Darrigol O. Worlds of flow. A history of hydrodynamics from the Bernoullis to Prandtl (794382), страница 54
Текст из файла (страница 54)
Cf. Kutzbach [1979] pp. 158-71 (circulation theorem), 206-18(J. Bjerknes's model), Khrgian [1970] pp. 215-16, Friedman [1989].93Cf. Kutzbach [1979] pp. 21 8-20. Two notable exceptions are Wenger [1 922] and Bernhardt [1973], who gavecompetent reviews of Helmholtz's meteorological works.WORLDS OF FLOW178perhaps they wished to glorify the founders of the newer schools of meteorology, especiallythe Bergen school, at the expense of earlier investigators. Yet there is no doubt thatHelmholtz anticipated some central concepts of modem meteorology.944.6 Wave formation4.6.1From atmospheric waves to water wavesBoth for organ pipes and for the general circulation of the atmosphere, an essentialproperty of Helmholtz's surfaces of discontinuity is their instability, which allows mixingof the layers in contact.
In the atmospheric case, the two layers usually have differenttemperatures, and therefore different densities. Consequently, Helmholtz noted in hismemoir of 1 888, the instability is similar to the one induced by wind blowing on a quietsea. For moderate winds, the surface of the sea oscillates periodically. For larger velocities,whirls are formed and the tips of the waves break into foam and droplets.
Helmholtzimagined similar turbulence to occur at the contact surface between two atmosphericlayers and to permit intimate mixing of their contents.95From Mount Rigi, Helmholtz had seen stratified and whirling clouds that directlysuggested the analogy between atmospheric and sea waves. He justified this analogy in1 889, in a sequel to his paper on atmospheric motion. By a similarity argument, he showedthat the scale of the waves varied as the square of the wind velocity, and that similarwaveforms occurred when the ratio of the kinetic-energy densities of the two media wasthe same in the reference system for which the waves are stationary.
These rules imply thattypical waves in the atmosphere are much larger than waves on the ocean, since the densityratios are much smaller in the atmospheric case. For example, waves of one meter in lengthon the ocean correspond to waves of about two kilometers in length between two layers ofthe atmosphere under the same wind (relative velocity) and with a temperature differenceof 10° Celsius. 96Helmholtz thus related waves in the sky to the better-known waves on the sea.
Thetheory of the latter kind of waves was fairly developed, thanks to the efforts of Britishphysicists. For example, in 1871 William Thomson had given a theory of small waves on acalm sea, including the influence of capillarity. He1mholtz must have been aware ofpart ofthis work, since he helped Thomson measure the minimum velocity of such waves duringa yacht trip.
Thomson's calculations included the effect of a horizontal wind, and showedthat in the linear approximation initially-small waves grew indefinitely when the windvelocity exceeded a certain, small limit. In other words, a plane water surface becameunstable under a sufficiently strong wind.9794ln K.brgian's words, 'Helmholtz's works .
. . are referred to only quite rarely now, but it should beremembered that these studies helped lay the basis for present day synoptic meteorology' (K.brgian [1970] p. 208).95Helmholtz [1 889] pp. 305-6.96/bid. pp. 3 16-22. As Helmholtz noted on p. 3 1 0n, Luvini [1888] pp. 370-1 independently introducedatmospheric waves and billows.97See above, pp. 87-S, for the fishing line; see below, pp.
188-90, for the wind wave instability.VORTICES179Helmholtz did not know the latter aspect of Thomson's ripple studies.98 He was,nevertheless, correct in regarding the production of finite waves by wind as an openmathematical question. 99 With his usual analytical power, he attacked this formidablenonlinear problem. Confining himself to two-dimensional, periodic waves of steady formwith irrotational flow, he applied the conformal method of his memoir on discontinuousfluid motion.
He thus determined the profile and the velocity of the waves to third order intheir relative height (height over wavelength), with the following results. 1 00Under a given wind, the wavelength can vary within certain limits that grow withthe wind strength. For a given wavelength, the remaining characteristics of the waveare completely determined. The longest possible waves are slowly-propagating, lowamplitude sine waves. Shorter waves are higher, faster, and more abrupt. 1 01 At the lowestwavelength, the proftle of the waves becomes discontinuous. Around the resulting ridge,the infinite velocity results in violent projections of water.
