different velocities. Under specific conditions these billows may become like the breakers and caps of waves of the ocean when driven by the wind. The hypothesis that these aerial breakers correspond to our troughs of low pressure and the storms experienced in the lower atmosphere seemed very plausible. As these billows are formed between upper and lower air currents of great extent, which themselves represent a large portion of the horizontal circulation between the poles and the equator, it results that if von Helmholtz’s suggestion and Hann’s hypothesis are correct then all general storms must be considered as essentially a part of the general circulation rather than as caused by the vertical circulation over any locality. It must occur to everyone to adopt the intermediate view that, on the one hand, the local vertical circulation, with its clouds, rain, hail and snow, and evolution of latent heat, and, on the other hand the waves and whirls in the general circulation, mutually contribute toward our storms and fair weather. It only remains to allot to each its proper importance in any special case.
Undoubtedly aerial billows, and the clouds that must frequently accompany them, exist everywhere in the earth’s atmosphere. Perhaps their extent and importance are not properly appreciated. A voyage around the Atlantic Ocean in 1889–1890, made by Professor Abbe, specifically to study cloud phenomena, revealed many remarkable cases, such as the cumulus rolls that, extend in a remarkably symmetrical series from the island of Ascension westwards for 100 m. in the south-easterly trades, or the delicate fields of cirro-cumuli that extend from the islands of Santa Lucia. and Barbados for 200 m. eastwards under favourable conditions. The mixtures and vorticose motions going on within aerial billows to form these clouds have been interpreted by Brillouin. In the further elucidation of the mechanism of storms Hann showed that every study of observational material confirms the conclusion that the descent of denser cool dry air is as important as the ascent of warm moist air, and that although the evolution of latent heat within the clouds of a storm may explain the local cloud phenomena, yet it will not explain the storm as a whole. The first “norther or blizzard” that was charted at Washington in November 1871 was at once seen to be a case of the underflow of a thin layer of cold dry air descending from high altitudes above Canada on the eastern slope of the Rocky Mountains, but driven southward by an excess of centrifugal energy added to a moderate barometric gradient. It was seen that in such grand overturnings the descent of masses implies energy communicated by the action of gravity, but the whole mechanics of this process was not clear until the publication by Margules of his memoir Über die Energie der Stürme (Vienna, 1905), which will be referred to hereafter.
Mathematics have, almost without exception, assumed a so-called steady condition in the motion of the atmosphere in order to achieve a successful integration of the general equations of motion. The restrictions within which Helmholtz and others have worked, and the limits within which their results are to be accepted, have been analysed by Dr E. Herrmann in a memoir of which a translation is published in the bulletin of the American Mathematical Society for June 1896. Of course Herrmann’s own investigation is also based upon certain simplifying hypotheses, such as the absence of outside disturbing forces and of viscosity and friction, a homogeneous ellipsoidal surface, and a uniform initial temperature and rate of revolution corresponding to an initial state of equilibrium. If now the initial static equilibrium be disturbed by introducing a different distribution of temperature, viz. one that varies with altitude and latitude, but is uniform in longitude along any circle of latitude, then the first question is whether the atmosphere can settle down to a new state of static equilibrium. Herrmann shows that in general it cannot do so, but that the new state and the future states can only be those of motion and dynamic equilibrium. If, however, there be no external forces acting on the atmosphere, then in one case static equilibrium relative to the earth can occur, namely, when the new temperatures are so distributed in the atmosphere as to satisfy the equation
ρ r 2 w d V＝M,
in addition to the ordinary equations of elasticity, inertia and continuity previously given, and to those representing the boundary conditions, M being the total amount of inertia of the atmosphere relative to the axis of rotation. In general, the movements in the atmosphere must consist not only of an interchange between the poles and the equator, but also of east and west motions, and there must therefore be a different rate of diurnal rotation for each stratum. The second step in this inquiry is, Can these movements become perfectly steady with this unvarying or steady distribution of temperature? In other words, Can the temperature and the movements be so adjusted to each other that each shall remain invariable within any given zone of latitude? The reply to this is, that if they are to become thus adjusted they must satisfy a certain differential equation, which itself shows that steady motions and stationary temperatures cannot exist if there be any north or south component. Apart from the fact that Herrmann assumes no friction, it would seem that he has proved that steady motions and stationary pressures cannot exist in the atmosphere over a homogeneous spherical surface, and presumably the same result would follow of a rotating globe for the irregular surface of the actual globe. The motions of the real atmosphere must therefore consist of irregular and periodic oscillations and discontinuous whirls and rolls superimposed upon more uniform, regular progressions, but never repeating themselves. Consequently, the conclusions deduced by those who have assumed that steady conditions are possible must depart more or less from meteorological observations. There is a general impression that the belt of low pressure at the equator and the low areas at the poles and the high pressures under the tropics are pseudo-stationary, and really represent what would be steady conditions if we had an ideal smooth globe; but Herrmann’s researches show that the unsteadiness observed to attach to these areas under existing conditions would also attach to them under ideal conditions. They really have and must have irregular motions, and we, by taking annual averages, obtain an ideal annual distribution of pressure, temperature and wind that does not represent any specific dynamic problem. The averages represent what is considered proper in climatology, but are quite, improper and misleading from a dynamic point of view, and have no logical mechanical connexion with each other.
Closely connected, with this study of steady motions under a constant supply and steady distribution of solar heat comes the further question as to what regular variations in atmospheric pressure and wind can be produced by regular seasonal variations
