TURBULENT COLLISION-COALESCENCE OF CLOUD DROPLETS AND ITS IMPACT ON WARM RAIN INITIATION

Transcription

1 TURBULENT COLLISION-COALESCENCE OF CLOUD DROPLETS AND ITS IMPACT ON WARM RAIN INITIATION Lian-Ping Wang, 1 Bogdan Rosa, 1 and Wojciech W. Grabowski 2 1 Department of Mechanical Engineering, University of Delaware, Newark, Delaware 19716, U.S.A. 2 Mesoscale and Microscale Meteorology Division, National Center for Atmospheric Research, PO Box 3000, Boulder, Colorado , USA 1. INTRODUCTION In recent years, there is a renewed interest in studying the effects of air turbulence on the collision-coalescence growth of cloud droplets [1 6]. This is motivated by the need to resolve an open issue in cloud physics concerning the growth of cloud droplets in the size range from 10 to 50 µm in radius (i.e., the so-called size gap), for which neither the condensation nor the gravitational collision-coalescence mechanism is effective [5, 7]. Observations of radar reflectivity in tropical regions suggest that rain could form in cumulus clouds by warm rain process in approximately 15 to 20 minutes [8, 9]. Theoretical predictions based on the gravitational-coalescence mechanism alone, however, would require a time interval on the order of an hour for droplets to grow from 20 µm to 100 µm in radius (the actual time depends on the cloud water content, initial droplet size spectrum, updraft speed, etc., see Pruppacher and Klett [7]). Therefore, there appears to be a factor of 2 or more difference between the predicted growth time and the observed growth time. The onset of drizzle-size ( 100µm in radius) raindrops is still poorly understood in many precipitating cloud systems. It has long been speculated that the effect of air turbulence could play an important role in closing the size gap [10], although several other alternative explanations have also been proposed, including growth by ultragiant particles, entrainment-induced spectral broadening, and effects of preexisting clouds [7, 11]. Resolving this open issue quantitatively is crucial in view of the fact that warm rain accounts for 31% of the total rain fall and 72% of the total rain area in the tropics [12]. Here we shall focus on the effects of air turbulence on droplet growth by collision-coalescence. The central issue is the magnitude of the enhancement of the gravitational collection kernel due to the air turbulence, and whether the enhancement can significantly impact rain initiation. The collection kernel is defined as the number of collisions per unit time between droplets of two different sizes, divided by the corresponding number of pairs involved. We will show that, despite the complexity of the problem, recent quantitative studies begin to address these long-standing issues with confidence. 2. TURBULENT COLLISION-COALESCENCE Corresponding author: Department of Mechanical Engineering, 126 Spencer Laboratory, University of Delaware, Newark, Delaware Phone: (302) ; Fax: (302) ; Electronic mail: LWANG@ME.UDEL.EDU During the last 10 years, studies have emerged in both engineering and atmospheric literature concerning the collision rate of particles in turbulent flow. These studies suggest that the collection kernel of cloud droplets can be enhanced by several effects of air turbulence: (1) the enhanced relative motion due to differential acceleration and shear effects [1, 13 16, 34]; (2) the enhanced average pair density due to local clustering ( preferential concentration ) of droplets [14 17]; (3) the enhanced settling rate [18 20], and (4) the enhanced collision efficiency [21, 22]. The enhancement depends, in a complex manner, on the size of droplets (which in turn determines the response time and terminal velocity) and the strength of air turbulence (i.e., the dissipation rate, flow Reynolds number, etc.). For cloud droplets, the two key physical parameters are the droplet inertial response time τ p and the still-fluid droplet terminal velocity v T. For the background air turbulence, the dissipation-range motions are characterized by the Kolmogorov time τ k and the Kolmogorov velocity v k. The Stokes number St, the ratio τ p /τ k, emphasized in early studies of particle-laden flows, is not the only parameter governing the interaction of droplets with air turbulence. The nondimensional settling velocity, S v v T /v k, is the second key parameter, typically one order of magnitude larger than St [23, 24]. This implies that the gravitational sedimentation determines the interaction time between the cloud droplet and the small-scale flow structures. Most of the published results on droplet clustering and collision rate from numerical simulations and theoretical studies assume no sedimentation and, as such, are not directly applicable to cloud droplets. For the case of strong sedimentation, a new governing parameter F p τ p v 2 T /Γ (where Γ is the circulation of vortical structures) has been found to better characterize the interaction of droplets with small-scale vortical structures and its effect on the mean settling rate [19, 25]. Recent systematic studies of the collection kernel for cloud droplets have been undertaken through either direct numerical simulation [22, 25 28] or a kinematic/stochastic representation of turbulence [21, 29, 30]. These studies provide not only quantitative data on the turbulent collision kernel, but also a better understanding of interaction between cloud droplets and air turbulence. An important observation emerging from these studies is that the enhancement factor, defined as the ratio of turbulent collision rate to the gravitational collision rate in still-air, is typically between one and five. Since the level of enhancement is moderate and the differences between various approaches are often comparable to the enhancement itself, quantification becomes extremely important.