This is the frothing of waves.Under any given wind, an obvious solution of the equations of motion corresponds toflat, undisturbed water. If some of the steady waves under the same wind have a smallerenergy, then this solution is unstable. Helmholtz proved that the instability occurred forarbitrarily-small winds and for all permitted wavelengths except those closest to thefrothing point. 102 Hence the faintest wind can produce waves on calm water.
If the windgrows, the height of these waves increases. The shortest ones break into foam, because theywere already on the verge of instability. Another cause of breaking is the superposition ofwaves of different length and velocity. Remembering his old acoustic works, Helmholtzdid not fail to notice that nonlinear superposition generated waves of longer wavelength,1 3just as the ear generates combination tones.
04.6.2A minimum principleIn order to reach his two main conclusions, the instability of a plane water surface under aconstant wind and the breaking for high waves, Helmholtz used truncated power series of98Cf. Rayleigh to Helmholtz, 29 Oct. 1889, HN.99As was discussed in Chapter 2, pp. 70-2, 83, Stokes and Rayleigh studied water waves of fmite height, butonly in the absence of wind. Stokes [1880c] used an analytical method somewhat intermediate between those ofHelmholtz [1868dj and [1 889].
Helmholtz was probably aware of this paper, since he knew a result of Stokes[l&&Oc] published next to [1880c] in the same volume, namely, the 120 degrees of the highest possible wave(Helmholtz [1 889] p. 328, where Stokes, however, is not named).100Helmholtz [1889] pp. 323-8. There is some confusion in Helmholtz's notation (besides numerous typographical errors). For example, in some formulas the letter b stands for the velocity potential, and in others for thevelocity. Helmholtz published only an outline of his calculations. Details are found in manuscripts, as well ascalculations for other types of wave (HN, #682, #884).101 By an unfortunate slip, Helmholtz stated the opposite on p. 328 of his paper [1 889]; however, he gave theright variations on p. 331, in conformance with his equations.102Helmholtz [1889] pp. 329-32.
This does not contradict Thomsen's earlier result, because Thomsen's windthreshold vanishes if capillarity is neglected. The existence of a lower limit for the wavelength seems to contradictthe calculation in Thomson [187la], according to which (capillarity being neglected) the plane water surface isunstable under any perturbation of small wavelength. In fact, it does not, because the growth of these perturbations does not necessarily lead to stable finite-height waves of the same wavelength.103Helmholtz [1889] p. 332.WORLDS OF FLOW1 80the relative height of the waves.
Since breaking occurs precisely when these series diverge,the second conclusion was rather fragile. Moreover, Helrnholtz had only exhibited oneclass of steady solutions to the hydrodynamic equations. His conclusions did not necessarily apply to more general solutions. 104 The following year, he offered a more rigorousapproach based on a new variational principle, akin to the principle of least action onwhich he was trying to base all physics in these years.105In the wind-over-water problem, the water surface is steady if and only if the pressure isthe same on both sides of the surface.
For irrotational flow, Helmholtz found thiscondition to be equivalent to the stationarity ofthe difference V -T between the potentialand the kinetic energy of the motion under infinitesimal deformations of the surface, thetotal air and water fluxes being kept constant. This condition is similar to the condition ofstatic equilibrium, whlch requires the stationarity of the potential energy. Helmholtzextended this analogy to the discussion of stability: while in the static case the equilibriumis stable if and only if V is a minimum, in the steady-wave case the motion is stable if andonly if V is a minimum.106TBy ingenious qualitative reasoning, Helrnholtz determined how the shape of the surfacerepresenting the variations of V -T (with respect to the parameters of the waves) changedwith the wind velocity.
He found that, for a given wavelength, no minimum could occur ifthe wind velocity was too hlgh. In other words, stable, steady waves of a given length areonly possible if the wind velocity does not exceed a certain limit. Nor can they occur if thewave velocity and the wind velocity with respect to the waves are both below certain limits.These theorems agreed with Helmholtz's earlier, less rigorous result about the finite rangeof wavelengths that corresponds to a given wind strength.107Lastly, Helrnholtz discussed the energy and momentum conditions for the initial formation and the growth of waves.
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
