Darrigol O. Worlds of flow. A history of hydrodynamics from the Bernoullis to Prandtl (794382), страница 52
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A similar reasoning appliesto the case when the vortex is no longer parallel to the wall. Then the vortex moves in thedirection of the fluid motion in the sharp angle made by its axis and the wall (of course,there is no motion when the vortex is perpendicular to the wall).76For a tornado in the northern hemisphere, the lower and upper trade winds tilt thewhirling axis: the upper part of the tornado is dragged NE and the lower part SW.Consequently, the flow in the sharp angle made by the vortex and the surface of theEarth is directed NW.
In the absence of external winds, the tornado must move in the samedirection.Lastly, Helmholtz offered some vague considerations on the more complex storms inthe temperate zone. Here he departed from Reye, who believed in a subtropical origin ofall storms, independent of the general circulation system. Instead, Helmholtz sharedDove's opinion that mid-latitude perturbations were due to the encounter of polar air74Reihe [1 872] pp. 40-6 (instability and weather columns), 1 37-8 (rotation of the Earth); Belt [1859].
Cf.Kutzbach [1979] pp. 88-96. Reihe was unaware of Ferrel's works.75Helmho1tz [1876] pp. 151-7. Cf. Kutzbach [1979] pp. 96-9.76Helmholtz [1858] p. 127, [1876] p. 159.WORLDS OF FLOW1 72and equatorial air. This is not to say that he followed Dove in every respect. The patriarchof German meteorology still believed that the mechanical conflict between the two aircurrents was the cause of mid-latitude storms, whereas Helmholtz knew well that onlythe thermal contrast between the two air masses could account for the energy of thestorms.77He!mholtz proposed that the equatorial air rose over the polar air, cooled downadiabatically, and thus precipitated rain.
Wherever air rose, a depression occurred andinduced whirling winds. The modem reader may recognize here some elements of thepolar-front theory of mid-latitude storms.784.5 Trade winds4.5.1A revelationTen years elapsed before Helmholtz returned to meteorology. The incentive was anobservation he made during a vacation in the Swiss Alps. His short report for the BerlinAkademie began as follows:Clouds and thunderstorm formation.On one day of the first half of September this year Mount Rigi offered a clearperspective toward Jura.
At a height somewhat lower than the viewing point-theRigi's Kanzli-there was the quite regular, upper horizontal limit of a heavier andmore turbid air layer; this limit was indicated by a thin layer of small clouds whichwent from North to South in narrow stripes, and which revealed the whirls formed byperturbation and rolling up of the limiting surface.Where the average mountaineer would only have seen one more pretty scene of nature,Helniholtz registered a concrete realization of a central hydrodynamic concept, namely, avortex sheet in the sky! (compare with Fig.
4.9).79From this observation, Helmholtz conceived that discontinuous motion played animportant role in atmospheric phenomena. He could thus explain a paradox that musthave been on his mind for some time. Hadley's theory of trade winds made frictionresponsible for the moderate value of the eastern or western component of these windsin comparison to the exceedingly high values that the Earth's rotation would by itselfimply.
This explanation seemed plausible for the lower trade winds, which experiencefriction on the surface of the Earth. However, it failed for the upper trade winds, which areexposed only to internal friction. As Helmholtz confirmed by an application of theprinciple of mechanical similarity, the effects of the viscosity of the air on atmospheric77Helmholtz [1876] p. 160. On Dove's views, cf. Khrgian [1970] pp. 168-71, Kutzbach [1979] pp.
1 1-16,81-2.78Helmholtz [1876] p. 160. In 1841, the American meteorologist Elias Loomis had proposed a similarmechanism, without the whirling and without the connection between the two currents and the general circulation.In the 1 870s, the Norwegian meteorologist Henrik Mohn also used a two-current thermal mechanism, but withoutthe cold front; Mohn assumed that the opposite cold and warm air currents mixed before the latter could rise.Cf. Kutzbach [1979] pp. 27-35 (on Loomis), 76-80 (on Mohn).79Helmholtz [1886].VORTICESFig. 4.9.173A rare cloud formation at Denver, Colorado, on 14 February 1953.
