Mag., Feb. 1890), and especially by Margules (Vienna Sitz. Ber. 1890–1893). The two latter have shown the truth of a proposition enunciated by Kelvin in 1882, without demonstration, to the effect that the free oscillation produced by a relatively small amount of tide-producing force will have an amplitude that is larger for the half-day term than for the whole-day term. They therefore explain the diurnal and semi-diurnal variations of the barometric pressure as simple pressural tides or waves of expansion, originally produced by solar heat, but magnified by the resonance between forced and free waves in an atmosphere and on a globe having the specific dimensions of our own. The analytical processes by which Laplace and Kelvin arrived at this special solution of the tidal equation were objected to by Airy and Ferrel, but the matter has been, as we think, most fully cleared up by Dr G. H. Ling, in a memoir published in the Annals of Mathematics in 1896. He seems to have shown that, although a literally correct result was attained by Laplace in his first investigation, yet his methods as presented in the Mécanique céleste were at fault from a rigorous analytical point of view. The process by which a diurnal temperature wave produces a semi-diurnal pressure oscillation, as explained by Rayleigh and Margules, may be stated as follows: The diurnal temperature wave having a twenty-four hours period is the generating force of a diurnal pressure tide, which is essentially a forced and small oscillation. The natural period of the free waves in the atmosphere agrees much more nearly with twelve than with twenty-four hours. In so far as the forced and the free waves reinforce each other, the semi-diurnal waves are reinforced far more than the other, so that a very small semi-diurnal term in the temperature oscillations will produce a pressure oscillation two or three times as large as the same term would in the diurnal period. These reinforcements, however, depend upon the elastic pressure within the atmosphere, just as does the velocity of sound. If the prevailing barometric pressures were slightly increased, the adjustment of the twelve-hours free wave of pressure to the forced wave of temperature could be so perfect that the barometric wave would increase to an indefinite extent. For the actual temperatures the periodicity of the free wave is about thirteen hours, or somewhat longer than the forced wave of temperature, so that the barometric oscillation does not become excessive. It would seem that we have here a suggestion to the effect that if in past geological ages the average temperature at any time has been about 268° C. on the absolute scale, then the pressure waves could have been so large as to produce remarkable and perhaps disastrous consequences, involving the loss of a portion of the atmosphere. A modification of this idea of resonance has been developed by Dr Jaerisch, of Hamburg (Met. Zeit., 1907), but the general truth of the Kelvin-Margules-Rayleigh theorem still abides.
The Thermodynamics of a Moist Atmosphere.—The preceding section deals with an incompressible gas, and therefore with simple, pure hydrodynamics. If now we introduce the conception of an atmosphere of compressible gas, whose density increases with altitude, so that rising and falling currents change their temperatures by reason of the expansion and compression of the masses of air, we take the first step in the combination of thermodynamic and hydrodynamic conditions. If we next introduce moisture, and take precipitation into consideration, we pass to the difficult problems of' cloud and rain that correspond more nearly to those which actually occur in meteorology. This combination has been elucidated by the works of Espy and Ferrel in America, Kelvin in England, Hann and Margules in Austria, but especially by Hertz, Helmholtz, and von Bezold in Germany, and by Brillouin in France. A general review of the subject will be found in Professor Bigelow’s report on the cloud work of the U.S. Weather Bureau and his subsequent memoirs “On the Thermodynamics of the Atmosphere” (Monthly Weather Review, 1906–1909).
