## 1. Introduction

The mechanism for warm-rain initiation is one of the unsolved problems of cloud physics. One aspect of the rain formation process that has not yet been satisfactorily explained by theory is the presence of broad droplet spectra in cumulus clouds (Warner 1969). A method for producing both smaller and larger droplets than those predicted by a closed, adiabatic parcel model of droplet formation and growth likely exists (Mordy 1959). This lack of agreement between theory and measurements exists even for droplet spectra measured in undiluted (adiabatic) parcels (Lee and Pruppacher 1977; Jensen et al. 1985). In addition, Hill and Choularton (1985) found that the largest droplets in a cloud are produced in the core regions of clouds where mixing effects are not expected to be large. Although some broadening of measured droplet distributions can be traced to instrumental artifacts (e.g., Cerni 1983), newer instruments continue to show broad spectra (Lawson et al. 1996). Recent measurements in adiabatic cores of cumulus clouds show that droplet spectra are broader than theory would predict (Brenguier and Chaumat 1996). This problem is critical because it is thought to be related to a “growth gap” separating the processes of droplet growth by condensation and growth by collision–coalescence (Cotton and Anthes 1989). It appears that to fully understand observations of cumulus clouds there likely is a mechanism for the broadening of cloud droplet spectra during the condensation growth process. It is particularly important that the mechanism produce droplets large enough to initiate the process of collision–coalescence.

Many hypotheses have been advanced to explore the broadening of cloud droplet spectra. These hypotheses have been reviewed by Beard and Ochs (1993). Here we wish to concentrate on some of the relevant hypotheses that point to turbulence as the cause of cloud droplet spectral broadening. The notion that cloud droplet spectra are broadened through the turbulent nature of clouds has been proposed by a number of researchers. Some of the earliest work was based on the concept of stochastic condensation: turbulent fluctuations in vertical velocity result in fluctuations in supersaturation and, therefore, droplet growth rate (Levin and Sedunov 1966). However, it was noted by Bartlett and Jonas (1972) that this effect is probably not significant due to the strong inverse correlation between the amount of supersaturation and total growth time. Hence, the droplets experiencing enhanced growth rates also experience shorter periods of growth and vice versa, resulting in little broadening of the droplet spectrum. Models that include mixing and entrainment are attractive because they help account for the production of large droplets and broad droplet spectra (e.g., Baker et al. 1980). Such models, however, do not explain the broad cloud droplet spectra observed in adiabatic core regions. Manton (1979) suggested that turbulence near cloud base, during the time of cloud droplet nucleation and initial growth, could account for some degree of broadening and could produce bimodal droplet spectra. Further work in this direction by Cooper (1989) has shown that fluctuations in the microphysical environment of growing cloud droplets can lead to significant broadening of droplet size spectra. Although the details of the above studies are different, it appears that the turbulent nature of cumulus clouds affects the formation and growth of cloud droplets by condensation through several mechanisms. Many of these are related to fluctuations in the vertical velocity field and to turbulence-induced variations in cloud microphysical properties such as the cloud droplet concentration. Whether turbulence effects alone can account for all or most of the observed features of warm-cloud droplet spectra and rain production times is still unclear (Beard and Ochs 1993).

In this paper we present a mechanism for the broadening of droplet spectra in clouds due to the influence of turbulence on local cloud droplet concentrations. We hypothesize that on small spatial scales cloud droplets are not randomly distributed, but are preferentially concentrated by coherent turbulent structures. Preferential concentration provides a theoretical basis for the recently observed inhomogeneous microstructure of cumulus clouds (Baker 1992). This preferential concentration, or “unmixing,” of cloud droplets may significantly alter the microstructure of a cloud by creating regions of relatively high and low supersaturation, thereby impacting the bulk microphysics. Here we describe a model of cloud droplet growth by condensation. The model takes into account, in a simple yet physically based manner, the effects of preferential concentration of cloud droplets by turbulence. The model accounts for some degree of droplet spectral broadening *independent* of entrainment, fluctuations in updraft velocity, variations of temperature or liquid water content at cloud base, variations in aerosol distribution, presence of giant nuclei, multiple cycling of droplets through clouds, etc. The present mechanism of droplet spectral broadening due to preferential concentration is always taking place in a cloud due to its inherent turbulent nature. This fundamental process may be combined with one or more of the above processes and in some cases may significantly enhance the effects of such processes.

Preferential concentration likely affects other aspects of cloud microphysics as well. For example, Pinsky and Khain (1996, 1997) studied the effects of preferential concentration on the collision–coalescence process in clouds. Their work considered droplet trajectories in a field of frozen turbulence. They observed that preferential concentration significantly increased the collision–coalescence rate and possibly increased the collection efficiency for smaller droplets. Their results are in agreement with recent dynamic simulations that show enhancements in the collision rate of more than one order of magnitude due to preferential concentration (Sundaram and Collins 1997). We note that the use of frozen turbulence by Pinsky and Khain cannot be easily extended to the present study of supersaturation fluctuations since this is inherently a time-dependent phenomenon; thus an alternative approach is used (see section 3b).

