Meteorol Atmos Phys 92, 191–204 (2006) DOI 10.1007/s00703-005-0157-4

The Institute of Earth Sciences, The Hebrew University of Jerusalem, Givat Ram, Israel

Investigation of small-scale droplet concentration inhomogeneities in a turbulent flow B. Grits, M. Pinsky, and A. Khain With 10 Figures Received November 9, 2004; revised April 28, 2005; accepted June 29, 2005 Published online: January 19, 2006 # Springer-Verlag 2006

Summary This paper is focused on the numerical investigation of spatial characteristics of droplet concentration fluctuations, originating in a turbulent flow. The turbulent flow is simulated using a model of a statistically stationary, homogeneous and isotropic turbulent flow. We found that preferential droplet concentration with the maximum of spatial spectrum at about 1.5 cm is formed regardless of the droplet size (up to the droplet radius of 20 mm). The amplitude of fluctuations depends on the droplet radius and constitutes a significant part of mean concentration. Analogous simulations with non-inertial droplets (passive scalar) do not reveal such concentration fluctuations. Our main conclusion is, therefore, the following: preferential droplet concentration at the cm-scale forms in a turbulent flow due to droplets inertia.

1. Introduction Several mechanisms of the formation of droplet concentration inhomogeneity in clouds were suggested in literature. Droplet fluctuations at scales above about 1 m are usually attributed to the fluctuations of vertical velocity at the cloud base, to the entrainment of droplet free air through the cloud boundaries, etc. (e.g., Kabanov et al, 1970; Baker and Latham, 1979; Cooper, 1989; Korolev and Mazin, 1993; Korolev, 1994; Hudson and Svensson, 1995) and to in-cloud nucleation of droplets (Korolev, 1994; Pinsky and Khain,

2002). It is often assumed that turbulent diffusion leads to drop concentration mixing and homogenization at smaller scales. At the same time, laboratory studies (Fessler, 1994; Aliseda et al, 2002), direct numerical simulation (D N S ) (Fung, 1993; Wang and Maxey, 1993; Sundaram and Collins, 1997; Reade and Collins, 2000; Wang et al, 2000; Vaillancourt et al, 2002) and analytical considerations (Elperin et al, 1996; Pinsky and Khain, 1997a; Pinsky et al, 1999; Jeffery, 2001; Falkovich et al, 2002) suggest the existence of another powerful mechanism leading to the formation of small centimeter-scale drop ‘‘clusters’’. Statistical methods, developed by Pinsky and Khain (2001), and Kostinski and Shaw (2001) used to analyze series of droplet arrival times measured by Fast Forward Scattering Spectrometer Probe (F F SS P ) (see Brenguier et al, 1998) allowed the authors to reveal centimeter scale drop concentration fluctuations in a set of cloud traverses. The root mean square (r.m.s.) amplitude of fluctuations for Stokes number range 0.001–0.1 was found by Pinsky and Khain (2001; 2003) to vary between 5% and 30% of the mean concentration. The formation of centimeter-scale drop concentration fluctuations is usually attributed to the inertia of droplets moving within a turbulent flow. Under the influence of centrifugal

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forces (actually, under the influence of accelerations and shears of a turbulent flow), inertial drops deviate from the tracks of air parcels. As a result, the droplet velocity flux turns out to be a divergent flux within a non-divergent turbulent airflow. According to theoretical evaluations (Maxey, 1987; Pinsky and Khain, 1997a), variations in the divergence of a cloud droplet flux are of a one centimeter-scale, being determined by corresponding scales of turbulent velocity shears. Maxey (1987) derived an equation for droplet velocity divergence, which for small droplets (r<20 mm) reduces to the expression @Vaj @Vai : r
~ V ¼ 
d @ri @rj

ð1Þ

Here Vai are the air velocity components and
d is the characteristic droplet relaxation time, defined as Vt
d ¼ ; ð2Þ g Vt being the droplet terminal fall velocity. From Eq. (1), one can see that the divergence amplitude increases with the droplet size (through Vt ). At the same time, the spatial structure of the divergence field, defined by spatial derivatives of the airflow velocity, depends only on the airflow structure. Pinsky et al (1999) calculated the spatial spectrum of the drop velocity flux divergence for different values of the dissipation rate. A pronounced maximum at the centimeter scale was found. The dominance of this particular scale is related to the fact that velocity shears (see Eq. (1)) increase with a decrease of their scales within the turbulent inertial subrange down to scales of about one centimeter. At scales below 1 cm, viscosity effects become significant, leading to a rapid decrease in the shear spectrum. As a result, the maximum of a drop flux divergence is located at scales of about one centimeter. It is natural to expect that such divergence should lead to the formation of an inhomogeneous field of drop concentration within clouds at the centimeter scale too. Such inhomogeneity in its turn may influence the microphysical processes in clouds. In the investigations of preferential concentration (see the papers, mentioned above), the main

