Darrigol O. Worlds of flow. A history of hydrodynamics from the Bernoullis to Prandtl (794382), страница 51
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As soon as unstable equilibrium interferes,this condition is no longer met. Hence chance still exists in our [predictive] horizon;but in reality chance only is a way of expressing the defective character of ourknowledge and the 'roughness of our combining power.'As an example of an unstable system with high sensitivity to the initial conditions (astoday's physicists would put it), Helmholtz gave a vertical, rigid bar that can freely rotatearound its lower, fixed extremity. Clearly, the smallest departure from the vertical positionhas dramatic consequences on the future of this system. Helmholtz judged that manymeteorological situations led to such instabilities.
67There already existed, however, successful applications of mechanical and thermalprinciples to the atmosphere. The most trivial that Helmholtz mentioned was the seasonal64Cf. Garber [1976] pp. 53-7, Kutzbach [1979] pp. 22-7 (on Espy in the 1 840s), 45-58 (on Thorn son, Reihe, andPes1in in the 1 860s).65Thomson [1865]; Reye [1864], [1872]. Cf.
McDona1d [1963a], Garber [1976] pp. 56-7, Kutzbach [1979]pp. 46-58.66He1mho1tz [1876].61Ibid. pp. 140, 1 62, 151. Hehnho1tz borrowed the bar on a fingertip from Reye [1872] p. 40.VORTICESFig. 4.8.169Whirling winds over burning reed bushes. From Reye [1872].variation of average temperature, obviously a consequence of the average height of theSun over the horizon. More generally, meteorologists could hope to explain regularities inspatial and temporal averages.
For instance, it was known that trade winds arise from thecombined effect of the heating of equatorial air and the rotation of the Earth. According toGeorge Hadley's theory of1735, the hot air around the equator rises into the upperatmosphere and the resulting rarefaction of the lower atmosphere induces low-altitudewinds converging toward the equator. During the latter motion, the air moves towardlower latitudes for which the linear velocity of the surface of the Earth is larger. Consequently, the air flow relative to the Earth is shifted toward the west.
The resulting windsare NE above, and SE below the equator. Hadley further noted that the effect of therotation of the Earth would be much too large were it not diminished by friction on theground. Lastly, he described the inverse effects in the upper atmosphere, resulting in SWwinds in the northern hemisphere and NW winds in the southern hemisphere.
Beyond thetrade-wind belt, these upper trade winds cool down and fall back on the Earth, which68explains the dominance of western winds in mid-latitudes.Helmholtz improved this theory by taking into account a mechanical effect overlookedby Hadley, namely that, when air moves toward the equator, its angular momentum isconserved, and therefore its absolute linear velocity diminishes as the inverse of its distancefrom the axis of the Earth. This effect adds to Hadley's relative velocity. Helrnholtz alsooffered an explanation of equatorial calms: the lower trade-wind air can only rise after itsrelative westward motion has been halted by friction, so that its centrifugal force becomes68Helmholtz [1 876] p.
142; Hadley [1735]. For a critical assessment of this theory, cf. Lorenz [1 967] pp. 1-3.170WORLDS OF FLOWas high as possible (for a relative westward motion, the absolute rotation of the air issmaller than that of the Earth). 69Helmholtz described similar effects around the poles. The descent of cold air at the polesimplies a diverging flow of air in the lower atmosphere and a converging one in the upperatmosphere. Owing to the rotation of the Earth, the resulting winds are NE around theNorth Pole and SE around the South Pole.
Helmholtz thereby explained Dove's observation that Germany was alternately exposed to these dry cold winds and to warm, humidSW winds from the equator.704.4.3Tropical stormsHaving described this global circulation system, Helmholtz turned to its perturbations.Those were easiest to explain, he judged, where they were the rarest, that is, in the tropicalzone. He borrowed from Reye the basic scenario for the formation of these storms.7 1Reye himself owed the basic idea of a thermal origin of storms to an Americanmeteorologist, James Espy, who developed his views in the 1830s. According to Espy,hot saturated air at the ground level is in an unstable equilibrium. Any local perturbationinduces an ascending motion of this air in a column.
