Vol. 5, No. 10/October 1988/J. Opt. Soc. Am. A 1717
Rainbows in strong vertical atmospheric electric fields
Stanley David Gedzelman
Department of Earth and Planetary Sciences, City College of New York, New York, New York 10031
Received August 26, 1987; accepted May 6, 1988
The effects of strong vertical atmospheric electric fields and electrically charged raindrops on the primary rainboware considered. A vertical electric field alters the bow by stretching the raindrops vertically. The effects may be
visible only for the extremely large electric fields (E > 105 V m'1) sometimes encountered in thunderstorms. The
electric field raises and brightens the top of the primary bow but reduces the relative intensity and spacing betweenthe supernumeraries. Electrical charges do not have any visible effect on the rainbow because they significantlyincrease the ellipticity of only the largest and most highly charged drops, which do not contribute to the bow's top.
INTRODUCTION
Since antiquity' the rainbow has been attributed to reflec-tion of sunlight from spherical objects. When Descartessuccessfully explained the rainbow in terms of a minimumscattering angle by using geometrical optics, he used a centu-ries-old idea that rainbows are produced by sunlight strikingspherical drops.2 Two centuries later, Airy calculated thediffraction pattern created by spherical drops in his treat-ment of the primary rainbow and its supernumerary bows.3
The idea that discrepancies between observed bows andcalculations based on Descartes' work might be due to non-spherical drops was first suggested by Venturi.4 Some timeafter Airy's classical papers, it was demonstrated that rain-drops are not perfect spheres. Surface tension acts to holddrops together in a volume possessing the minimum surfacearea, but falling drops are flattened and distorted. Thelarger the falling raindrop, the greater its ellipticity anddegree of distortion. Small raindrops can be approximatedby oblate spheroids, while large drops are decidedly flat-tened on the bottom and resemble hamburger buns.5
The flattening of raindrops was first observed photo-graphically by Magnus in 1859,6 but this fact was not widelyappreciated until Lenard published results of his classicalinvestigation of falling raindrops in 1904.7 Shortly there-after, in 1907, Mobius calculated the minimum deviationangle of the rainbow for drops with elliptical cross sectionsand confirmed his calculations experimentally with glasscylinders and globes.8 M6bius showed that the greater theellipticity of the drops, the greater the Descartes minimumdeviation angle for the top of the bow. Elliptical dropswould thus lower the top of the bow.
The idea of Venturi and work of M6bius attracted almostno attention until 1960. Then, Volz showed that certainaspects of the rainbow could be explained by considering thejoint effects of deviations from sphericity of the raindropsand the broad spectrum of raindrop sizes that exists in everyrain swath.9 "10 Airy's solution for the location of the intensi-ty peaks for light striking spherical drops is highly size de-pendent. By integrating Airy's solution for spherical dropsover several realistic drop-size distributions, Volz showedthat the intensity peaks for the various-sized drops over-
lapped, and this effectively wiped out the supernumeraries.Volz then suggested that supernumeraries are possible onlybecause the larger drops are not spherical and therefore donot contribute to the top of the bow. The greater flatteningof the larger drops imposes an upper limit on the drop sizethat can contribute to the top of the bow. He noted that thisrestriction does not apply on the vertical portion of therainbow arc, where Poey" had long since observed that thesupernumeraries vanish. This is because the vertical sidesof the bow are produced by light passing through the hori-zontal cross sections of the drops, all of which are circularand all of which contribute.
Volz also made the point that whenever the larger dropscould not contribute to the bow, the bow's intensity would bereduced. Fraser suggested that the vertical portion of thebow is often observed to be the brightest part of the arcprecisely because that is the only part where light from eventhe large drops contributes to the bow.12 K6nnen integrat-ed the rainbow's intensity over a Marshall-Palmer drop-sizedistribution in a calculation that also included the finite sizeof the solar disk and showed that drop distortion reduces thepeak intensity of the rainbow's top by about a factor of 4.13
In 1983 Fraser pointed out that the varying ellipticity ofraindrops not only is a necessary condition for the existenceof the supernumerary bows but also determines their preciselocation.14 Fraser included the effect of varying ellipticityof the drops and showed that the supernumeraries exhibit aminimum scattering angle for drops with radii of -0.25 mm.This results in a spacing between the first two supernumer-aries that is almost constant along a given bow and dependsonly slightly on wavelength. The spacing also varies some-what with solar elevation angle, the smallest value (0.75 degfor yellow light and a flat drop-size distribution) occurringwhen the Sun is at the horizon or at an elevation angle of 42deg, and the largest (0.81 deg), when the Sun is 21 deg abovethe horizon. Fraser's findings were confirmed both photo-graphically and by K6nnen's integration.