attention is drawn to the dependence of the strength of this phenomenon on the Stokes number (St), i.e., on the particles size. For example, Sundaram and Collins (1997), Reade and Collins (2000), and Wang et al (2000), calculate radial distribution function (number of droplets inside a spherical shell between radii r and r þ dr, divided by the expected number of droplets given a uniform droplet field) as a function of St and find maximum at St  1. For this St their studies predict the amplitude of concentration fluctuations to be several hundreds percents. The St number range usually studied is from 0.2–0.4 up to 4–8. At the same time the typical cloud droplets (10 mm–20 mm) correspond to much smaller St numbers (0.03–0.12). The main questions interesting for cloud microphysics, concerning the cm-scale concentration fluctuations, are the spatial and time spectra and r.m.s. amplitude of these fluctuations. There is actually no systematic investigation of such spectra, using numerical simulations, especially for conditions typical of clouds (St  1). Besides, there is still no consensus concerning the amplitude of concentration fluctuations in such conditions (Grabowski and Vaillancourt, 1999; Vaillancourt and Yau, 2000; Jeffery, 2001). We formulated the goals of this paper as follows: (1) to show that the mechanism, responsible for the formation of concentration fluctuations, is the inertial response of droplets to the drag force in a turbulent flow; (2) to evaluate spatial spectrum and r.m.s. amplitude of fluctuations. We perform this investigation by means of numerical simulations of droplet tracks within a turbulent flow generated by a 2-D turbulent flow model. 2. Description of simulations 2.1 A model of the turbulent flow The turbulent air velocity field was simulated using a model of homogeneous and isotropic turbulent flow. This model is introduced in Pinsky and Khain (1995; 1996) in details. Here we give only a brief description of the model. Description of the models of turbulent field preceding our model (see, for example, Kraichnan, 1970; Turfus and Hunt, 1986, and Maxey, 1987) and comparison with these models are given in Pinsky and Khain (1995).

Investigation of small-scale droplet concentration

The velocity components of the flow are calculated in this model as the sum of a large number of harmonics, corresponding to different wave numbers and different directions in space. Different harmonics are non-correlated with each other. The longitudinal structure function proposed by Batchelor (1951) has been chosen for the description of a locally isotropic turbulence in the inertial and viscous turbulent ranges. The resulting velocity field is statistically homogeneous and isotropic, and obeys the continuity equation (Pinsky and Khain, 1995). The velocity harmonic power follows the Kolmogorov (5=3) law in the inertial range with high accuracy. Besides, in the present work we allowed the amplitude of each harmonic to vary with time, thus simulating the time dependence of the turbulent flow (see Appendix A). The resulting velocity field is statistically stationary. We have used the Kolmogorov scaling law,
l ¼ l2=3 "1=3 , valid within the inertial range, for the determination of the characteristic times of harmonics with different wavelengths. In Fig. 1, the characteristic times
l , and the Kolmogorov time scale,
K, are plotted together in the log–log plot. The fact that
K coincides almost exactly with the straight line, formed by
l, indicates that we have not made a crude error, using the Kolmogorov law near the
K. Since the drop velocity flux divergence is maximal at the centimeter scale, one can expect drop-

Fig. 1. Log-log plot of the characteristic time scales vs. spatial scales, calculated from the Kolmogorov scaling law. The first point represents the Kolmogorov scale

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let concentration fluctuations to be the strongest on this scale as well. In the study we investigate the evolution of the structure of a droplet spot (hereafter denoted as ‘‘droplet cloud’’) inserted into the turbulent flow. Turbulent vortices with scales significantly exceeding 1 centimeter do not contribute to the divergence at the centimeter scale. These vortices transport small-scale droplet concentration inhomogeneities with no significant influence on their inner structure. Turbulent vortices distort the droplet cloud if its size is comparable with the size of vortices. To eliminate this distortion, we exclude turbulent vortices over 5 cm from our simulations. This scale is large enough as compared to the 1 cm scale and, at the same time, small as compared to the cloud size, which in our simulations was of the order of 70 cm. As a result, we use 21 wave number harmonics for the description of a turbulent velocity field corresponding to scales ranging from 7 mm up to 5 cm. Therefore, the external turbulent scale is equal to about 5 cm. The dissipation rate in all our simulations is set equal to " ¼ 0.01 m2s3 , typical of early cumulus clouds (Mazin et al, 1989). At the same time, the Reynolds number is small, R ¼ 18, because the largest spatial mode in our simulations is 5 cm. In spite of the fact that this value is many orders less as compared to typical values of cloud turbulence Reynolds number, the results we obtain can be attributed to cloud conditions. This point requires an explanation. The rate of droplets accumulation in high strain regions of centimeter scale is defined by the droplets velocity divergence. The spatial spectrum of this divergence at scales larger than 5 cm is less than 15% of its maximal value. Therefore, scales larger than about 5 cm do not contribute to the value of droplet velocity divergence and, consequently, the amplitude of concentration fluctuations. Thus, the cut off of the spatial scales larger than 5 cm in our simulations should not influence the results. On the other hand, reducing the external scale we reduce significantly Reynolds number. Therefore, we believe that our results reflect real values of droplets concentration fluctuations in atmosphere in spite of the fact that the Reynolds number is small. The applicability of the results to cloud conditions is discussed in the final section in more detail.