The impact of preferential concentration on cloud droplet condensational growth is governed by several fundamental processes, each with a characteristic timescale. We will introduce the controlling physical processes here and discuss them in greater detail in later sections of the paper. The characteristic processes are (i) small-scale turbulent fluctuations (including coherent vortex structures), (ii) droplet response to fluid velocity fluctuations, (iii) evolution of supersaturation (dependent on droplet concentration and vertical velocity), and (iv) diffusive decay of supersaturation gradients. The two significant variables that determine the timescales for these processes are the turbulent dissipation rate and the droplet radii. The relationship between the droplet response time and turbulence timescale determines the degree to which droplets preferentially concentrate. The supersaturation timescale specifies the required time needed for the supersaturation to respond to changes in the droplet concentration resulting from the preferential concentration. Finally, the molecular diffusion timescale is related to the rate at which the buildup of supersaturation gradients in the fluid is resisted. Preferential concentration is most significant when the turbulence timescale is of the order of the droplet timescale. For preferential concentration to significantly impact the droplet size spectrum, it is necessary to have both the diffusion and preferential concentration timescales longer than the supersaturation timescale.

In the next section, we shift our focus to the phenomenon of preferential concentration. In section 3 we introduce the role it can play in the condensational growth process of clouds. Model results for the specific case of a continental cumulus cloud are reported in section 4. Section 5 contains a discussion of these results along with a discussion of the significant model attributes in their atmospheric context. Conclusions are presented in section 6.

## 2. Turbulence-induced preferential concentration of droplets

Aerosol particles, due to their higher density relative to the continuous gas phase, will not remain uniformly distributed throughout a turbulent flow field. This phenomenon, often referred to as preferential concentration, has been an area of active research within the engineering community for approximately one decade (Maxey 1987; Squires and Eaton 1991; Sundaram and Collins 1997). Our purpose here is not to give a detailed review of the theory and evidence for preferential concentration, but to point out the fundamental nature of the phenomenon so that its role in cloud processes may be understood. For a thorough review of the state of this field of research, see Eaton and Fessler (1994).

The phenomenon of preferential concentration has been studied primarily using direct numerical simulation (Squires and Eaton 1991; Wang and Maxey 1993; Sundaram and Collins 1997). Direct numerical simulation (hereafter DNS) refers to the numerical integration of the three-dimensional, time-dependent Navier–Stokes equations. The numerical algorithm must fully resolve all fluid time and length scales, thus providing a simulation of the turbulent flow. Figure 1 shows the particle positions in a slice from the computational domain of a typical DNS. Figure 1a is the initial (random) particle concentration field and Fig. 1b is the concentration field after several eddy turnover times. Notice the dramatic change in the particle concentration field with time. Figure 1b contains large regions with essentially zero particle concentration and other regions with several-fold increases in its concentration. This is referred to as preferential concentration (Squires and Eaton 1991). There has been experimental confirmation of this phenomenon (Eaton and Fessler 1994), including recent observations suggesting there are inhomogeneities in droplet concentration on the scale of 5 mm to 5 cm in cumulus clouds (Baker 1992).

The mechanism responsible for concentrating the particles is shown schematically in Fig. 2. In effect, intense vortical structures within the turbulent flow field act as centrifuges, separating the heavier particle phase from the lighter fluid phase. The concentration of particles in the “high strain” regions of the flow can be many times higher than the average concentration, and conversely the concentration of particles within the intense vortices can be greatly reduced by this mechanism. It is important to note that not all droplet sizes preferentially concentrate to the same degree. Indeed, the degree of preferential concentration is a strong function of the physical properties of the particles and the turbulence properties. Below is a brief discussion of the important turbulence scales and particle properties that influence the preferential concentration mechanism.

### a. Turbulence scales

*τ*

_{η}) and length (

*η*) scales for the Kolmogorov eddies are given by (Tennekes and Lumley 1972)

*ν*

_{f}is the fluid kinematic viscosity. To provide some feeling for the numerical values of these scales, we note that under typical conditions found in cumulus clouds the average energy dissipation rate varies over the range 0.001 to 0.1 m

^{2}s

^{−3}. The corresponding range of Kolmogorov length scales is 0.4–1.25 mm and time scales of 0.01–0.1 s. Turbulent kinetic energy dissipation rates are on the order of 0.01 m

^{2}s

^{−3}in the adiabatic cores of cumulus clouds (Paluch and Baumgardner 1989; Meischner et al. 1997).

Recent studies of turbulent flows recognize that superimposed on the near-Gaussian background turbulence are localized coherent motions that persist for times that are much longer than predicted by classic scaling laws. These persistent motions fall into the broad category generally referred to as coherent structures. Here, we refer to the dissipative structures that appear as vortex “tubes” or “worms” that permeate isotropic turbulence (She et al. 1990, 1991). Vortex tubes are relevant to the present discussion because they are thought to be responsible for ejecting particles into high strain regions of the flow. There is still no comprehensive theory that fully describes the vortex tubes; nevertheless, it is known that they have several attributes that are easily identified. In general, the vortex tubes are thin filaments with diameter/length ratios much smaller than unity. They are stabilized against viscous dissipation by large-scale straining motions that stretch the filaments along their axes, producing vorticity.

