The distribution of raindrops speeds



[1] A general discussion on the drops speeds and their distribution in rainfall accounts quantitatively for the anomalous features discovered by Montero-Martinez et al. (2009), who demonstrate that a non negligible fraction of drops have an anomalously large fall velocity owing to their diameter, the ratio between the observed, and expected speeds being all the more large that the drop diameter is small. While these authors attribute this ‘anomaly’ to breakup between colliding drops, we show on the basis of detailed laboratory experiments that it can be fully accounted for by a scenario which has successfully represented the distribution of drops sizes in rainfall, namely the spontaneous breakup of large drops. In particular, this scenario predicts that a fraction of drops in rainfall should have a velocity approximately 2.5 times larger than their equilibrium fall velocity given their size, consistent with measurements. The trajectories of the bursted drop fragments, and the distribution of the norm of their velocity at bursting are determined. The average fragment velocity induced by bursting is found to be much smaller that the velocity of the initial mother drop. Moreover, the fraction of drops affected by the velocity ‘anomaly’ is estimated.

1. Introduction

[2] The terminal velocity u(d) of a drop with density ρ and diameter d in the millimeter range falling by gravity g in air with density ρa is

equation image

or u(d) = equation image, with viscous corrections for smaller diameters [Mason, 1971; Pruppacher and Klett, 1997]. The drag coefficient CD is approximately equal to 0.5 for a sphere, and to 1 for a flat disk [Roos and Willmarth, 1971; Briffa, 1981]. The fall velocity decays like u(d) ∼ equation image for decreasing sizes d as long as Re = ρau(d)d/ηa > 103, and even faster (u(d) ∼ ρ g d2/ηa) in the viscous range, with ηa the dynamic viscosity of air. An intermediate scaling (u(d) ∼ d) holds for drizzle.

[3] Given that the drop size distribution in the falling rain is well approximated by a decaying exponential [Marschall and Palmer, 1948] with an average size 〈d〉 function of the rate of rainfall, and if a rigid one-to-one correspondence between the fall speed u and the drop diameter d of the kind in equation (1) where existing, then the volume averaged distribution of the fall speeds (see Leijnse and Uijlenhoet [2010] for alternate averaging weights) would be a Rayleigh distribution

equation image

with 〈u2〉 = g′〈d〉. Figure 8 of Niu et al. [2010] is qualitatively consistent with equation (2), indicating that most of the drops are probably well described by a simple scenario attributing to a given drop a single fall velocity, function of its size. However, the distribution of raindrops speeds in nature is likely to be slightly more complicated.

[4] Although detailed measurements of drops speeds are scarcer than those of drops sizes, a very interesting, though intriguing observation has been made recently [Montero-Martinez et al., 2009]. The measurements of the fall speed conditioned to the drops diameter made by these authors demonstrate that a non negligible fraction of drops have an anomalously large fall velocity in the sense of equation (1), the ratio between the observed, and expected speeds being all the more large that the drop diameter is small. The authors attribute this anomaly to breakup between colliding drops.

[5] Concomitantly, a quantitative, mechanistic interpretation of Marschall-Palmer law on the basis of single drop fragmentation, irrespective of any other effect, was proposed [Villermaux and Bossa, 2009], a proposal which has been equally favorably welcomed [Kostinski and Shaw, 2009; Niu et al., 2010], and disputed [Barros et al., 2010; Villermaux and Bossa, 2010]. Here, we show that the anomaly discovered by Montero-Martinez et al. [2009] is actually a prediction of the single drop fragmentation scenario.

2. Distributions of Fragments Sizes, and Initial Velocities

[6] A liquid lump as those found at the clouds base [Hobbs and Rangno, 2004] falling by its own weight will fragment provided it is big enough: ‘big’ means that its size d0 should be such that the Weber number We = ρau(d0)2d0/σ is larger than about 6 (where u(d) is the terminal velocity in equation (1) based on d0, and σ is the liquid surface tension). The fragments resulting from the breakup of a single drop are distributed in size according to

equation image

where Γ(n) = ∫0tn−1 et dt is the Gamma function. The index n refers to the roughness of the toroidal rim bordering the inflated bag before breakup (see Figures 1 and 2): a smooth rim has n = ∞ and produces a single size population of fragments. A strong corrugation means that the amplitude of the cross-section diameter fluctuations along the rim is of the order of its mean radius. In that case, the parameter n is of order of a few units, typically 4–5 [Villermaux et al., 2004; Eggers and Villermaux, 2008; Zhao et al., 2011]. The average fragment diameter 〈d〉 is related to the initial connected drop diameter d0 and to the corrugation index n by