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2.2 The droplet motion equation The drop motion equation may be written as (Pinsky and Khain, 1997b): d~ V 1 ¼  ð~ V ~ Va  ~ V t Þ: ð3Þ dt
d Here ~ V, ~ V a and ~ V t are drop velocity, air velocity and droplet terminal velocity respectively. The finite difference analog of this equation, written using the midpoint method (Press, 1986), is used to calculate droplet tracks within a turbulent flow generated by the model described above. We performed simulations for 10 mm, 15 mm and 20 mm-radii droplets for two cases: without drop sedimentation (~ V t ¼ 0) and with drop gravitation sedimentation (~ V t 6¼ 0) taken into account. The time step of integration is chosen to be 0.25
d (0.25 103 sec), which is much smaller than the characteristic time of the minimal turbulent scale (0.18 sec). To eliminate the contribution of Poisson fluctuations (see below Sect. 2.4), we performed analogous simulations for non-inertial droplets passively advected by air (hereafter non-inertial droplets are referred to as passive scalar). The evolution equations for the passive scalar are equations of air motion: dx ¼ Uðx; z; tÞ; dt ð4Þ dz ¼ Wðx; z; tÞ: dt Here Uðx; z; tÞ and Wðx; z; tÞ are air velocities in horizontal and vertical directions, respectively. The data on the passive scalar behavior also allowed us to reveal the influence of boundary effects related to the penetration of drop-free filaments into the droplet cloud (see Sect. 2.4). 2.3 The simulation procedure We perform two-dimensional simulations (in the x–z plane) of droplets motion during 20 seconds (100.000 droplets of the same size) in a turbulent flow, generated by the model described above. The Gaussian form of the initial probability distribution function (p.d.f.) of droplet spatial distribution, Pðx; zÞ, was used. The initial diameter of the droplet cloud (defined by the diameter of 4,  being the standard deviation of Gaussian

distribution) is set equal to 64 cm. According to the theory of turbulent diffusion, the cloud of passive scalar should maintain the Gaussian form of Pðx; zÞ during expansion with time (Monin and Yaglom, 1975, p. 562). We assumed that in our simulations clouds of inertial droplets also maintain Gaussian Pðx; zÞ, and checked this hypothesis using the Kolmogorov test (see Appendix B). Due to the angular symmetry of the problem, instead of Pðx; zÞ we work with radial p.d.f. PðrÞ for droplet to be situated at distance r from the center. In case Pðx; zÞ is Gaussian, PðrÞ obeys the Rayleigh distribu2 2 tion: PðrÞ ¼ r2 er =2 . Among three cases (inertial droplets without sedimentation, inertial droplets with sedimentation and passive scalar) we found that in case gravitational force is taken into account, as well as in case of passive scalar this hypothesis is acceptable with a 2% significance level (the commonly used levels are between 1% and 10%). In case no gravity force is taken into account, this hypothesis can be accepted with the significance level equal to 0.3%, which is a more crude but still a sufficiently good result. These results show that the form of Pðx; zÞ is really close to the Gaussian one during droplets cloud expansion. We also compared directly the theoretical (Rayleigh) and real p.d.f. PðrÞ. We obtained parameter  of the theoretical curve by means of the maximum likelihood method (see Appendix C). The experimental curve should be purified from small-scale fluctuations since we are testing the mean profile. We can obtain such a curve by fitting the original experimental curve using the Rayleigh distribution. As a result, we obtain theoretical and experimental Rayleigh curves, which are to be compared. The r.m.s. difference between the two curves does not exceed 1.5%. To evaluate drop concentration fluctuations, we calculate fluctuation time realizations of the number of droplets in a certain volume surrounding a fixed point (the Eulerian approach). For this purpose 2000 points have been randomly chosen within the inner part of the droplet cloud. To investigate the spatial characteristics of droplet concentration fluctuations, we surround each point by a set of 25 concentric circles of different radii. Figure 2 shows the geometrical scheme used for these calculations. The diameters of the circles range from 0.6 mm to 6 cm, thus forming a set of

Investigation of small-scale droplet concentration

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Fig. 3. Time series for the 1 cm diameter circle of the 20 mm droplet Fig. 2. An example of the set of circles around one of the 2000 fixed points. r is the distance of circles center from the droplet cloud center, rc is the radius of the given circle

spatial scales. We calculate time variations of the drop amount within each circle. For each circle with diameter d we calculate the deviation of the number of particles within this circle, Nd, from the mean value hNd i. For hNd i, that we calculate analytically (assuming the Gaussian profile), we derived the following formula (see Appendix D):
 ð N rc 0 ðr 0 Þ2 þ ðrÞ2 hNd ðrÞi ¼ 2 r exp   0 22  0  rr ð5Þ  I0 dr 0 ; 2 where N is the total number of droplets (N ¼ 100.000 in our case), r is the distance from the circle center to the droplet cloud center, rc ¼ d2 is the circle radius (see Fig. 2), I0 is the modified Bessel function of zero order and  is the standard deviation of Gaussian distribution of droplets at time t. Since the droplet mean concentration varies with r, both Nd and hNd i depend on the distance from the droplet cloud center. If we do not compensate this dependence, the deviation ðNd  hNd iÞ will depend both on the local concentration fluctuation and on r. Therefore, we normalize this deviation by the mean value: Cd ðr; tÞ ¼

Nd ðr; tÞ  hNd ðr; tÞi ; hNd ðr; tÞi

ð6Þ

where r is the vector from the cloud center to the circle center and r is the length of this vector. We regard Cd ðr; tÞ as a homogeneous and stationary variable (in the statistical sense). Figure 3 shows an example of a Cd ðr; tÞ realization for the circle with the 1.5 cm in diameter. One can see that fluctuations of Cd ðr; tÞ are approximately statistically stationary during the calculation time. We calculate Nd and hNd ðrÞi with the time increment of 0.01 sec. This time increment was chosen based on the following considerations. The maximum velocity is determined by the velocity corresponding to the largest harmonic, which in our case is equal to about 8 cm s1 . This harmonic advects 1 cm-scale concentration fluctuations (the scale we are interested in) through a fixed point during 0.12 sec. Therefore, to investigate this process, we choose the time step to be one order less, i.e., 0.01 sec. 2.4 Analysis of the data Fluctuations, described by the function Cd ðr; tÞ, include three components: fluctuations due to drops inertia, natural Poisson fluctuations and boundary effects. The Poisson noise arises because of the finite number of particles we deal with. As for boundary effects, lateral mixing of drop-free air with the droplet cloud causes formation of filaments, which penetrate into the expanding cloud. Fluctuations of droplet concentration, we are going to investigate, are due to