Any force that can alter the shape of raindrops will alsoaffect the rainbow. Volz conjectured that acoustically in-duced drop oscillations might account for a curious observa-tion reported by Laine that rainbows can be affected bythunder.9 According to Minnaert, Laine noticed that15
1718 J. Opt. Soc. Am. A/Vol. 5, No. 10/October 1988
Each time it thundered ... the boundaries of thecolour in the rainbow became obliterated. Thechange was particularly noticeable in the spurious (orsupernumerary) bows: the space between the violetand the first spurious bow disappeared entirely andthe yellow grew brighter. It was as if the whole rain-bow vibrated.
This optical effect did not occur simultaneouslywith the lightning, but several seconds later, togetherwith the sound of the thunder.
In the discussion session following Volz's presentation,Vonnegut proposed that these changes in the rainbow mightbe caused by oscillations of raindrops that were due to thechanging electric field during and after lightning flashes.9Nevertheless, Volz pointed out that the timing of thechanges in the bow was observed to match the arrival of thethunder rather than of the lightning, and there the matterrested.
In this paper we return to the question of whether atmo-spheric electrical forces can visibly alter the rainbow. Elec-trical forces have been shown both theoretically and experi-mentally to alter the shape of raindrops.'6.'7 A verticalelectric field polarizes all drops and stretches them vertical-ly. Thus the effect of a vertical electric field on falling dropsis opposite that of the aerodynamic forces. The extremelylarge vertical electric fields sometimes found in conjunctionwith thunderstorms are able to produce significant changesin drop shape and therefore have potentially measurableeffects on rainbows. A net electric charge on the drops alsoaffects their shape, enhancing any preexisting departuresfrom sphericity by reducing the effective surface tension.For realistic values of charges it is only the largest dropswhose shapes are significantly affected. Since the largedrops do not contribute to the top of the rainbow arc, electri-cal charge should not affect the visible appearance of thebow.
MATHEMATICAL MODEL
The rainbow model used in this paper represents a general-ization of the works of Fraser and K6nnen to the case forwhich electrical forces are included. We begin by present-ing a formula for the approximate shape of a drop falling atterminal velocity as a function of its equivalent volumespherical radius r, its charge Q, and the ambient electric fieldE (assumed vertical). The drops are assumed to be spher-oids with ellipticity e = (a - b)/(a + b), where a is thehorizontal semiaxis and b is the vertical semiaxis. Dropswith e > 0 are oblate spheroids (saucer shaped) wider thanthey are high, while drops with e < 0 are prolate spheroids(cigar shaped) elongated in the vertical. The effect of theellipticity on the minimum scattering angle of the rainbowray, based on the work of M6bius, is then included. Thelight intensity for each size of drop is then calculated as afunction of angle by using the Airy integral. Finally, asimulation of the rainbow intensity in a shower is obtainedby integrating over the drop-size distribution and over thefinite size of the solar disk.
The expression for drop shape used here was obtained byexpanding Eq. (9) of Ref. 16 in a Taylor series expansion inpowers of e and is given by
e= gr' - 3/2(rEE 2)
T _ (4er 2)2]
where e = 8.85 X 10-12 C2 N-' m-2 is the permittivity, p = 999kg m-3 is the density difference between air and water, g =9.8 m sec- 2 the acceleration of gravity, and T = 7.275 X 10-2kg sec- 2 is the surface tension.
Equation (1) is a valid approximation for drop shape onlyfor small values of ellipticity and hence for small drops. Itneglects the progressive flattening of the bottoms of largerdrops. Nevertheless, Eq. (1) still represents a reasonableapproximation for use in the rainbow problem, since largeellipticity and asymmetry become serious only for dropswith r > 1 mm, which do not contribute significantly to thetop of the bow.
Aerodynamic forces squash all freely falling drops. Anelectric field polarizes electrically neutral drops and stretch-es them in a direction parallel to the field lines. Sinceaerodynamic forces act to squash all freely falling drops, avertical electric field reduces the oblateness by stretching alldrops vertically. Drops smaller than r = (3/2)EE2I(pg) (or r= 0.2 mm when E = 4 X 105 V m-') will actually be prolatespheroids.