The expected lifetime of the coherent structures, here designated as *τ*_{s}, is a vitally important parameter that will impact the degree of preferential concentration (as will be shown in the next section). It is therefore desirable to obtain a scaling relationship for this parameter. Unfortunately, there is not very much known about the eddy lifetime of the coherent structures per se. It is generally thought that the presence of structures gives rise to the statistical intermittency observed in the higher-order velocity moments and their derivatives. There have been a number of experimental and theoretical attempts to quantify the scaling relationships for intermittency (Meneveau and Sreenivasan 1991; Sreenivasan and Antonia 1997); however, despite the important contributions of this work, there is still very little understanding about the characteristic life expectancy of a given vortex structure. Ruetsch and Maxey (1992) considered this question with DNS and proposed that *τ*_{s} is proportional to the large eddy turnover time *T*_{e}. Recent experimental observations of vortex filaments using microbubbles in nearly isotropic turbulence (Douady et al. 1991; Villermaux et al. 1995) are in qualitative agreement with Ruetsch and Maxey (1992). Unfortunately, none of these studies definitively answers the question since they were all limited to relatively modest Reynolds numbers as compared to the values found in the atmosphere. Given the present uncertainties, we elect to treat the coherent structure lifetime *τ*_{s} as a parameter and vary it over a reasonable range. More precise scaling of *τ*_{s} will require additional analysis of coherent structures in turbulent flows at appropriately high Reynolds numbers.

### b. Particle scales

*a,*the radius of the particles;

*ρ*

_{p}, the density of the particles; and

*n,*the number density of particles. Alternatively, we can replace one of the physical parameters (e.g., density) by the particle response time defined as

*μ*

_{f}is the dynamic viscosity of the fluid in which the particle resides and

*τ*

_{p}is the characteristic time for a particle to respond to accelerations imposed by the surrounding fluid. It is convenient to represent the three new parameters in a nondimensional form. The dimensionless response time or particle Stokes number is defined as

*φ*is the volume fraction of particles.

In clouds the volume fraction is extremely low so that dilute conditions prevail throughout. In this limit, *φ* acts purely as a multiplicative factor but otherwise has no influence on the processes. Likewise, the dimensionless droplet radius *a*

### c. Preferential concentration: The Rankine vortex model

*η*and the outer-edge velocity is assumed to be

*υ*

_{η}≡ 6.8

*η*/

*τ*

_{η}based on scaling from the Kolmogorov spectrum. A particle with a response time

*τ*

_{p}is then embedded in the vortex at an initial radial position

*r*

_{0}and with zero velocity. The dimensionless equations (normalized by

*η*and

*τ*

_{η}) in Eulerian coordinates that describe the motion of the particle as a function of time are as follows:

*θ*and

*r*are the angular and radial coordinates of the center of the particle, respectively;

*υ*

^{p}

_{θ}

*υ*

^{p}

_{r}

*υ*

_{θ}is the angular velocity of the fluid (note: the radial velocity of the fluid

*υ*

_{r}is identically zero in a Rankine vortex); and St retains its original definition [see Eq. (3)]. The preceding equations are derived from a force balance on the particle assuming the dominant force exerted by the fluid is Stokes drag (Maxey and Riley 1983; Manton 1974). This implicitly assumes that the Reynolds number, Re

_{p}, based on the fluid properties and the particle diameter remains small. An upper bound for Re

_{p}can be found by considering a

*fixed*particle in the vortex. In this limit, the Reynolds number is bounded by 68

*a*

*υ*

_{θ}. Thus, if

*υ*

_{θ}⩽ 1, and we assume

*a*

_{p}⩽ 1.4. Because the Reynolds number for a moving particle will be much lower than this rather conservative bound, we conclude that Stokes drag is the dominant term in the equation of motion for the particles and that neglecting finite-Reynolds-number corrections to Eqs. (8) and (9) is justified. Note that because the maximum drop size calculated in this study is moderately small (i.e., radius below ∼8

*μ*m), effects due to gravitational settling can also be neglected. Finally, the density ratio

*γ*for a cloud droplet is small and therefore is neglected in this analysis.

*radial*direction, will cause a resistance to the acceleration [see Eq. (9)]. The net effect is the particle will circulate rapidly but move radially very slowly. Asymptotic analysis yields the following relationship for the radial position of a particle as a function of time:

*υ*

^{p}

_{θ}

^{2}/

*r*is small [see Eq. (9)]. Once again, asymptotic analysis can be used to determine approximately the position of the particle as a function of time

*maximum*preferential concentration in earlier DNS studies (Sundaram and Collins 1997). Indeed, the above model, despite its simplicity, appears to capture the essential physics. Figure 3a also demonstrates the qualitative role that the structure lifetime

*τ*

_{s}may have in controlling the degree of preferential concentration. We anticipate two effects as

*τ*

_{s}increases: (i) the degree of preferential concentration of particles of a given Stokes number will increase and (ii) the range of Stokes numbers that will experience preferential concentration will broaden.

An alternative view is given in Fig. 3b, which shows the percentage of particles of a given Stokes number that remain in the vortex as a function of time, assuming an initially uniform distribution of particles. Naturally, the percentage that remains decreases with time as particles are irreversibly lost from the vortex. Notice that particles with Stokes numbers near ⅔ are ejected the most rapidly; nevertheless particles with moderately low or moderately high Stokes numbers are still ejected at a finite rate. This suggests that eventually all particles will preferentially concentrate, given a sufficiently large value of *τ*_{s}.