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drops inertia only. Therefore, we have to eliminate the Poisson noise, as well as fluctuations caused by boundary effects. This problem was solved as follows. Let us denote the deviation of a number of droplets from the mean value due to Poisson fluctuations as Np0 , due to drops inertia as Nin0 , and due to filaments as Nfl0 . One can write then: Nd ¼ hNd i þ Np0 þ Nin0 þ Nfl0 : For normalized deviation Cd ðr; tÞ after some rearrangements we can write: hCd 2 ðr;tÞi ¼ ¼

hðNd  hNd iÞ2 i hNd i hN 0 p 2 i hNd i2

þ

rffiffiffiffiffiffiffiffi

2

hNin0 2 i hNd i2

Fig. 4.
¼

þ2

hNin0 Nfl0 i hNd i2

þ

hNfl0 2 i hNd i2

hNfl0 2 i  100%, hNd2 i

measured within 4 inner rings

of different radii

:

The Poisson noise is not correlated with other fluctuations. Therefore the Poisson fluctuations enter only as hN 0 p 2 i. Let us first regard the Poisson fluctuations, for which hN 0 p 2 i ¼ hNd i. Therefore, one can eliminate the Poisson fluctuations simply subtracting the corresponding term: hNin0 Nfl0 i hNfl0 2 i 1 hNin0 2 i þ2 þ : ¼ hCd ðr; tÞi  hNd i hNd i2 hNd i2 hNd i2 2

ð7Þ Another way to eliminate the Poisson fluctuations is to calculate hCd 2 ðr; tÞi for the passive scalar and to subtract it from hCd 2 ðr; tÞi for inertial droplets. This is possible since, once again, the Poisson fluctuations are not correlated with other fluctuations. The resulting difference will include only the terms due to inertia and boundary effects. Both methods gave us actually the same results. Concerning the influence of boundaries, this influence decreases toward the center of the cloud. The distance from the center of a droplet cloud, at which boundary effects become significant, may be evaluated in the following way. In case of the passive scalar the only term0 2 that hN i remains in the right-hand side of (7) is hNfl i2 . In d simulations with the passive scalar we divided the whole area into seven concentric rings, with the radius increment of 4 cm (first 0–4 cm,

second 4–8 cm, etc.). For each ring hCd 2 ðr; tÞi 02 hNfl i 1 was hNd i  hN i2 r ffiffiffiffiffiffiffifficalculated. Figure 4 shows the d

value
¼

hNfl0 2 i  100% hNd2 i

for the 4 inner rings.

We can see that in the first three rings
does not exceed 7%, lying mostly within 5%. Therefore, we performed estimations of concentration fluctuations in the central area of the cloud within 12 cm, where hCd 2 ðr; tÞi  hN1d i  hN 0 2 i Gd  hNini2 . Far from the boundaries, where d boundary effects are negligible, Gd describes normalized mean square concentration fluctuations due to inertia. This is the value we used in our further calculations. Let us regard how different scales of concentration fluctuations influence Gd. Let us consider a circle with a certain diameter d. The averaged effect of scales less than d on drop concentration fluctuations vanishes, while that of scales larger than d – does not. Figure 5 illustrates this statement. Thus, Gd is determined by all frequencies (time scales) and spatial scales over d. Therefore, the difference between the spectra of neighboring circles roughly describes the contribution of scales, ranging from the diameter of the smaller circle to that of the larger one. The spatial spectrum of droplet concentration fluctuations FðkÞ may be estimated as follows: FðkÞ ¼

ðGd1  Gd2 Þ ; ðk1  k2 Þ

ð8Þ

Investigation of small-scale droplet concentration

Fig. 5. The way two different wavelengths, larger than the circle and smaller than it, influence it 2 where k1 ¼ 2 d1 and k2 ¼ d2 are the wave numbers corresponding to the smaller d1 and larger d2 diameters, respectively, and k is their algebraic mean. The r.m.s. q amplitude of concentration flucffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ð1 tuations, I ¼ 0 FðkÞdk  100%, may be eval-

uated as Gd for d ¼ 0. 3. Results We first present the results for the case when gravitational settling of droplets is excluded. According to the theory, the evolution of the pffi passive scalar cloud size should follow the t law when the cloud size is much larger than the external turbulent scale (5 cm in our case) (Monin

Fig. 6. Time dependence of the spot diameter for 20 mm, 15 mm and 10 mm-radii droplets in case droplet sedimentation is not taken into account