In this process the air expands andcools down, so that water is condensed. The formation of clouds is thus explained.Moreover, the condensation frees the caloric of the vapor and thus prevents the risingair from cooling too fast. As Espy could determine from his own experiments on theadiabatic expansion of saturated air, this heating of the air column by condensation wassufficient to keep the density of the rising air smaller than that of the surrounding air andthus sufficient to sustain the rising motion.72This ascending convection of warm, saturated air not only accounted for the formationof clouds, but it also provided an explanation of storms.
The warm column of ascendingair, Espy noted, needed to be fed at its base by converging air. The correspondinghorizontal winds were those observed in storms. Their radial pattern was still acceptablein Espy's times, owing to the imprecision and confusion of data. In around 1 860, anotherAmerican meteorologist, William Ferrel, recognized the importance of the effects of theEarth's rotation on atmospheric motion, gave its precise mathematical expression, andmodified Espy's theory accordingly. Espy's converging winds had to be deflected, in acounterclockwise manner in the northern hemisphere. The resulting motion was a whirlwith growing rotation toward the center of the storm.
7369Helmholtz [1 876] pp. 142-5. In the late 1 850s, the American meteorologist William Ferrel had already givena correct mathematical formulation of the effect of the Earth's rotation on the motion of the atmosphere(Kutzbach [1 979] pp. 35-41). As his works were largely unknown in Europe, Hehnholtz's use of the conservationof angular momentum in this context may have been original.70Helmholtz [1876] p. 146. In the 1 850s, Ferrel had already described the 'cold polar vortex' (Kutzbach [1979]pp. 39-40; Khrgian [1970] pp. 239-40). James Thomsen's general-circulation system of 1857 was similar toHelmholtz's (Khrgian [1970] pp.
239, 242). For a critical history of general-circulation systems, cf. Lorenz[1967] pp. 59-78.71 Helmholtz [1876] p. 148.72Espy [1841]. Cf. McDonald [1963b], Garber [1976] pp. 53-5, Kutzbach [1979] pp. 22-7.73Ferrel [1 860]. Cf. Kutzbach [1979] pp. 35-41, and p. 37n for literature on the introduction of the Coriolisforce.VORTICES171Reye's theory of whirling storms was a modernized version of Espy's theory, based on athermodynamic analysis of the saturated adiabatic process and on a major insight by theAustralian-based meteorologist Thomas Belt. According to Belt, the cause of storms wasthe unstable accumulation of hot, low-density air near the ground. Similarly, the first stepof Reye's theory of storms was the formation of an inferior layer of hot, saturated air,owing to solar heating. The hydrostatic equilibrium of this air with the upper, dry air isunstable because saturated air is more expandable than dry air: if some saturated airbegins rising through drier air, then it will continue to do so, because its density diminishesfaster for the same decrease in pressure.
When the equilibrium is broken at a given place,the hot humid air rises in a vertical column. The corresponding depression induces a radialconverging motion of the lower air. Due to the rotation of the Earth, this air whirls aroundthe center. 74Helmholtz reproduced this theory with a superior understanding of the relevant dynamics. For example, he explained the whirling motion by the conservation of angularmomentum.
A ring of air entering the depression without initial rotational velocity withrespect to the Earth has an absolute rotational velocity (increasing with the latitude) anda corresponding angular momentum. When the ring converges, its angular momentumis a constant, so that its absolute angular velocity must grow. Consequently, it gainsa rotational velocity with respect to the Earth. In the northern hemisphere, the rotationseen from the sky is counterclockwise. 75Most originally, Helrnholtz deduced the direction of motion of a tornado.
In his paper onvortex motion, he had shown that a linear vortex parallel to a plane wall moved in a parallelplane in the direction of the fluid flow between the vortex and the wall. The reason is thatthe velocity field is the same as it would be if the fluid were unlimited and had anothervortex, the mirror image of the real vortex with respect to the wall (the flow in the medianplane of the fictitious two-vortex system is parallel to this plane, in conformance with theboundary condition of the real system). According to one ofHelmholtz's theorems, the realvortex must move with the velocity induced by its inirror image.
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