The Rankine vortex model described above forms the core of the proposed cloud formation model. The important element that is missing is the cloud droplet growth mechanism due to surface condensation. The physics of activation and growth will be discussed in detail in the next section.

## 3. Droplet growth model

In the previous section we have presented an overview of the factors controlling the small-scale distribution of cloud droplets in a turbulent medium. In this section we present the model used to investigate the effects of preferential concentration on condensational growth of cloud droplets. This model is based on the theory of cloud droplet activation and growth by condensation in an ascending, unmixed (adiabatic) cloud parcel.

### a. Supersaturation evolution and droplet growth

*w*is the constant updraft velocity,

*s*is the supersaturation in percent (

*e*/

*e*

_{s}− 1) × 100%,

*e*is the partial pressure of water vapor,

*e*

_{s}is the equilibrium vapor pressure of water,

*α*and

*β*are considered constants, and the sum is over the radii

*a*

_{i}of all existing cloud droplets in a unit volume of cloud (the integral radius). This equation leads to the well-known supersaturation development curve consisting of a rapid, nearly linear rise in supersaturation in time followed by a drop to a nearly steady-state value.

*N*

_{CCN}is the number of activated CCN per cubic centimeter at supersaturation

*s,*and

*C*and

*k*are constants that depend on the nature of the background aerosol. The initial size of the activated droplet (in micrometers) is approximated following Twomey (1977) as

*μ*m in diameter (to exclude giant nuclei effects).

*G*is considered a constant.

### b. Turbulence effects

Next we develop the formalism to include the effects of preferential concentration on the growth of cloud droplets by condensation. The basis for the following argument is that preferential concentration affects cloud droplet spectra through the formation of pockets of enhanced supersaturation. As discussed in section 2, the structure timescale *τ*_{s} is thought to be at least an order of magnitude larger than *τ*_{η}. Values of *τ*_{s} in the atmosphere are presently unknown, but it is reasonable to assume that they are one or two orders of magnitude larger than atmospheric values of *τ*_{η}. Therefore, we assume that regions of high vorticity can persist for times (*τ*_{s}) on the order of 10 s. The timescale for variations in supersaturation, *τ*_{sat}, is given by (*β* Σ *a*)^{−1}, which is typically between 1 and 10 s. It is apparent, therefore, that the supersaturation of a high-vorticity turbulent structure will grow to a larger value than the surrounding fluid because it contains fewer droplets. If significant preferential concentration occurs, regions of low vorticity will have larger than average number concentrations of cloud droplets and therefore a lower supersaturation and droplet growth rates. Regions of high vorticity, on the other hand, will be populated with a significantly smaller number of cloud droplets, allowing the supersaturation to be larger. Droplets in these regions will experience enhanced growth rates or, if supersaturations are high enough, new droplets may be activated.

Based on the above physical arguments we artificially divide the model into two distinct “zones” representing high- and low-vorticity parcels of fluid respectively. The assumption that each zone is of the same volume is an approximation based on the observation that the regions of high and low vorticity shown in Fig. 1b are of roughly the same size. The high vorticity region is assumed to be a Rankine vortex, and, therefore, droplets move from the high-vorticity region according to the mechanism described in section 2. When a droplet leaves the Rankine vortex it enters the second, or low-vorticity, zone. This process continues until time *τ*_{s} elapses, after which the Rankine vortex is assumed to decay and the zone of low vorticity becomes a Rankine vortex. In other words, a single zone oscillates between a state of zero vorticity and the vorticity of the Rankine vortex presented in section 2 with a period of twice the structure time *τ*_{s}. The supersaturation in each region is allowed to developed independently as the number of droplets in each region changes. Because both zones are rising at the same updraft speed, only changes in the second term on the right-hand side of (14) can give rise to fluctuations in the supersaturation.

A numerical model was constructed using Eqs. (7)–(9) and (14)–(17). Although this model of turbulence and cloud droplet interactions can be applied generally, in this paper we apply it to the specific case of a typical continental cumulus cloud. Constants in Eqs. (4) and (5) were assigned typical values (*α* = 5.5 × 10^{−2} m^{−1}, *β* = 3.1 × 10^{2} m^{−1} s^{−1}, *C* = 650 cm^{−3}, *k* = 0.7, and *G* = 9 × 10^{−13} m^{2} s^{−1}), and a constant vertical velocity (*w* = 2 m s^{−1}) was used. These values were chosen to allow qualitative comparison with the simple droplet growth model described by Rogers and Yau (1989). The equations were solved using the Euler method with a time step of 1 × 10^{−3} s. Equations (8) and (9) were solved implicitly for stability. The radius of the eddy was assigned a value of 10 *η.* The eddy lifetime, *τ*_{s}, was assigned a value of 5, 10, and 15 s. The model was initialized with no droplets. Activated droplets were randomly distributed within their parcels of origin. After each *τ*_{s} seconds the kinematic structures of the parcels were reversed, with the resulting effect that droplets then moved in the opposite sense. The model was run first without preferential concentration (ε = 0), and then with preferential concentration and at three values of turbulent dissipation (0.001, 0.01, and 0.1 m^{2} s^{−3}). This range of dissipation rates is representative of measurements in adiabatic cores of cumulus clouds (Paluch and Baumgardner 1989; Meischner et al. 1997).