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and Yaglom, 1975, p. 543). Figure 6 shows time dependence of the cloud diameter, D, for 20 mm, 15 mm and 10 mm-radii droplets. Note that the cloud consisting of larger droplets expands faster. The analysis of these curves shows that they may be fitted by the curve having the form D2 ðtÞ D2 ð0Þ ¼ 2ct with 0.1%–0.2% accuracy. In both directions the diffusion follows 2ct law and diffusivity coefficients in both directions are equal when there is no gravity. Thus, according to our results, the diffusion of inertial p droplets up to the ffi radius 20 mm also follows the t law. The particle diffusivity coefficient c is estimated as 40.54 cm2 s1 , 39.56 cm2 s1 and 37.56 cm2 s1 for droplets of 20, 15 and 10 mm-radii, respectively. The difference may be attributed to the influence of drop inertia. The velocity of inertial droplets deviates from the air velocity in a turbulent media. For example, their velocity is directed outward from the regions of high vorticity. In a turbulent medium the displacements due to these velocity deviations are chaotic, they cause diffusion of droplets, apart from ordinary turbulent diffusion of non-inertial droplets. Velocity deviations of inertial droplets increase with their mass, therefore a cloud consisting of more massive droplets diffuses faster. The curve for the cloud, consisting of passive scalar droplets (not shown here) almost coincides with the curve for 10 micron droplets (i.e., coefficient of turbulent diffusion of passive scalar is 37.5 cm2 s1 ). The increase of the rate of turbulent diffusion with the droplet size agrees with the results obtained previously by other authors (for example, analytical studies of Pismen and Nir (1978), Reeks (1977), and Wang and Stock (1993) and numerical simulations of Squires and Eaton (1991). We calculated the dependence of fluctuations power on the wave number (the spatial spectrum) (Fig. 7). The r.m.s. amplitude I of fluctuations is 48.6%, 26.6% and 12.2% of the mean concentration for droplets of 20 mm, 15 mm and 10 mmradii, respectively. At the same time the spatial structure of the concentration fluctuations field does not reveal any marked dependence on the drop radius: all three curves show that the maximum lies near 1.5 cm. The latter may be attributed to the fact that spatial characteristics of the concentration fluctuations field are defined by the structure of the turbulent flow (see Eq. (1)). The only difference is that the small-scale tail is

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Fig. 7. Spatial spectrum for the case droplet sedimentation is not taken into account (for 20 mm, 15 mm and 10 mm-radii droplets)

longer for larger radius droplets, which can be attributed to the fact that the process of droplets accumulation between turbulent eddies is more intensive in case of larger droplets. Therefore, the probability that small-scale inhomogeneities can be formed is also higher in this case. In connection with the scale of maximal fluctuations, we want to note also the following point. The fact that spatial structure of droplet velocity divergence field is determined by the structure of the turbulent flow means that the scale of maximum fluctuations should depend on flow parameters, dissipation rate and viscosity coefficient. This is obvious also from the expression for energetic spectrum of droplet velocity divergence (Pinsky et al, 1999), including these two parameters. Thus one may expect that the scale of maximum fluctuations should depend on these parameters. For example, it should depend on the droplet Stokes number through the Kolmogorov time scale of turbulent flow. Figure 8 presents the function FðkÞ (see (8)) for the passive scalar. Since we deal with discrete set of points, moving with fluid, this curve is in fact the derivative of normalized mean square Poisson fluctuations with respect to wave number. It reflects the discreetness effects and we need it for a pure technical reason: to eliminate the fluctuations caused by these effects from fluctuations of inertial droplets. There is no sense in comparing Fig. 8 with spatial spectrum of continuously distributed passive scalar (see, for example, S. Corrsin, 1951). Another important

Fig. 8. Function FðkÞ for the passive scalar

point is that both the fluctuations and the mean concentration decrease (!) with the spatial scale. Since the mean square fluctuations is equal (for Poisson distribution) to the mean value hN 0 2 i hNp0 2 i ¼ hNd i, hNp i2 ¼ hN1d i ! 1, when spatial d scale goes to zero. Therefore the increase of FðkÞ in Fig. 8 with wave numbers does not mean that the fluctuations themselves grow at small scales. Comparison of Fig. 8 with the spatial spectra of inertial droplets proves that cm-scale fluctuations are due to the inertia of these droplets. In case the terminal velocity of droplets is taken into account, the cloud containing larger droplets expands more slowly (Fig. 9). The diffu-

Fig. 9. Time evolution of the droplet cloud diameter for 20 mm, 15 mm and 10 mm-radii droplets in the case with terminal velocity

Investigation of small-scale droplet concentration

sion coefficient c is 28.54 cm2 s1 , 33.36 cm2 s1 and 35.94 cm2 s1 for droplets of 20 mm, 15 mm and 10 mm-radii, respectively. Here we once more assumed that diffusivity coefficients in both directions are equal. This is not true in the presence of gravitational force, but the difference between them is of order 5% for 10 mm–20 mm radii droplets (see, for example, Wang and Stock, 1993). We believe the decrease of diffusivities when compared to the case without gravity can be attributed to the fact that more massive droplets remain in turbulent eddies during shorter periods, which leads to a decrease in their displacements. For example, a cloud consisting of very heavy droplets will not diffuse at all. These droplets will simply fall down, slightly vibrating around their vertical trajectories. We also show in this figure the curve for the passive scalar. The decrease of the rate of turbulent diffusion with an increase in the droplet size when droplet sedimentation is taken into account agrees with the results obtained in previous studies on this subject (for example, analytical study of Csanady, 1963, numerical simulations of Squires and Eaton, 1991, and experimental measurements of Wells and Stock, 1983). The presence of drop sedimentation does not change the spatial structure of droplet concentration fluctuations: all three curves of the spatial spectrum indicate the peak at 1.5 cm in case sedimentation is taken into account as well

Fig. 10. Spatial spectrum for the case droplet sedimentation is taken into account (for 20 mm, 15 mm and 10 mmradii droplets)