The fair-weather electric field (102 V m-l) produces negli-gible drop stretching. Only the largest atmospheric electricfields (E > 105 V m'-), which are sometimes attained inthunderstorms,' 8 can visibly alter the rainbow. Althoughfields larger than -5 X 104 V ml' become progressively moredifficult to produce and sustain because of corona dischargefrom hydrometeors, fields as large as -5 X 105 V m'1 (be-yond which the atmosphere breaks down as an insulator) canapparently be maintained for distances as great as severalhundred meters in and below thunderstorms.'9' 20
Electric charges increase the ellipticity of the drops byopposing the binding force of surface tension. Highlycharged drops will explode when the electrostatic repulsionexceeds the surface tension.2' The limiting value of thecharge is known as the Rayleigh limit, and it occurs when thedenominator of Eq. (1) becomes zero. The electric chargeson drops in thunderstorms are observed to be highly vari-able, but a good estimate for their maximum value is ob-tained from the assumption of conduction charging, i.e., Q =6 X 10-12 Er2 C.22,23 Even with the largest observed atmo-spheric electric fields (E = 4 X 105 V m-') and for drops withr = 10-3 m, the effective surface-tension term-the denomi-nator of Eq. (1)-is changed by a factor of only 1.4 X 10-4.The effect on the eccentticity for the smaller drops responsi-ble for the rainbow top is insignificant. As a result, realisticvalues of charge have almost no bearing on the rainbow andwill no longer be considered here.
When the influence of charge is neglected, the approxi-mate equation for the ellipticity of freely falling raindrops inthe presence of a vertical electric field reduces to
e = 0.05r2- 0.00067rE2 . (2)
M6bius found that the difference AO between the geometri-cal optics scattering angle of the primary rainbow for spher-oids and spheres is given by
AO = 180 e sin(M)cos3(o) cos(2h - 42°), (3)7r
Stanley David Gedzelman
Vol. 5, No. 10/October 1988/J. Opt. Soc. Am. A 1719
where h is the solar elevation angle and : is the angle ofrefraction of the Descartes ray."32 4 We then proceed bydisplacing the Airy integral for the intensity of the rainbowas a function of scattering angle by an amount AO for thespheroidal drops. Marston has experimentally documentedthese shifts in the Airy peaks for nonspherical drops.2 5 Therainbow intensity is then found by integrating the displacedAiry integrals over a Marshall-Palmer raindrop-size distri-bution with n/no = exp(-6 X 103r) and finally integratingthe resulting intensity over the angular width of the solardisk. The exponent in the Marshall-Palmer distributioncorresponds to a modest rainfall rate of 4.4 mm h-'.26 Forlarger rainfall rates the exponent is smaller, and there is agreater percentage of large drops. Calculations are per-formed for the top of a monochromatic bow with n = 4/3,which results in a scattering angle of the primary geometri-cal optics bow for spherical drops of 0 = 1380 and 3 = 40.2.Unless otherwise specified, h = 0. Results are identical for h= 42 and only slightly different for all intermediate values of
150
RESULTS
Figure 1(a) was first obtained by Fraser'4 to illustrate thatthe supernumeraries in showers are produced by drops asso-ciated with a minimum of the scattering angle arising out ofthe combined effects of drop ellipticity and the variouspeaks of Airy's theory. Figure 1(a) shows the scatteringangles of the first three peaks of the Airy integral for sphe-roidal drops as a function of drop size when E = 0. Most ofthe drops responsible for the main peak near the top of thebow are slightly less than 0.2 mm in radius, while the super-numeraries are produced by drops whose radii lie in the sizerange from about 0.2 to 0.3 mm. Figure 1(a) also suggeststhat the distance between the first two supernumerariesnear the bow's top is 0.75°, although this depends some-what on drop-size distribution, solar elevation angle, andwavelength.
The effects of a strong vertical electric field on the top ofthe rainbow are seen by comparing Fig. 1(a) with Fig. 1(b).Figure 1(b) shows the first three peaks of Airy's integral forspheroidal drops with a vertical electric field of E = 4 X 105 Vm-1. The electric field decreases the minimum value of thescattering angles of the first three Airy peaks by reducing theoblateness of the drops. This elevates the top of the bow.Figure 2 shows the minimum scattering angle of the firstAiry peak as a function of electric field strength. The stron-ger the electric field, the smaller this angle. Thus the scat-tering angle of the peak intensity at the top of the primaryrainbow may provide a measure of the intensity of the elec-tric field.