## 4. Results

In this section we present results from simulations with the model described in the previous section. Results from a single parcel system (classic theory) are contrasted with those from two parcel systems of varying dissipation rates and structure times.

Figure 4 compares the time evolution of the total number of droplets in each parcel, as well as the total number of droplets in the system for ε = 0.01 m^{2} s^{−3} and *τ*_{s} = 10 s, to that of a single parcel (ε = 0, dotted line). For the single parcel case, the number concentration increases initially as droplets activate. After 10 s the parcel reaches its maximum supersaturation, which subsequently decreases to its quasi-steady-state value with an associated cessation of activation events (Fig. 5). On the other hand, the total number concentration in the preferentially concentrating system continues to increase after 80 s as new droplets are activated. The number of droplets in each of the two parcels fluctuates around average values since droplets are continuously being thrown out of or collected by the parcel depending on whether or not the parcel is a high-vorticity structure. As more droplets grow into the appropriate Stokes number range, the degree of preferential concentration increases. Thus, higher supersaturations are reached in the high-vorticity parcel due to the decreased concentration of droplets. Eventually higher supersaturations are reached than the peak supersaturation at cloud base and new droplets are activated above cloud base.

Figure 5 shows the supersaturation dependence on the two critical parameters (dissipation rate and structure timescale). As the dissipation rate increases, and thus the degree of preferential concentration, fluctuations about the values produced by the classic theory increase. These fluctuations increase in magnitude for increasing structure timescales. Values up to three times the maximum of the classic theory are produced for large dissipation rates and structure timescales. Even at typical dissipation rates each parcel experiences a mean supersaturation greater than that of the classic theory (1.3–2.1 times). Essentially, this is because the Rankine vortex model does not account for fluctuations in the source term (*αw*) of the supersaturation equation. Hence, the parcels can never become subsaturated. On the other hand, the maximum supersaturation is only constrained by the number of droplets in each parcel, which can drop to almost zero at large structure times.

Figure 6 shows the calculated droplet spectra after 150 s for different values of dissipation rate and structure time. The droplet spectrum resulting from the classic theory is included for comparison. There is little change in the droplet spectrum for the ε = 0.001 m^{2} s^{−3} case (for all three values of *τ*_{s}). The spectrum broadens by a few tenths of microns. This is also true for the ε = 0.01 m^{2} s^{−3}, *τ*_{s} = 5 s case. All the other cases show broadening of the spectrum. The ε = 0.1 m^{2} s^{−3} cases always broaden, but the maximum radius is smaller than without preferential concentration. In this case the high supersaturations attained result in the activation of a relatively large number of droplets compared to the classic case. This increased competition effectively slows down the condensational growth of all of the droplets. The ε = 0.01 m^{2} s^{−3} cases show not only significant broadening on the small end, but also produce droplets larger than would be predicted without preferential concentration. We note that droplets are only grown to roughly 15 *μ*m in diameter because gravitational sedimentation is not considered in this model.

## 5. Discussion

In the previous sections we have presented a conceptual model to investigate the effects of small-scale turbulence on the condensational growth of populations of cloud droplets. It should be stressed that we are not attempting to develop a comprehensive model that includes the many complexities of a developing cumulus cloud. Rather, it is intended to illustrate the potential effects that preferential concentration has on the local supersaturation field and ultimately on the droplet size spectrum.

### a. Comparison with two existing theories

The theory presented is similar in spirit to two other theories that consider the effects of turbulence-induced fluctuations in the temporal and spatial distributions of cloud droplets on small scales (Cooper 1989; Srivastava 1989). In order to illustrate some important aspects of our model, we will compare it briefly with the models of Srivastava and Cooper.

Among other factors, Srivastava (1989) considers fluctuations in the local supersaturation field about individual droplets due to randomness in droplet spacing. The model presented there is based on the idea that the value for supersaturation used in the droplet growth equation [our Eq. (17)] is a *local,* or microscopic, value, while the value used in the supersaturation time evolution equation [our Eq. (14)] is an *average,* or macroscopic, value. In our model we do not consider the local diffusion and conduction fields surrounding individual droplets, but we do attempt to account for the fact that the value of supersaturation used in Eqs. (14) and (17) varies on the smallest spatial turbulence scales in a cloud. This is done by treating the supersaturation evolution in the coherent vortices separately from the supersaturation evolution in the surrounding low-vorticity fluid. In addition, by taking into account the physics of cloud droplet motion in turbulence we show that the spatial fluctuations in cloud droplet concentration are much stronger than would be expected from a random spatial distribution. These two facts combined yield a strongly fluctuating supersaturation field and, as a result, significant broadening of droplet size spectra.

The theory of Cooper (1989) “breaks the otherwise close link between supersaturation and vertical velocity.” It does so by accounting for the fluctuations in supersaturation that are created by fluctuations in the integral radius and are not necessarily related to fluctuations in vertical velocity. Our model describes a physically based mechanism for producing very strong spatial and temporal fluctuations in the integral radius on the smallest turbulent scales in a cloud. The fluctuations can be strong enough, in fact, that Cooper’s approximation that *I*′ ≪ *I,* where *I* is the mean integral radius and *I*′ is the fluctuation about the mean, will be violated. As a result, we have calculated explicitly the supersaturation as a function of time rather than using the quasi-steady-state methodology of Cooper.

