Darrigol O. Worlds of flow. A history of hydrodynamics from the Bernoullis to Prandtl (794382), страница 53
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For two such layers to be inbetween the pressure P at a point of the layer, the distancethe Earth, and its distance d from the axis of the Earthequilibrium the pressures at their contact surface must be the same. This condition yields arelation between r and d which defines the trace of this surface on a meridian plane.84Helmholtz required this equilibrium to be stable in the sense that the imbalance ofpressure caused by a protrusion of the surface must counteract this protrusion. Thiscondition implies that the potential temperature must be higher in the layer that is closerto the celestial pole. When the ratec? /e is an increasing function of the distance p from82Helmholtz [1888] pp. 308 (quote), 298 (rings).83Ibid.
p. 293; Bezold [1888] p. 1 1 89. Cf. Garber [1976] pp. 59-62, Kutzbach, [1979] pp. 1 43-4. Helmholtzcorresponded with Bezold and supported him for the Buys-Ballot medal of meteological merits in 1893 (JuliusHann won), cf. Horz [1997] pp. 201-24.84Helmholtz [1888] p. 298.VORTICES175the axis, which is true in the normal state of the atmosphere, stability also implies that theseparating surface should depart from the ground at an angle between the horizon andthe polar direction (see Fig.4.10). Hence, thin layers, or strata of homogenous air, lean tosuch an extent that their lower part is further from the Earth's axis than their higher part.85From this simple rule, Helmholtz drew essential conclusions.Suppose that the stratum appears on Earth as an easterly wind. Then its rotation isslower than that of the Earth.
Friction on the Earth's surface therefore increases theabsolute rotation of the lower part of the stratum. This part of the stratum being alsothe furthest from the axis, the effect of friction will remain confined to it, because theexcess of centrifugal force pulls the air away from the axis (and because internal friction is86Consequently, the stratum slides toward the equator and its velocity becomesnegligible).more and more heterogeneous: whereas the momentum of the upper part remains unchanged and corresponds to higher and higher easterlies, that of the lower part, near theground, increases until the wind vanishes. According to Helmholtz, this process feedshigh-momentum air into the zone of calms, which consequently grows to touch the upperpart of the converging easterly strata (see Fig.4.1 1). Since momentum is conserved for thetwo kinds of air thus coming into contact (internal friction again being negligible), a·87surface of discontinuity is born.4.5.3Corrections and additionsHelmholtz expressed himself in so condensed a manner that his arguments are sometimesdifficult to follow.
For example, he claims to be explaining the formation of a surfaceof discontinuity from an initial, continuous state of motion, whereas in fact he startss:sFig. 4.1 0.Permitted inclination of an air stratum(thick line) according to Helmholtz's atmosFig. 4.1 1 .Formation of a discontinuity surface(thick line) bordering the zone of calms.pheric circulation theory.85Ibid. pp. 299-301.86Helmholtz compared this process with the heating of a volume of air from its top, which affects only theupper air layer because of the lack of convection.87Helmholtz [1888] pp. 304-5.176WORLDS OF FLOWwith a state of contiguous air layers with different angular momenta.
At first glance,it would seem that he has only proved that existing discontinuities can be amplified.In reality, his stronger claim for the genesis of discontinuities holds. As he made clear atsome point, he included the case of infinitely-thin strata in his analysis. The abovereasoning extends to this case, and still leads to the discontinuity at the upper limit ofthe calm zone.Another difficulty of Helmholtz's memoir is that it blurs the distinction betweenintuitive arguments, empirical data, and strict dynamical deduction. For example, shouldthe existence of the zone of cahns be regarded as empirically given, or does it result fromdynamical reasoning? In this paper, with its emphasis on the formation of discontinuities,Helmholtz seems to be taking the first option, whereas in his earlier discussion of tradewinds in 1 875 he explained the calm zone as an indirect consequence of the centrifugalforce.At least in one case, Helmholtz dangerously confused intuition and deduction.He argued that the mixed air produced by instability at the border between the lowerand upper trade winds had to move toward the equator, because its intermediate temperature and velocity belonged to lower altitudes and latitudes.
As he admitted thefollowing year, this conclusion was wrong. A rigorous treatment of the conditionsof equilibrium of the mixed air with the two mixing layers implies an ascending motionof this air. 88During this upwards expulsion of the mixed air, originally-remote parts of the twomixing layers come into contact. Owing to the conservation of momentum, the shiftedparts of the polar-side layer lose velocity, whereas those on the equatorial side gainvelocity. Hence the discontinuity surface is renewed, even if the remote parts of the layerswere originally at rest with respect to the Earth.
