earth and the friction, but in the latter also on the intensity of the ascending current of air. In passing from the outer to the inner surface the wind experiences a sudden change of angle, so that the directions of the winds are not continuous, although the movement and the barometric pressures are assumed to be continuous. This latter peculiarity does not occur in nature, and is undoubtedly an analytical result peculiar to Oberbeck’s method of treating the fundamental equations.
An improvement in the mathematical analysis was introduced by Dr F. Pockels of Göttingen in a memoir published in the Met. Zeit., 1893, pp. 9–19. He deduces equations showing the continuous changes of temperature, pressure, gradient, wind direction, and velocity from the centre of the cyclone to the outer edge of the anticyclone, or, more properly, the peri-cyclone; these, therefore, may reasonably be supposed to have their counterparts in nature. Such mathematical solutions, however, are based upon the assumption that we are dealing with a comparatively small portion of the earth’s surface, which may be considered as a plane having a uniform diurnal rotation and a uniform coefficient of friction. Moreover, the movements in the cyclones and anti-cyclones are assumed to be steady and permanent by reason of the perfect balance of all the forces involved therein. Of course these conditions are not exactly fulfilled, but in general Pockels shows that his theoretical results agree fairly well with the observed conditions as to wind and pressure. He computes the actual distribution of these elements under the assumption that the centre of the anti-cyclone is at latitude 55·5, and that the coefficient of friction is 0·00008, whereas viscosity proper would require only 0·0002. An elegant mathematical presentation of these studies in cyclonic motion is given by W. Wien, Lehrbuch der Hydrodynamik (Leipzig, 1900).
Notwithstanding the fact that these difficult mathematical investigations still lead us to unsatisfactory results, they are yet eminently instructive as showing the methods of interaction of the various forces involved in the motions of the atmosphere. We must therefore mention the interesting attack made by Oberbeck upon the problem of the general circulation of the atmosphere. His memoir on this subject was published in the Sitzungsberichte of the Academy of Sciences at Berlin in 1888. The fundamental assumption in this memoir implies that there is a general and simple system of circulation between the equatorial and the polar regions, but the eventual solution of the problem leads Oberbeck to two independent systems of winds, an upper and a lower, without any well-defined connexion at the polar and equatorial ends of these two currents, so that after all they are not rigorously re-entrant. Among the hypotheses introduced in the course of his mathematical work, the most important, and perhaps the one most open to objection, is that the distribution of temperature throughout the atmosphere in both the upper and lower strata can be represented by the equation T＝A + B (1−3 cos2 θ). Undoubtedly this equation represents observations in the lower strata near the surface of the earth, but the constants that enter into it, if not the form itself, must be changed for the upper strata. The solution arrived at by Oberbeck gives the following equations representing the components of the movement of the atmosphere toward the zenith V, toward the north N, and toward the east O:—
| V＝C(1−3 cos2 θ) 𝑓σ |
| N＝−6 C cos θ sin θ φ σ |
| O＝D[sin θ(1 − 3 cos2θ)𝑔σ + 6 cos2θγσ]. |
In accordance with these equations he deduces the general circulation of the atmosphere as follows: In the lower current the air flows from the polar regions eastward until it reaches the parallel of 30° or 40°; it then turns directly towards the equator, and eventually westward, until at the equator it becomes a strong east wind (or a so-called west current). In the upper layer the movement begins as an east wind, turns rapidly to the north at latitude 20° or 30°, and then becomes a south-west wind (or north-eastern current) in the northern hemisphere, but a north-west wind (and south-eastern current) in the southern hemisphere. Of course in the higher strata of air the currents must diminish in strength. In a second paper in the same year, 1888, Oberbeck determines the distribution of pressure over the earth’s surface as far as it is consistent with his system of temperatures and winds. His general equation shows that as we depart from the equator the pressure must depend upon the square and the fourth power of the cosine of the polar distance or the sine of the latitude, and in this respect harmonizes with Ferrel’s work of 1859, although more general in its bearings. By comparing his formulae with the observed mean pressure in different latitudes, Oberbeck obtains the general angular velocity of the air relative to the earth, i.e. 0·0292 (sin2φ−0·0836), which is quite small and is a maximum (4·6 metres per second) at latitude S. 56° 27′. H. Hildebrandsson (1906) showed that observations do reveal an east wind prevailing above the equatorial belt of calms.