An important element of Cooper’s theory is the calculation of droplet spectrum dispersion. Figure 7 shows the droplet spectrum dispersion, plotted as a function of time, for three of the cases discussed in section 4. Here, the dispersion is defined as the standard deviation of droplet radius divided by the mean droplet radius (*σ*/*a*_{ave}). The droplet spectrum dispersion for the classic (nonpreferential concentrating) theory drops rapidly to much less than 0.1. The two remaining curves are for *τ*_{s} = 10 s and ε = 0.01 and 0.1 m^{2} s^{−3}; they show the large effect of preferential concentration on droplet spectral broadening. Typical cloud drop dispersions in the atmosphere can vary from 0.1 to 0.3 (Jensen et al. 1985; Politovich 1993), and, for comparison, Fig. 7 shows dispersions of roughly 0.15–0.2 at 200 m above cloud base.

One factor that Cooper considers is the correlation of vertical velocity fluctuations and integral radius fluctuations. We have not accounted for any fluctuations in the term *αw* [Eq. (14)] in our model, and it should be noted therefore that the broadening observed in the spectra shown in Fig. 6 is independent of any large-scale vertical velocity fluctuations. The effect of the latter combined with droplet preferential concentration will likely be significant, as expected from the results of Cooper (1989) and Politovich (1993), but has not been considered here.

### b. Coherent structure timescale

The essence of this model lies in the relative magnitudes of the various timescales for the relevant physical processes. In this respect, the model is similar to the inhomogeneous mixing model of Baker et al. (1980). For example, inhomogeneous mixing is effective on scales at which the time for turbulent and diffusive mixing of two parcels (one of cloud air and one of entrained air) is large compared to the time for droplet response (evaporation and/or growth). An analogous situation exists in the case of preferential concentration. If it is to influence droplet spectra, droplets must be preferentially concentrated for times on the order of or larger than the time necessary for local supersaturations to adjust and for significant droplet growth to take place.

Perhaps the key variable in our conceptual model of droplet growth in turbulence, and the one that is least understood, is the size and lifetime of the coherent structures responsible for the centrifuging of droplets into regions of low vorticity. Based on relatively low Reynolds number DNS studies, we have scaled the coherent structures to 10*η,* where *η* is the Kolmogorov microscale. Unfortunately, the extrapolation of this scaling to the relevant atmospheric context covers several orders of magnitude in Reynolds number. To compound the problem, we are aware of little atmospheric data on the small-scale nature of turbulence and coherent structures in clouds. Some intriguing observations are those by Baker (1992) and Brenguier (1993) of centimeter-scale inhomogeneities in droplet concentration. Indeed, the centimeter scale is roughly 10*η* for typical values of the turbulent kinetic energy dissipation rate (ε) in clouds.

*D*is the diffusion coefficient of water vapor in air (roughly 2 × 10

^{−5}m

^{2}s

^{−1}) and

*λ*is the diameter of a vortex tube. For coherent structures of sizes 1–5 cm the diffusion timescale varies from several seconds to a minute in magnitude. If the preferential concentration (i.e., fluctuations in integral radius) is occurring on spatial scales less than approximately 1 cm, the resulting fluctuations in supersaturation will be strongly damped by molecular diffusion since the timescale for such fluctuations is typically several seconds. On the other hand, for coherent structures greater than ∼1 cm in diameter, diffusion effects are much smaller than the driving supersaturation fluctuations and may be ignored to first order, as in the model results presented in this paper.

^{2}s

^{−3}case, structure timescales greater than about 10 s result in highly asymmetric (in amplitude) oscillations in supersaturation. This is because the physical processes governing the increase and decrease in supersaturation are distinct. While a fluid parcel experiences high vorticity (i.e., it is part of a coherent structure) cloud droplets are flung out at a rate dependent upon the strength of the turbulence (ε) and the cloud droplet size. For typical levels of turbulence and drop sizes, nearly all droplets have been spun out of the coherent vortex in a time less than the time over which the coherent structure exists. At this point, the supersaturation increases in the parcel at approximately the rate

*αw*[see (14)] because the only sink for the available water vapor is activation. Therefore, the supersaturation increases nearly linearly until the fluid parcel ceases to exist as a coherent, high-vorticity structure. Then, as a low-vorticity entity, it receives cloud droplets from adjacent regions of high vorticity and the supersaturation is rapidly depleted until a lower limit is reached. That lower limit on supersaturation is the quasi-steady-state supersaturation defined by

From the above arguments it is apparent that, while the supersaturations may reach values significantly higher than that predicted by classic theory, few droplets will benefit from this high vapor availability. However, DNS results indicate that our model is oversimplified. In Fig. 1, several individual particles can be observed in otherwise largely vacated regions. This implies that certain individual droplets may indeed benefit from the large supersaturations in these regions; hence, our model may underestimate the growth of the large droplet tail of the spectrum.