As Helmholtz explained, in his first paperhe had shown how and where discontinuities were formed in an originally continuouslymoving atmosphere. He could now show that the mixing process at a surface of discontinuity renewed this surface instead of destroying it. 89A short section of the 1 888 memoir sketched the production of surfaces of discontinuityaround the poles. Due to the cooling of the Earth near the poles, cold air strata divergefrom the pole at low altitude. Owing to the rotation of the Earth, these strata appear asnorth-easterlies.
As was shown for the lower trade winds, their inferior parts experiencefriction on the Earth's surface and the resulting increase of the centrifugal force dragsthem further south. Owing to the inclination of the strata, this cold air remains close tothe surface of the Earth, in conformance with the fact that in northern Germany thenorth-east winter winds do not reach the summits of mountains. 'At the front border ofthese easterlies advancing into warmer zones,' Helmholtz went on, 'the same circumstances that produce discontinuities of motion between upper and lower currents in theadvancing trade winds are effective, bringing about a new cause for the formation ofvortices.'9088He1mho1tz [1888] p. 306; He1mho1tz [1889] pp. 312-15.90He1mho1tz [1888] pp.
307-8.89He1mho1tz [1889] p . 315n.VORTICES4.5.4177AnticipationsReading these lines, modem meteorologists could speculate that Helmholtz introduced thenow fundamental notion of the polar front. The extreme concision of his statement doesnot allow any such judgment. The main purpose of his paper was to find a mechanism fordamping the winds induced by the rotation of the Earth. Unlike some of his followers, hedid not have in mind a theory of storms based on surface discontinuities.
Most likely,he still believed that mid-latitude storms were too complex to be subjected to dynamicalanalysis.Yet there is no doubt that Helmholtz was the first to realize the essential importance ofsurface discontinuities in meteorology, before horizontal and vertical field measurementsmade them clearly visible toward the end of the century. With some delay, his 1 888 paperwas a major source of inspiration for the meteorologists who applied the concept to thetheory of storms. Early in this century, the Austrian Max Margules integrated Helmholtz'sdiscontinuity surfaces in his atmospheric energetics and generalized Helmholtz's formulafor the slope of surfaces of discontinuity. Subsequently, Felix Exner and his Vienneseschool of meteorological dynamics heavily relied on Margules's extensions of Helmholtz'smeteorological concepts.91The Norwegian meteorologist Vilhelm Bjerknes also owed much to Helmholtz.
This istrue for two of his breakthroughs in dynamical meteorology and weather forecasting. Hiscirculation theorem, giving the rate of variation of the vorticity for a compressible fluid,was a simple extension of Helmholtz's theorem on the conservation of the vorticity inincompressible fluids. His atmospheric kinematics emphasized the singularities of thevelocity field that Helmholtz first discovered. The concept of a 'cold front', which is socentral to modem meteorological forecasting, occurred to him while studying Helmholtz's1888 paper in his Leipzig seminar.92Helmholtz's works were not the only resources exploited by Margules, Bjerknes, andother founders of modem meteorology. As Kutzbach has shown, the various thermaltheories of cyclones, their late-nineteenth-century difficulties, the enormous improvementof the quality and quantity of weather data, and observations of the higher atmospherewere all important factors of progress.
Against this view, later meteorologists have usuallyregarded the polar-front theory as a sharp break from the past and ignored the manycontinuities with the past, including its Helmholtzian roots.93 They may have been blindedby the spectacular progress in weather forecasting that this theory brought about. Or91Margules [1906]; Exner [1925] has many references to Helmholtz's works, on pp. 92 (mechanical similarity),203-10 (air rings), 214n (general circulation), 234-5 (empirical verification of Hehnholtz's stability conditions),334 (Helmholtzian origin of Bjerknes's polar front).
On the observation of surface discontinuities, cf. Kutzbach[1 979] pp. 175 (Bigelow), 1 81-3 (Shaw), 194-7 (Margules). On Margules's extensions of Helmholtz's results,cf. ibid. pp. 197-9.92Bjerknes [1898] for the circulation theorem; Bjerknes et al. [1910] for the kinematics; Bjerknes et al. [1933] pp.784 (reading Helmholtz and the polar front), 785 (reading Helmholtz and the wave theory of cyclones). The firstfrontal cyclone model was published by Bjerknes's son Jacob, who used Margules's generalization of Helmholtz'sslope formula for the surface of discontinuity.
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