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(Fig. 10). However, the r.m.s. amplitude in this case is smaller: 39.9%, 25.8% and 11.1%, respectively. This is another proof of the fact that sedimentation reduces effective time droplets spend in turbulent eddies. It also explains, why amplitude reduction is much more pronounced in the case of more massive droplets (20 mm). Vaillancourt et al (2002) in their D N S simulations also obtained a decrease in the amplitude of concentration fluctuations when sedimentation was taken in account. The values of r.m.s. amplitudes of droplet concentration fluctuations we obtained in our investigation are close to the experimental results (Pinsky and Khain, 2001, 2003). For example, Pinsky and Khain (2001) received r.m.s. amplitude of concentration fluctuations 25% of the mean concentration for St ¼ 0.07 (St ¼ 0.07 corresponds to 15 mm in our simulations). 4. Discussion and conclusions In this study, we develop and use a new method of analysis of droplet concentration fluctuations in a turbulent flow. A spot of inertial droplets (droplet cloud) is placed into a turbulent velocity field generated by a model of a turbulent flow. The velocity field generated by the model obeys turbulent correlation laws in the inertial and viscous ranges. The model predicts maximum droplet flux divergence at centimeter scales as follows from the strain spectra of the generated turbulent flow. The model includes small spatial scales and the longitudinal structure function, valid in both the inertial and viscous ranges. All this together with a continuous description of the velocity field makes it possible to describe smallscale details of droplet trajectories, which is important when exploring the process of droplets accumulations at these scales. Time variations of the velocity field depending on spatial scales, makes it possible to reproduce realistic time periods during which droplets tend to concentrate and periods when droplets tend to scatter. This factor turns out to be of high importance since the process of droplet accumulations in areas between turbulent eddies depends strongly on the characteristic time scales of these eddies. A special approach was successfully used to eliminate the noise caused by Poisson fluctua-

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tions and boundary effects. The normalization removes the dependence of droplet concentration fluctuations on the mean concentration. As a result of the analysis the spatial spectrum of droplet concentration fluctuations induced by droplet inertia was calculated for droplets of different mass. The results of the analysis show that: (i) Inertial droplet concentration inhomogeneities form in a turbulent flow at the centimeter scale. (ii) It is the turbulent-inertia mechanism that leads to droplet concentration fluctuations at the cm-scale. No maximum was found for the passive scalar (non-inertial particles). (iii) The spatial spectrum for the 10 mm, 15 mm and 20 mm-radii droplets indicate the maximum located at about 1.5 cm both in the cases with and without gravity-induced sedimentation. This scale is close to that of the velocity flux divergence (Pinsky et al, 1999). (iv) The r.m.s. amplitude of fluctuations was found to be significant, it varies from about 12% for 10 mm-radii drops to 40% for 20 mm-radii drops. An increase in the r.m.s. amplitude is approximately proportional to the droplet radius square. It is natural, since the r.m.s. amplitude of fluctuations is defined by the time integral of the droplet velocity flux divergence, which in its turn is proportional to the droplet radius square (Pinsky et al, 1999). (v) The amplitude of droplet concentration fluctuations slightly decreases in the presence of sedimentation. This effect is more pronounced for massive droplets. (vi) The rate of turbulent diffusion of droplets is smaller than that of the passive scalar in case droplet sedimentation is taken into account. The larger the droplets are the slower they diffuse. On the contrary, when there is no sedimentation, inertial droplets diffuse more quickly as compared to the passive scalar and the rate of diffusion increases with the droplet size. We recognize limitations related to the utilization of a 2-D model that simulates a statistically stationary, homogeneous and isotropic turbulent flow and does not reproduce intermit-

tent structures, typical of real cloud turbulence with very high Reynolds numbers. Nevertheless the values of amplitude of droplet concentration fluctuations we obtained in this study are close to the estimations, obtained by Pinsky and Khain (2001; 2003) using a statistical analysis of series of drop arrival times measured in several tens of clouds. They are also close to the results of analytical investigations. For example, the analytical results of Falcovich et al (2002) predict 20%–40% r.m.s. amplitude for 15 mm–20 mm droplets. Our method is able to reproduce the features of droplet diffusion in turbulent field, like different dependence of diffusion rate on droplets size with and without sedimentation, reported in literature (Csanady, 1963; Reeks, 1977; Pismen and Nir, 1978; Wells and Stock, 1983; Squires and Eaton, 1991; Wang and Stock, 1993). As a whole, we believe our results reflect a probable pathway for the formation of concentration inhomogeneities in clouds. The calculation were performed for the dissipation rate " ¼ 0:01 m2 s3 typical of weak cumulus clouds. For stronger cumulus clouds dissipation rate is higher and can reach 0.1 m2 s3 (Mazin et al, 1989; Weil et al, 1993). As it was shown by Pinsky et al (1999), r.m.s. of the divergence is proportional to the dissipation rate: pffiffiffiffiffiffiffiffi 4 "d hðdiv~ V Þ2 i1=2 ¼ 240
G

d ¼ pffiffiffiffiffi 2 rd2 ; 9 15 v a

ð9Þ

where G ¼ "
ð30vÞ1 and d and a – densities of droplet substance and air, respectively; v is the kinematic air viscosity; rd – droplet radius. The analogous calculation for a two-dimensional turbulent flow gives: pffiffiffiffiffiffiffiffi ð10Þ hðdiv~ V Þ2 i1=2 ¼ 192
G

d : The turbulent model used in the simulations indicates similar dependence of the divergence on ". We suppose, therefore, a significant increase in the drop concentration fluctuation with the increase in the turbulence intensity. Comparison of Eqs. (9) and (10) shows that the difference between r.m.s. drop velocity divergence in a fully 3-D approach and in the model used is small (the difference is about 10%). Therefore we suppose that our results give correct order magnitude of r.m.s. concentration fluctuations.