### c. Further considerations

In addition to the droplet spectral broadening by enhanced droplet growth, our model also suggests that significant activation is possible well above cloud base. It is indeed possible for the supersaturation in the high vorticity parcel of air to grow larger than the maximum observed in the classic theory (see Fig. 5). When this occurs, new droplets can be activated, broadening the spectrum toward smaller size. Therefore, it is conceivable that the effects of preferential concentration of cloud droplets by turbulence could account for the enhanced small droplet concentrations often observed in cloud droplet spectra (bimodal spectra). It should be stressed that these results are only valid in the *adiabatic core* of the updraft: no mixing was introduced.

Our results further indicate that there seem to be optimum turbulent conditions for maximum droplet spectral broadening. This is primarily because the Stokes number of a droplet of a given size is directly related to the local turbulent kinetic energy dissipation rate ε [Eqs. (1) and (3)]. Therefore, as ε decreases, the droplet Stokes number decreases until the droplet is no longer within the optimal range for preferential concentration (Fig. 3). As the dissipation rate increases, droplets preferentially concentrate more easily because the Stokes number of each droplet increases. As the degree of preferential concentration increases, however, higher supersaturations are attained in the high-vorticity structures, thereby leading to secondary activation of droplets. For large ε the average concentration of droplets can increase enough to limit the growth of all droplets due to increased competition for water vapor. While the spectrum still broadens, the mean radius then becomes significantly smaller (Fig. 6).

## 6. Summary and conclusions

Our simple parcel model indicates that the preferential concentration of cloud droplets due to turbulence helps to account for the existence of broad droplet spectra in clouds. Specifically, it provides a mechanism for the activation of new cloud droplets above cloud base and the existence of larger droplets than those predicted by pure condensation-growth theory. Further research is needed to learn how significant this effect is in real clouds and whether it alone, or in conjunction with other droplet spectral broadening theories, can explain observed spectra. Especially intriguing is the potential explanation of broad droplet spectra observed in unmixed, adiabatic updraft cores.

The key ideas proposed in this paper may be summarized as follows: (i) Cloud droplets are not randomly dispersed in a turbulent cloud. Rather, they are preferentially concentrated into regions of low vorticity, leaving regions of high vorticity relatively free of droplets. The phenomenon of preferential concentration occurs on spatial scales of centimeters for typical turbulence levels for cumulus clouds. (ii) Turbulence-induced fluctuations in cloud droplet concentration induce large fluctuations in the supersaturation field. As a result, individual cloud droplets grow by vapor condensation at different rates depending on their location in the turbulent flow field. (iii) New cloud droplets can be activated in regions of strong turbulence. This can lead to significant activation taking place well above cloud base and in adiabatic regions of the cloud. (iv) The effects of preferential concentration on cloud droplet growth by condensation depend strongly on the temporal and spatial scales of coherent vorticity structures in the turbulent flow field.

To better understand the extent to which cloud droplets are preferentially concentrated in real clouds and the effect this has on cloud microstructure will require further research. As a first step, we are in the process of modifying a two-phase DNS code to include a scalar supersaturation field and the physics of cloud droplet activation and growth. Although the DNS will be limited to low Reynolds numbers compared to those in clouds, it will provide further insight into what physical processes are the most significant in modifying cloud droplet spectra. For example, it will allow us to investigate the relative volumes of high and low vorticity regions and the associated particle residence times in the volumes. Besides more research using numerical simulations, further observations of cloud microstructure are needed. An improved understanding of small-scale turbulence in clouds is crucial to understanding the significance of preferential concentration in cloud physics. In particular, there is a need for further measurements of turbulence parameters in adiabatic core regions of clouds. Finally, it is also possible to observe effects of turbulence-induced preferential concentration in the laboratory. It is likely that under the controlled conditions of laboratory experiments, much progress can be made in determining the Reynolds number dependence of preferential concentration.

## Acknowledgments

We are grateful to Dr. John C. Wyngaard for his helpful discussions and to Dr. Jean-Louis Brenguier and the anonymous reviewers for their comments and suggestions. RAS was supported by a NASA Global Change Research Fellowship.

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Schematic view of particle response to two velocity fields characteristic of turbulent flow. Particle inertia causes particles to be flung out of regions of high vorticity and to accumulate in regions of high strain.

Citation: Journal of the Atmospheric Sciences 55, 11; 10.1175/1520-0469(1998)055<1965:PCOCDB>2.0.CO;2

Schematic view of particle response to two velocity fields characteristic of turbulent flow. Particle inertia causes particles to be flung out of regions of high vorticity and to accumulate in regions of high strain.

Citation: Journal of the Atmospheric Sciences 55, 11; 10.1175/1520-0469(1998)055<1965:PCOCDB>2.0.CO;2

Schematic view of particle response to two velocity fields characteristic of turbulent flow. Particle inertia causes particles to be flung out of regions of high vorticity and to accumulate in regions of high strain.

Citation: Journal of the Atmospheric Sciences 55, 11; 10.1175/1520-0469(1998)055<1965:PCOCDB>2.0.CO;2

Particle response to a Rankine vortex. (a) Ejection time (nondimensional) as a function of Stokes number for initial particle locations of one-tenth and one-half of the vortex radius. (b) Percentage of particles remaining in the vortex as a function of time (nondimensional) for particle Stokes numbers 0.1, 1, and 10, assuming random initial particle locations. Both (a) and (b) show that preferential concentration is most pronounced for particles of St ∼ 1.