Contemporary with Oberbeck’s admirable memoirs are those by Professor Diro Kitao, of the university of Tōkyō, who, as a student of mathematics in Germany, had become an expert in the modern treatment of hydrodynamic problems. In three memoirs published by the Agricultural College of the university of Tōkyō in the German language in the years 1887, 1889 and 1895, he develops with great patience many of the minutiae of the movement of the earth’s atmosphere and cyclonic storms. The assumptions under which he conducts his investigations do not depart from nature quite so far as those adopted by other mathematicians. Like Ferrel, he adheres as closely as possible to the results of physical and meteorological observations; and although, like all pure mathematicians, he considers Ferrel as having departed too far from rigorous mathematical methods, yet he also unites with them in acknowledging that the results attained by Ferrel harmonize with the meteorology of the earth.
The fact is that the solution of the hydrodynamic equations is not single, but multiplex. Every system of initial and boundary conditions must give a solution appropriate and peculiar to itself. The actual atmosphere presents us with the solution or solutions peculiar to the conditions that prevail on the earth. Entirely different conditions prevail on Jupiter and Saturn, Venus and Mars, and even on the earth in January and July, and therefore a wholly new series of solutions belongs to each case and to each planet of the solar system. It matters not whether we attempt to resolve our equations by introducing terrestrial conditions expressed by means of analytical algebraic formulae, and integrate the equations that result, or whether we adopt a graphic process for the representation of observed atmospheric conditions and integrate by arithmetical, geometrical or mechanical processes. In all cases we must come to the same result, namely, our resulting expressions for the distribution of pressure and wind will agree with observations just as closely as our original equations represented the actual temperatures, resistances and other attending conditions. In the last portion of Kitao’s third memoir he gives some attention to the interaction of two cyclonic systems upon each other when they are not too far apart in the atmosphere, and shows how the influence of one system can be expressed by the addition of a certain linear function to the equations representing the motions of the other. He even gives the basis for the further study of the extension of cyclonic storms into higher latitudes where conditions are so different from those within the tropics. Finally, he suggests in general terms how the resistances of the earth’s surface, in connexion with the internal friction or viscosity of the air, are to be taken into consideration, and shows under what conditions the assumptions that underlie his own solutions may, and in fact must, very closely represent the actual atmosphere.
The General Circulation of the Atmosphere.—If the meteorologist had a sufficient number of observations of the motions of the air to represent both the upper and lower currents, he would long since have been able to present a satisfactory scheme showing the average movement of the atmosphere at every point of its course, and the paths of the particles of air as they flow from the poles to the equator and return, but hitherto we have been somewhat misled by being forced to rely mainly on the observed movements of clouds. This motion has been called the general circulation of the atmosphere; it would be a complex matter even if the surface of the earth were homogeneous and without special elevations, but the actual problem is far different. Something like this general circulation is ordinarily said to be shown by the monthly and annual charts of pressure, winds and temperature, such as were first prepared and published by Buchan in 1868, and afterwards in Bartholomew’s Physical Atlas of 1899. We must not, however, imagine that such charts of averages can possibly give us the true path of any small unit mass of air. The real path is a complex curve, not re-entrant, never described twice over, and would not be so even if we had an ideal atmosphere and globe. It is a compound of vertical and undulatory movements in three dimensions of space, variable as to time, which cannot properly be combined into one average.
The average temperatures, winds and pressures presented on these charts suggest hypothetical problems to the student’s mind quite different from the real problems in the mechanics of the atmosphere—problems that may, in fact, be impossible of solution, whereas those of the actual atmosphere are certainly solvable. The momentary condition presented on any chart of simultaneous observations constitutes the real, natural and important problems of meteorology. The efforts of mathematicians and physicists have been devoted to the ideal conditions because of their apparent simplicity, whereas the practical problems offered by the daily weather chart are now so easily accessible that attention must be turned towards them. The most extensive system of homogeneous observations appropriate to the study of the dynamics of the atmosphere is that shown in the Daily Bulletin of International Simultaneous Observations, published by the U.S. Signal Service in the years 1875–1384, with monthly and annual summaries, and a general summary in “Bulletin A,” published by the U.S. Weather Bureau in 1893. The study of these daily charts for ten years shows how the general circulation of the atmosphere differs from the simple problems presented in the idealized solutions based on monthly and annual averages. The presence of a great and a small continent, and a great and small ocean, and especially of the moisture, with its consequent cloud and rain, must enter into the study of the problem of the general circulation. The most prominent features of the general circulation of the atmosphere are the system of trade winds, north-easterly in the northern tropics and south-easterly in the southern tropics, the system of westerly winds beyond the trade-wind region, namely, north-westerly in the north temperate and south-westerly in the south temperate zone, and again the system