The proper treatment of this subject began with the memoir of Kelvin on convective equilibrium (see Trans. Manchester Phil. Soc., 1861). The most convenient method of dealing approximately with the problems is graphic and numerical rather than analytical and in this field the pioneer work was done by Hertz, who published his diagram for adiabatic changes in the atmosphere in the Met. Zeit. in 1884. He considers the adiabatic changes of a kilogram of mixed air and aqueous vapour, the proportional weights of each being λ and μ respectively. In a subsequent elaborate treatment of the same subject by von Bezold in four memoirs published during 1889 and 1899, the formulae and methods are arranged so as to deal easily with the ordinary cases of nature which are not adiabatic; he therefore prepares diagrams and tables to illustrate the changes going on in a unit mass of dry air to which has been added a small quantity of aqueous vapour, which, of course, may vary to any extent. Both Hertz and von Bezold consider separately four stages or conditions of atmosphere: (A) The dry stage, where aqueous vapour to a limited extent only is mixed with the dry air. (B) The rain stage, where both saturated vapour and liquid particles are simultaneously present. (C) The hail stage, where saturated aqueous vapour, and water, and ice are all three present. (D) The snow stage, where ice vapour and snow itself, or crystals of ice, are present. The expressions aqueous vapour and ice vapour do not occur in Hertz’s article, but are now necessary, since Marvin, Fischer and Juhlin have been able to show that vapour from water and vapour from ice exert different elastic pressures, and must therefore represent different modifications of liquid water. According to Hertz, we may easily follow this mass of moist air as it rises in the atmosphere, if by expansion it cools adiabatically so as to go successively through the four preceding stages. For a few thousand feet it remains dry air. It then becomes cloudy and enters the second stage. Next it rises higher until the cloudy particles begin to freeze into snow, sleet or hail, which characterizes the third stage. When the water has frozen and the cloud has ascended higher, it contains only ice particles and the vapour of ice, a condition which characterizes the fourth or snow stage. If in this condition we give it plenty of time the precipitated ice or snow may settle down, and the cloudy air, becoming clear, return to the first stage; but the ordinary process in nature is a circulation by which both the cloud and the air descend together slowly, warming up as they descend, so that eventually the mixture returns to the first stage at some level lower than the clouds, though higher than the starting-point.
The exact study of the ordinary non-adiabatic process can be
carried out by the help of Professor Bigelow’s tables, and especially
by the very ingenious tables published by Neuhoff (Berlin, 1900), but the approximate adiabatic study is so helpful that in fig. 10 we have
traced a, few lines from Hertz’s
Fig. 10.—Diagram for Graphic Method of following Adiabatic Changes.
diagram sufficient to illustrate its
use and convenience. The reader
will perceive a horizontal line at
the base representing sea-level;
near the middle of this line is zero
centigrade; as we ascend above
this base into the upper regions
of the air we come under lower
pressures, which are shown by
the figures on the left-hand side.
The scale of pressures is logarithmic,
so that the corresponding
altitudes would be a scale of
equal parts. The temperature
and pressure at any height in
the atmosphere are shown by
this diagram. If the air be saturated
at a given temperature, then
the unit volume can contain only
a definite number of grams of
water, and this condition is represented
by a set of moisture lines,
indicated by short dashes, showing
the temperature and pressure
under which 5, 10 or 20 grams of water may be contained in
the saturated air. Let us now suppose that we are following the
behaviour of a kilogram mass of air rising from near sea-level, where
it has a pressure of 750 millimetres, a temperature of 27° C., and
a relative humidity of 50%. A pointer pressing down upon the
diagram at 750 millimetres and 27° C. will represent this initial condition.
A line drawn through that point parallel to the moisture
lines will show that if this air were saturated it would contain about
22 grams of water; but inasmuch as the relative humidity is only
50%, therefore it actually contains only 11 grams of water, and an
auxiliary moisture line may be drawn for this amount. If now the
mass rises and cools by expansion, the relation between pressure and
temperature will be shown by the line α α. When this line intersects
the inclined moisture line for 11 grams of water we know that
the rising mass has cooled to saturation, and this occurs when the
pressure is about 640 millimetres and the temperature 13·2° C.
By further rise and expansion a steady condensation continues,
but by reason of the latent heat evolved the rate of cooling is diminished
and follows the line β β. The condensed vapour or cloud
particles are here supposed to be carried up with the cooling air,
but the temperature of freezing or zero degrees centigrade is soon
attained—as the diagram shows—when the pressure is about 472
millimetres. At this point the special evolution of latent heat of
freezing comes into play; and although the air rises higher and more
moisture is condensed, the temperature does not fall because the
water already converted into vapour and now becoming ice is giving
out latent heat sufficient to counteract the cooling due to expansion.
This illustration from Hertz’s diagram therefore shows that the
curve for cooling temperature coincides with the vertical line for
freezing, and is represented on the diagram by the short piece β γ.
By this expansion due to ascent the volume is increased while the
temperature is not changed; therefore, the quantity of aqueous
vapour has increased. When the ascending mass has reached the
level where the pressure is 463 millimetres it has also reached the
moisture line that represents this increase in aqueous vapour. As
this shows that the aqueous particles have now all been frozen, and
as the air is now continuously rising, while its temperature is always
below freezing-point, therefore at levels above this point the vapour
that condenses from the air is supposed to pass directly over into the
condition, of solid ice. Therefore from this point onwards the falling
temperatures follow along the line γ γ, and continue along it indefinitely
From these considerations it follows that the clouds
above the altitude of freezing temperatures are essentially snow
crystals, and if the air rises slowly there may be time for the water
and ice to settle down towards the ground; in this case the quantity