equation image

giving, in the corrugated limit for n = 4,

equation image

a relationship quantitatively consistent with the proportionality of 〈d〉 on d0 for small d0 reported by Villermaux and Bossa [2009, Figure 4a], Dai and Faeth [2001], and Zhao et al. [2011]. The fragments which, by definition, are smaller (on the average by a factor 6.3) than the mother drop thus have for some time (which we determine below), a fall velocity of the order (we give the correction precisely below) of that of the drop from which they originate; that velocity is larger than the one corresponding to equation (1) given the fragment size. An instantaneous measurement collecting blindly both the drops having relaxed to their equilibrium velocity in equation (1), and those which have just been released from a big mother drop will therefore present a broad scatter reflecting these two concomitant effects. Since the equilibrium fall velocity is ∼equation image, the distribution of the overall fall velocities normalized to their expected value u(d) in equation (1) given their measured size d will thus present an anomalous peak located around an average given by

equation image

The measurements reported by Montero-Martinez et al. [2009, Figure 2] indicate that most of the drops comply to equation (1) (with a weak dispersion), but that a measurable fraction of them do bear a much larger velocity, constructing a secondary peak in the distribution centered around a ratio as in equation (6) of about 2.8, a value not inconsistent with the one anticipated above. Moreover, more big drops producing more fragments are liable to break in heavy rains, consistently with the observed trend of Montero-Martinez et al. [2009] that the relative intensity of the secondary peak is increasing with the rate of rainfall that is, owing to Marschall-Palmer's law, with the average drop diameter [Villermaux and Bossa, 2009].

Figure 1.

An initially spherical drop with d0 = 4 mm has flattened to a pancake shape, then inflates a bag bordered by a thick corrugated rim, which ultimately fragments. The time interval between the images is about 10 ms. As the drop deforms, the fluid particle which will be constitutive of the future fragments are themselves accelerated, thus setting their escape velocity v.

Figure 2.

Series of consecutive snapshots showing how the dynamics of the fragments can be complicated, and broadly distributed immediately after breakup (d0 = 6 mm, sequence lasting about 50 ms). The lines are the trajectories of a few selected fragments. Small fragments have a larger escape velocity (see equation (11)), and relax faster (equation (14)) to their equilibrium velocity (equation (1)). In any case, the magnitude of the fragments escape velocities at breakup v is much smaller than the fall velocity of the mother drop (equation (11)). Note also that even at the scale of a bursting drop where the fragments spatial density is large, collisions between neighboring drops are essentially inexistent.

[7] This scenario can be further assessed by examining its prediction for the fragment velocity distribution at breakup, and for the permanence time of this distribution. We rely on the set of experiments presented by Villermaux and Bossa [2009] consisting in allowing millimetric water drops to deform, destabilize and fragment in mid-air after being dripped in the centerline far field of a counter current ascending air jet. Fragments velocities and trajectories are measured from time resolved movies of the breakup phenomenon for a large collection of drops, retaining only the in-focus fragments in the plane perpendicular to the visualization direction. In the reference frame of the air (i.e., the immobile atmosphere in natural rainfall), the velocity u of a fragment is dominated by that of the mother drop u(d0) directed downwards, plus a correction v resulting from the dynamics of the bursting

equation image

and we note

equation image

in the sequel. The fragments come from the breakup of the corrugated rim bordering the deformed drop in its ultimate ‘bag shape’ (Figure 1). As the bag inflates under the stagnation pressure ρau2, the fluid particles constitutive of the rim are accelerated according to

equation image

where v = ∣v∣ is the norm the fragments velocity, their direction being essentially distributed isotropically (Figure 1). The acceleration period lasting for the mother drop bursting time τb [Chou and Faeth, 1998; Dai and Faeth, 2001; Villermaux and Bossa, 2009]

equation image

which is the time it takes for the initial drop to deform, and breakup. Water evaporation in standard conditions is negligible on this timescale. From (9) and (10), one expects that the initial average fragment velocity 〈v〉 will be related, in the mean, to the fragment diameter d as

equation image

Smaller fragments are faster, but the magnitude of their intrinsic velocity remains much smaller than that of the mother drop u; the data in Figure 3 are compatible with 〈v〉/u ≈ 0.015 d0/d, consistently with equation (11).The magnitude of the velocity difference between the released fragments, and the surrounding air is thus essentially given by u based on d0, the correction induced by the breakup itself being negligible, hence the success of the estimate in equation (6). The distribution of fragments initial velocities q(v), such that q(v)dv = p(d)dd is

equation image

representing fairly (with n = 4 and CD = 0.5) the distribution measured from the busting of drops with various diameters d0, as seen in Figure 4.