Figure 1(b) shows that the electric field also flattens theminimum in the scattering angles of the three Airy peaks as afunction of drop radius and shifts that minimum to largerdrops. A flatter minimum produces a brighter bow since itincreases the effective range of drop sizes contributing to thebow. Figure 3 shows the equivalent spherical radius of thedrops that produce the minimum scattering angle of the firstthree Airy peaks, 0, as a function of the electric field. Dropsnear this size produce the predominant contribution to themain intensity peak and first two supernumeraries near the
145
140
135
r (mm)
(a)
0.2 0.4 0.6 0.8 1.0r (mm)
(b)
Fig. 1. (a) Scattering angle 0 of the first three peaks of the Airyintegral for the top of the rainbow as a function of drops' equivalentspherical radius for electric field, E = 0. The lowest curve corre-sponds to the main maximum. 0
r is the geometrical rainbow scat-tering angle. (b) Same as (a) but for E = 4 X 105 V m'1.
139.0lll
138.9_
138.8 -
138.7 - e~~~138.6 - f
138.50 I 2 3 4
E (10 V m ')Fig. 2. Scattering angle 0 of the main maximum of the top of therainbow as a function of the electric field E. This is the minimumvalue of the first Airy peak as a function of raindrop size.
.l~~~~~~~~~~~~~
Stanley David Gedzelman
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1720 J. Opt. Soc. Am. A/Vol. 5, No. 10/October 1988
0.4
0.31
-E 0.2 _
0.1 _
0 I 2 3 4
E (105V m-)Fig. 3. Equivalent spherical radius r (millimeters) of drops pro-ducing the minimum scattering angle for the first three Airy peaksas a function of the electric field E.
I .0
0.8
0.61-
I0.4 -
0.2 _
0 I 2 3 4E (105 V m-')
Fig. 6. Peak intensity Iof the top of the rainbow in relative units asa function of the electric field E for a Marshall-Palmer drop-sizedistribution and a finite Sun.
0.8
8e0
0.7
0.6L0 2 3 4
E (10 5 V m-')Fig. 4. Angular spacing 60 between the first two supernumerariesas a function of the electric field E for a point Sun assuming thatthey correspond to the minimum scattering angles for the secondand third peaks of the Airy integral.
1.0
I
135 138 140eo
Fig. 5. Intensity I of the top of the rainbow in relative units as afunction of scattering angle 0 for E = 4 X 105 (solid curve) and E = 0(dashed curve) obtained by using a Marshall-Palmer drop-size dis-tribution and a finite Sun.
rainbow top. Even for the largest measured electric field, E= 4 X 105, the spheroidal assumption is still reasonable forsuch small drops, and so the present theory is viable.
Comparison of Figs. 1(a) and 1(b) suggests further that alarge electric field reduces the spacing of the main intensitymaximum and of the supernumeraries near the top of thebow. This is a direct outgrowth of the shift of the minimumscattering angle of the first three Airy peaks toward largerdrops. Figure 4 shows that spacing between the minimumscattering angles of the second and third Airy peaks de-creases with increasing electric field strength. The spacingof the supernumeraries may thus serve as an independentmeasure of the electric field.
We now include the effects of drop-size distribution andthe finite size of the solar disk on the rainbow. The mono-chromatic intensity of the top of the rainbow as a function ofdeviation angle is shown for both E = 0 (dashed curve) and E= 4 X 105 V m-1 (solid curve) in Fig. 5. The large electricfield has displaced the entire pattern and increased the in-tensity of the first maximum by about 30%. The electricfield compresses the entire pattern somewhat, and this re-sults in a relative reduction of contrasts between the higher-order peaks and valleys. Thus for large electric fields thesupernumeraries should appear subdued.
The reasons for these changes have already been touchedon. As the electric field increases it produces a broaderminimum in the scattering angle of a given Airy peak as afunction of equivalent spherical drop radius. The bowbrightens as more raindtops contribute. The maximum in-tensity of the top of the bow is shown as a function of E inFig. 6 and increases monotonically with E. This effect is notobserved for the higher-order peaks since they are producedby larger drops that are fewer in number. Furthermore, asthe spacing between the peaks is reduced, the finite size ofthe Sun more effectively smudges intensity contrasts.
SUMMARY AND CONCLUSIONS
We have considered the effects of electric charges and avertical electric field on the rainbow. The electrical forcesalter the rainbow by changing the shape of the drops. Ob-served values of drop charges in thunderstorms are utterly
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Stanley David Gedzelman
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inadequate to produce noticeable changes in the bow. Onlythe extremely large electric fields sometimes encountered inthunderstorms are able to elongate the vertical dimension ofthe drops sufficiently to produce measurable changes in thebow. Increasing the field strength intensifies the top of thebow and displaces its position upward but decreases theintensity contrasts of and spacing between the supernumer-aries.
ACKNOWLEDGMENT
The clarity of this paper was greatly improved by the thor-ough and incisive comments of an anonymous reviewer.
The author is also with the Lamont Doherty GeologicalObservatory of Columbia University, Palisades, New York10964.
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