Investigation of small-scale droplet concentration

Pinsky et al (1999) calculated spatial spectrum of droplet velocity field divergence for different dissipation rates in 3-D flow and found that the maximum varies between 1 cm for " ¼ 0:05 m2 s3 and 2.5 cm for " ¼ 0:001 m2 s3 . The 2-D model reproduces the main features of 3-D flow, like (5=3) energy spectrum of velocity and velocity strains maximum near 1 cm scale. Therefore we suppose that shortcomings of our 2-D approach do not influence the location of the spatial spectrum maximum of concentration fluctuations. The influence of small-scale concentration fluctuations on microphysical processes in clouds may be significant due to significant amplitude of these fluctuations. For example, Pinsky and Khain (1997c) demonstrated the possibility of a significant acceleration of collision processes when small-scale spatial inhomogeneities of droplet concentration were taken into account. At the same time, the influence of concentration inhomogeneities on microphysics processes depends crucially on two factor: (a) spatial correlation in the locations of the areas of enhanced droplet correlation of droplets of different size, and (b) the characteristic time scales of these inhomogeneities. We have not performed simulations with multi-disperse droplet populations in this study. We assume, however, that small cloud droplets tend to concentrate in the same volumes of the turbulent flow, because the spatial structure of the droplet concentration is defined by the spatial structure of turbulent shear field, which is the same for all droplets. This expectation is supported by the result of the present study that sedimentation does not influence significantly the amplitude of droplet concentration fluctuations. Nevertheless, this expectation has to be justified in future simulations with multidisperse drop populations. As concerns the characteristic time scales for droplet concentration non-homogeneities, they should be at least comparable with the characteristic time scales of condensation or coagulation processes to provide some reasonable effect. To investigate the characteristic time scales of the inhimogeneities of droplet concentration one should use the Lagrangian approach. In this approach fluctuations of a droplet concentration around moving droplets will be analyzed. In this

201

way the characteristic time during which these droplets move within zones of enhanced (or decreased) droplet concentration may be evaluated. Appendix A Time dependence of the velocity field in the turbulent flow model The velocity components are defined in the following way (Pinsky and Khain, 1995, 1996): Ux ðx; z; tÞ ¼

K X

k

k¼1

M X

k sin ’m fak;m ðtÞ

m¼1

 cos ½k ðx cos ’m þ z sin ’m Þ þ bk;m ðtÞ sin ½k ðx cos ’m þ z sin ’m Þg; Uz ðx; z; tÞ ¼ 

K X k¼1

k

M X

k cos ’m fak;m ðtÞ

m¼1

 cos ½k ðx cos ’m þ z sin ’m Þ þ bk;m ðtÞ sin ½k ðx cos ’m þ z sin ’m Þg: Here fk g is the set of wave numbers. For a given wave number the set of ’m determines the set of random harmonics with a different direction of the wave vector. Coefficients k are defined proceeding from the form of the spatial correlation function. The values ak;m ðtÞ and bk;m ðtÞ are the sequences of normal random numbers with the following properties: hak;m ðtÞi ¼ hbk;m ðtÞi ¼ hak;m ðtÞbl;n ðtÞi ¼ 0; hak;m ðtÞal;n ðtÞi ¼ hbk;m ðtÞbl;n ðtÞi ¼ km;ln : Since all velocity harmonics are non-correlated, the velocity time correlation function is equal to the sum of time correlation functions of separate harmonics, Uk,m. The latter is (assuming correlations of ak,m and bk,m to be equal): hUk;m ðx; z; tÞUk;m ðx; z; t þ
Þi ¼ 2k 2k sin 2 ’m fhak;m ðtÞak;m ðt þ
Þi cos 2 ½k ðx cos ’m þ z sin ’m Þ þ hbk;m ðtÞbk;m ðt þ
Þi sin 2 ½k ðx cos ’m þ z sin ’m Þ g ¼ 2k 2k sin 2 ’m hak;m ðtÞak;m ðt þ
Þi: Therefore, the velocity time correlation function, corresponding to one wave number, is: hUk ðx; z; tÞUk ðx; z; t þ
Þi ¼ 2k 2k

M X

sin 2 ’m

m¼1

hak;m ðtÞak;m ðt þ
Þi; or, assuming hak;m ðtÞak;m ðt þ
Þi to be the same for all m: hUk ðx; z; tÞUk ðx; z; t þ
Þi ¼ hak;m ðtÞak;m ðt þ
Þi M X 2k 2k sin 2 ’m m¼1

¼ hak;m ðtÞak;m ðt þ
Þi2k 2k

M : 2

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B. Grits et al

For
¼ 0 this result reduces to hUk2 ðx; z; tÞi ¼ 2k 2k M2 , since hak;m ðtÞak;m ðtÞi ¼ 1. Time evolution of ak,m and bk,m is simulated by means of an autoregression (AR) sequence of the first order (Hamilton, 1994): ak;m ðn þ 1Þ ¼ dkak;m ðnÞ þ ck ðn þ 1Þ: Here ak;m ðnÞ is defined at discrete time n and  is the sequence of non-correlated stochastic numbers with zero mean and unity dispersion. The time correlation function of such a process is hak;m ðnÞak;m ðn þ
Þi ¼

c2k c2k dk
 e
=
k ; 2 1  dk 1  dk2 ðdk ¼ e1=
k Þ;