Figure 3.

The magnitude v of the fragments velocity normalized by the mother drop velocity u. The data, representing an ensemble of more than 1000 fragments, were collected from several drops with various diameters d0 ranging from 4 to 8 mm. The solid line is 〈v〉/u ≈ 0.015d0/d.

Figure 4.

Distribution of the magnitude v of the fragments velocity normalized by the mother drop velocity u. Same data as in Figure 2. The solid line is equation (12).

3. After Breakup: Fragments Trajectories and Persistence

[8] The initial velocity distribution q(v) holds immediately after breakup. Soon, the fragments whose velocity, dominated by u, is initially too large according to their size d < d0 to comply to the steady state equilibrium (1), relax towards their free fall velocity prescribed by their diameter. Neglecting again gravity for small fragments, the vertical motion of a fragment equation image = v · ez is described by

equation image

with z(t = 0) = 0 the position of the fragment as it is released from the fragmenting rim with vertical velocity equation image(t = 0) = v0. Introducing equation image0 = v0/u, equation image = t/τ and equation image = z/(), with

equation image

one has from equation (13) the expected fragment vertical trajectory

equation image

which fits well the observed ones, as seen in Figure 5. We have checked that the radial velocity component of the fragments equation image = v · er is approximately constant over the corresponding time interval. The time τ sets the permanence time of the initial velocity distribution q(v), after what the fragments relax to their equilibrium fall velocity prescribed by their size. Note that the relaxation is faster for a smaller fragment, as seen in Figure 5. For a millimetric fragment, τ is of the order of a second, and in any case long compared to the bursting time of the mother drop τb in equation (10), these two timescales being in the ratio (d/d0)equation image. It is thus not surprising that a measurable fraction of the drops recorded in rainfall are interpreted to have an ‘anomalous’ velocity. That fraction is certainly larger at the clouds base, where spontaneous bursting is the rule, than at the ground level, where the drops size distribution is essentially frozen, and where bursting is a rare event.

Figure 5.

Vertical trajectories z(t) of (left) fragment 1 with d = 3 mm and v0 = −0.55 m/s, and (right) fragment 2 with d = 0.8 mm and v0 = −0.35 m/s issued from the bursting drop in Figure 3 with d0 = 6 mm and u = 8 m/s. The lines are from equation (15) with time τ in equation (14) equal to 1.3 s and 0.4 s, respectively.

[9] However, we now finally give a tentative estimate of the fraction of those drops still bearing the mother drop velocity when probed at random in rainfall, at the ground level. A drop in free fall is likely to break spontaneously if its Weber number We is larger than about 6, which means that drops larger than dm = equation imagea ≈ 6 mm (with a = equation image) will fragment [Villermaux and Bossa, 2009]. We rely on Marschall-Palmer distribution for describing the drop size content in rain, and estimate the relative fraction of the drops larger than dm as exponentially small (this is probably a comfortable overestimate)

equation image

for an average drop size 〈d〉. The number N of fragments from a bursting drop (averages are computed making use of equation (3)) is

equation image

so that the probability f that a drop picked up at random has not relaxed to its equilibrium velocity is that it belongs to the swarm of the N fragments released by a mother drop whose bursting probability is given by equation (16), times the ratio of the permanence time of its anomalous velocity τ to the cumulated times of the breakup event, namely the mother drop bursting time plus the permanence times of the fragments as

equation image

on average. The fraction f is a very sharp function of the average drop size, and goes to zero very rapidly as 〈d〉 decreases, that is as the rate of rainfall R decays (we recall that 〈d〉 ∼ R2/9). From equation (18), f drops to zero for 〈d〉 below 1 mm or so. This trend is consistent with the observation made by Montero-Martinez et al. [2009] who note strong variations in f (from 50 percent down to a few percent if not zero), as R varies from large to low rates.

[10] The breakup scenario originally evoked by Montero-Martinez et al. [2009], and more precisely the single drop fragmentation version of it thus seems to account also for the distribution of raindrops velocities, its predictions regarding this facet of the phenomenon of rainfall resorting to the very same physical ingredients than those which have been necessary for explaining the drop size content in rain.


[11] This work has been supported by the Office national d'études et recherches aŽrospatiales (ONERA) under contract F/20215/DAT-PPUJ and Agence Nationale de la Recherche (ANR) through grant ANR-05-BLAN-0222-01.

[12] The Editor thanks the two anonymous reviewers for their assistance in evaluating this paper.