Particle response to a Rankine vortex. (a) Ejection time (nondimensional) as a function of Stokes number for initial particle locations of one-tenth and one-half of the vortex radius. (b) Percentage of particles remaining in the vortex as a function of time (nondimensional) for particle Stokes numbers 0.1, 1, and 10, assuming random initial particle locations. Both (a) and (b) show that preferential concentration is most pronounced for particles of St ∼ 1.

Particle response to a Rankine vortex. (a) Ejection time (nondimensional) as a function of Stokes number for initial particle locations of one-tenth and one-half of the vortex radius. (b) Percentage of particles remaining in the vortex as a function of time (nondimensional) for particle Stokes numbers 0.1, 1, and 10, assuming random initial particle locations. Both (a) and (b) show that preferential concentration is most pronounced for particles of St ∼ 1.

Time evolution of the number of droplets in each of the two fluid “zones,” as well as the total number of droplets for ε = 0.01 m^{2} s^{−3}. Also shown is the total number of droplets for ε = 0 (classic theory). The total number of droplets increases during the first ∼10 s due to cloud-base activation. The activation of new droplets beginning at ∼80 s results from the formation of high supersaturations as a result of preferential concentration.

Time evolution of the number of droplets in each of the two fluid “zones,” as well as the total number of droplets for ε = 0.01 m^{2} s^{−3}. Also shown is the total number of droplets for ε = 0 (classic theory). The total number of droplets increases during the first ∼10 s due to cloud-base activation. The activation of new droplets beginning at ∼80 s results from the formation of high supersaturations as a result of preferential concentration.

Time evolution of the number of droplets in each of the two fluid “zones,” as well as the total number of droplets for ε = 0.01 m^{2} s^{−3}. Also shown is the total number of droplets for ε = 0 (classic theory). The total number of droplets increases during the first ∼10 s due to cloud-base activation. The activation of new droplets beginning at ∼80 s results from the formation of high supersaturations as a result of preferential concentration.

The supersaturation time evolution of a single fluid “zone” for 150 s (to 300 m above cloud base). Results are shown for ε = 0, 0.001, 0.01, and 0.1 m^{2} s^{−3} and for *τ*_{s} = 5, 10, and 15 s. The supersaturation fluctuations are caused by the varying droplet concentration in the parcel due to turbulence. The magnitude of the supersaturation fluctuations increases with increasing ε and *τ*_{s}.

The supersaturation time evolution of a single fluid “zone” for 150 s (to 300 m above cloud base). Results are shown for ε = 0, 0.001, 0.01, and 0.1 m^{2} s^{−3} and for *τ*_{s} = 5, 10, and 15 s. The supersaturation fluctuations are caused by the varying droplet concentration in the parcel due to turbulence. The magnitude of the supersaturation fluctuations increases with increasing ε and *τ*_{s}.

The supersaturation time evolution of a single fluid “zone” for 150 s (to 300 m above cloud base). Results are shown for ε = 0, 0.001, 0.01, and 0.1 m^{2} s^{−3} and for *τ*_{s} = 5, 10, and 15 s. The supersaturation fluctuations are caused by the varying droplet concentration in the parcel due to turbulence. The magnitude of the supersaturation fluctuations increases with increasing ε and *τ*_{s}.

Cloud droplet spectra corresponding to the conditions in Fig. 5 (after 150 s of growth). Droplet spectral broadening and large droplet production are optimal for *ε* = 0.01 m^{2} s^{−3} and *τ*_{s} = 10 s. Secondary activation becomes more pronounced as *ε* and *τ*_{s} are increased.

Cloud droplet spectra corresponding to the conditions in Fig. 5 (after 150 s of growth). Droplet spectral broadening and large droplet production are optimal for *ε* = 0.01 m^{2} s^{−3} and *τ*_{s} = 10 s. Secondary activation becomes more pronounced as *ε* and *τ*_{s} are increased.

Cloud droplet spectra corresponding to the conditions in Fig. 5 (after 150 s of growth). Droplet spectral broadening and large droplet production are optimal for *ε* = 0.01 m^{2} s^{−3} and *τ*_{s} = 10 s. Secondary activation becomes more pronounced as *ε* and *τ*_{s} are increased.

The droplet spectrum dispersion (*σ*/*a*_{ave}) as a function of time for *τ*_{s} = 10 s and *ε* = 0, 0.01, and 0.1 m^{2} s^{−3}. Preferential concentration causes significant spectral broadening, largely due to the secondary activation of cloud droplets.

The droplet spectrum dispersion (*σ*/*a*_{ave}) as a function of time for *τ*_{s} = 10 s and *ε* = 0, 0.01, and 0.1 m^{2} s^{−3}. Preferential concentration causes significant spectral broadening, largely due to the secondary activation of cloud droplets.

The droplet spectrum dispersion (*σ*/*a*_{ave}) as a function of time for *τ*_{s} = 10 s and *ε* = 0, 0.01, and 0.1 m^{2} s^{−3}. Preferential concentration causes significant spectral broadening, largely due to the secondary activation of cloud droplets.