ðA:1Þ

where
k is the characteristic time scale of mode k. ck may be found based on the requirement that hak;m ðtÞak;m ðtÞi ¼ 1, pffiffiffiffiffiffiffiffiffiffiffiffiffi namely ck ¼ 1  dk2. Therefore hak;m ðnÞak;m ðn þ
Þi ¼ e
=
k and the velocity correlation is equal to hUðx; z; tÞUðx; z; t þ
Þi ¼

K X hUk ðx; z; tÞUk ðx; z; t þ
Þi k¼1

K MX ¼ 2 2 e
=
k : 2 k¼1 k k

ðA:2Þ

We defined dk by requiring
k to follow the Kolmogorov scaling law, which is
k ¼ ð2Þ2=3 "1=3 k2=3 (or, as the function of spatial scale l,
l ¼ l2=3 "1=3 (see Fig. 1)). Since hak;m ðtÞak;m ðt þ
Þi does not depend on t, the turbulent field is statistically stationary.

Appendix B Checking the hypothesis on the droplet probability distribution function by means of the Kolmogorov test To test if some experimental sample belongs to a concrete distribution function, it is customary to compare the hypothetical integral distribution function FðxÞ with the empirical one F  ðx; xðÞ Þ (for example, Bredley, 1968). The latter is defined as follows: n 1X F  ðx; xðÞ Þ ¼ uðx  xðkÞ Þ; n k¼1 where xðÞ is an ordered sample and u(z) is a step function:
0; z < 0; uðzÞ ¼ : 1; z 0: According to Kolmogorov, as a measure of the difference between empirical F  ðx; xðÞ Þ and hypothetical FðxÞ distribution functions the following statistics serves: d ðF  ; FÞ ¼ sup jFðxÞ  F  ðx; xðÞ Þj: n

Therefore, given the significance level and d ¼ f ðÞ, the hypothesis is rejected if dn ðF ; FÞ > d and accepted otherwise. Kolmogorov pffiffiffiffi found an approximate expression for , which, for N dn > 1 (N being the number of elements in a sample; in our case N ¼ 105 ), reads: 2

2e2Nd ¼ ; the expression we used for the calculation of d . The commonly accepted values of the significance level are between 1% and 10%.   1% means that the test for hypothesis is too crude (the hypothesis will be accepted actually for every sample), while   10% means that the hypothesis can be rejected too easily.

Appendix C Determination of parameter  of the Rayleigh distribution by means of the maximum likelihood method Let us we have NðN  1Þ independent points, each of which has the same probability density to be situated at a position ri, Wðri ; Þ.  is the parameter of this probability function. The joint probability function for N points will be: N Y Wðri ; Þ: WN ðr1 ; r2 ; . . . ; rN ; Þ ¼ i¼1

Then, for a given measured set fri g, according to the maximum likelihood method, the parameter  may be found from the requirement: @ ln WN ¼ 0: @ In our case: Wðri ; Þ ¼ Wðri ; Þ ¼ QN

i¼1 ri 2N

WN ¼


ri ri2 ; exp  2 22

ðrI ¼ jri jÞ;


N 1 X 2 and exp  2 r 2 i¼1 i

 Y N N 1 X lnðWN Þ ¼ ln ri  2N lnðÞ  2 r2 : 2 i¼1 i i¼1 Then: N @ ln WN 2N 1X þ 3 ¼ r 2 ¼ 0;   i¼1 i @ N 1 X  ¼ r2 : 2N i¼1 i

or ðC:1Þ

2

Appendix D

x

If dn ðF  ; FÞ exceeds some critical value, d , this hypothesis is rejected, otherwise it is accepted. This critical value is directly connected with the probability of erroneous rejection of the hypothesis (significance level), : Pfd ðF ; FÞ > d g ¼ : n



Calculation of the mean number of droplets within the circle Let us we have a circle with radius rc situated at a distance r from the center of a droplet cloud (see Fig. 2). Let r0 denote the distance from the circle center to some point within this

Investigation of small-scale droplet concentration circle and ’ be the angle between r and r0 . If p.d.f. of the droplet spatial distribution has the Gaussian form, then the probability for the droplet to be situated within the element r 0 dr0 d’ at a distance r from the cloud droplet center is equal r 02 2r 0 r cos ’þr2

 1 22 r0 dr0 d’. The probability to dPðr; r0 ; ’Þ ¼ 2 2 ‘ for the droplet to be situated within this circle is ðrc 2 ð 02 0 r 2r r cos ’þr2 1 22 ‘ r0 dr 0 d’ Pðr; rc Þ ¼ 2 2 0 0

ðrc

r02 þr2 1 ¼ ‘ 22 2 2

0

2ð

r 0 r cos ’ 2

‘

r0 dr 0 d’

0

 1 r0 r 0 dr 0 ; ‘ r I 0 2 2  0 0 where I0 r2r is a modified Bessel function of zero order. The mean value of the droplet concentration is then equal to hNd i ¼ NPðr; rc Þ, where N is the total number of droplets (100.000 in our simulations). ¼

ðrc

02 2 r þr2 2



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Corresponding author’s address: Alexander Khain, The Institute of Earth Sciences, The Hebrew University of Jerusalem, Givat Ram Campus, Jerusalem, 91904 Israel (E-mail: [email protected])