Photograph by P.E. Branstine, inColson [1954] p. 34.motion are extremely small or confined to the surface of the Earth. Yet, undamped uppertrade winds would give western winds as impossibly high as 1 30 m I s at a latitude of 30° . 80The key to the paradox, Helmholtz realized after contemplating a vortex sheet in the Swisssky, was discontinuous motion. The boundary between the upper and lower tradewinds hadto be a surface of discontinuity, analogous to those occurring in the mouth of an organ pipe.This unstable surface waved and eventually unrolled into a series 'of vortices. The twodifferent air strata thus became thoroughly mixed. In the case of organ pipes, the mixingallowed the transfer of momentum from the blown air to the oscillating air column. In theatmospheric case, it produced the required damping of the upper trade winds, as well as heatexchange between the upper and the lower air.
In Helmholtz's words: 8 1The principal obstacle to the circulation of our atmosphere, which prevents thedevelopment of far more violent winds than are actually experienced, is to be foundnot so much in the friction on the Earth's surface as in the mixing of differentlymoving strata of air by means of whirls that originate in the unrolling of surfaces ofdiscontinuity. In the interior of such whirls the originally separate strata of air arewound in continually more numerous and therefore thinner layers spiraling abouteach other; the enormously extended surfaces of contact allow a more rapid exchangeof temperature and the equalization of their movement by friction.80Helmholtz [1888] pp. 290-3. If the spatial variations of velocity are smooth, the large scale of atmosphericmotions implies a negligible effect of viscosity.
However, when there are abrupt variations of velocity (on thesurface of the Earth, or between two strata of air), the surface friction does not depend on the scale and can havea non�negligible damping effect.81Helmholtz [1888] p. 308. In modern terms, the discontinuity surface evolves into a fractal object.174WORLDS OF FLOW4.5.2 The thermal genesis of discontinuity surfacesIn order to establish this conclusion, Helmholtz needed 'to show how by means ofcontinually effective forces [owing to gravitation, solar heat, rotation of the Earth, andfriction on the Earth's surface] surfaces of discontinuity were formed in the atmosphere.'For this purpose, he introduced two idealizations. Firstly, he ignored the longitudinalvariations of the state of the atmosphere.
Secondly, he assumed that, within sufficientlythin layers of the atmosphere, the air was in (dry) adiabatic equilibrium and had a uniformmotion. Accordingly, the basic concept of his analysis was that of ring-shaped air layersrotating around the axis of the Earth with a uniform angular momentumand a 'heat content' e.82u per unit massBy the latter quantity, Helmholtz meant the temperature that a sample of the air wouldtake when brought adiabatically to the normal pressure. At least since his theory of thefoehn, and even more since his reading of Reihe, Helmholtz was aware of the importanceof adiabatic processes in atmospheric phenomena. Perhaps he also knew of Thomson'spaper on adiabatic convection and rate of temperature decrease with elevation.
By anargument similar to the one used for viscosity, he showed that ordinary thermal conduction was negligible in smooth, continuous atmospheric motions, so that only Thomson'sconvective equilibrium, and not ordinary thermal equilibrium, applied to homogenous airlayers. For the former kind of equilibrium, the 'heat content', and not the ordinarytemperature, is uniform. This concept, under a different name, quickly became central inmeteorological thermodynamics. Dove's successor at the head of the Prussian Meteorological Institute, Wilhelm von Bezold, called it 'potential temperature', which had noovertones of the old caloric theory.83From Euler's equation of motion and the law of adiabatic compression applied within agiven layer, Hehnholtz derived the relation1 2 G 1a OP ("- l)/" = 2 u 2 + r - 2 !12 d2 + {3d(4.19)r of this point from the center of(a and {3 are two constants, y is theratio of the specific heats at constant pressure and constant volume, G is the gravitationalconstant, and !1 is the rotational velocity of the Earth).
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