2 FIG. 1: Visualizations from the hybrid simulation approach [31]. The droplets are much smaller than the dissipation-range eddies of the background air turbulence so the disturbance flows due to droplets are represented analytically by Stokes flow solutions. The background air turbulence is solved numerically by integrating the Navier-Stokes equations. The left panel shows vortical structures of the background air turbulence; the center panel shows relative trajectories at the scale of droplets in the turbulent flow; and the right panel is a snapshot of flow vorticity surfaces and the locations of droplets in a subdomain of the computational region. 3. HYBRID DNS AND TURBULENT ENHANCEMENT Motivated by the issues explained above, we have developed a consistent and rigorous simulation approach to the problem of turbulent collisions of cloud droplets [22, 31]. The basic idea of the approach is to combine direct numerical simulation (DNS) of the background air turbulence with an analytical representation of the disturbance flow introduced by droplets (Fig. 1). The approach takes advantage of the fact that the disturbance flow due to droplets is localized in space and there is a sufficient length-scale separation between the droplet size and the Kolmogorov scale of the background turbulent flow. This hybrid approach provides, for the first time, a quantitative tool for studying the combined effects of air turbulence and aerodynamic interactions on the motion and collisional interactions of cloud droplets. The disturbance flow is coupled with the background air turbulence through the approximate implementation of the no-slip boundary conditions on each droplet. Dynamical features in three dimensions and on spatial scales ranging from a few tens of centimeters down to 10 µm are captured. Both the near-field and the farfield droplet-droplet aerodynamic interactions could be incorporated [32], with possible systematic improvements of their accuracy. The most important aspect of the approach is that dynamic collision events are detected, along with the direct and consistent calculations of all kinematic pair statistics related to the collision rate [15, 16, 22]. These unique capabilities allow application of the following general kinematic formulation of the collection kernel K 12 [14, 15, 22] K 12 = 2πR 2 w r (r = R) g 12 (r = R), (1) where the geometric collision radius R is defined as R = a 1 + a 2, with a 1 and a 2 being the radii of the two colliding droplets, w r is the radial relative velocity at contact which combines the differential sedimentation and turbulent transport, and g 12 is the radial distribution function (RDF) that quantifies the effect of droplet-pair clustering on the collision rate. This formulation is applicable to aerodynamically interacting droplets in a turbulent background flow, showing that the total enhancement factor η T of the collision rate is a product of the enhancement of the geometric collision, η G, and the enhancement of the collision efficiency, η E, [22]. We found that the net enhancement factor η T = η E η G is typically in the range of one to five [28]. It tends to be larger when the droplets are rather different in size or of nearly equal in size (Fig. 2). In the first limiting case, the gravitational hydrodynamic kernel is small due to small differential sedimentation. In the second limiting case, the gravitational hydrodynamic kernel may also be small due to small collision efficiency. Therefore, air turbulence plays an important role in enhancing the collision kernel in both limiting cases. The values of η E and η G from our HDNS have been tabulated in [25, 28]. 4. IMPACT ON RAIN INITIATION We next consider the question of how the above turbulent enhancements on the collection kernel alter the size evolution of cloud droplets. An analytical model has been developed for the geometric collision rate of cloud droplets based on the results from the hybrid simulation approach [33]. The model consists of a parameterization of the radial relative velocity w r similar to [34] for sedimenting particles, and a parameterization of the RDF g 12 following the work reported in [35]. Of significance is the fact that the above parameterizations for both w r and g 12 consider the effects of flow Reynolds number which cannot be fully represented by the hybrid simulations. For example, the parameterization for w r makes use of velocity correlations that are valid for both the dissipation subrange and the energy-containing subrange of turbulence [17]. The intermittency of small-scale turbulent fluctuations can be incorporated into the model for RDF [35]. Therefore, our parameterization of the collection kernel, while

3 η T a 2 /a 1 FIG. 2: The net enhancement factor, the ratio of the turbulent collection kernel and the hydrodynamic-gravitational collection kernel, as a function of a 2 /a 1 for a 1 = 30 µm. The largest enhancements occur at small size ratio or for nearly equal-size pairs, in qualitative (but not quantitative) agreement with the results in [21]. In the legend, ε is the flow viscous dissipation rate and R λ is the Taylor microscale Reynolds number of the simulated background turbulent air flow. The overall enhancements reported in [29, 30] appear to be quantitatively similar to the results shown here. guided by the results from the hybrid simulations, extends the capabilities of the simulations concerning the flow Reynolds number effect. Moreover, we include the effect of η E by interpolating the tabulated simulation results of η E from [28]. The ratio (η T ) of the resulting turbulent collection kernel to the Hall kernel [36] is shown in Fig. 3 for a typical condition of cloud turbulence. The Hall kernel [36], a hydrodynamical gravitational kernel independent of air turbulence, is used as a base to compare the relative impact of turbulence. Several important inferences can be made from Fig. 3. First, a noticeable enhancement occurs for droplets less than 100 µm. Second, the overall enhancement is moderate with a value ranging from 1.0 to 4.0 for most regions or an average value of about 2 for droplets in the bottleneck range. The magnitudes are one to two orders of magnitude smaller than assumed in [3] for reasons given herein and also in [24]. An important fact is that the enhancement factors shown in Fig. 3 are similar to those reported recently in [29, 30], where dramatically different approaches were employed. Third, the enhancement is more uniform for droplets less than 60 µm than other unrealistic turbulent kernels [3]. In general, it is the average enhancement over the bottleneck range that determines the overall impact of turbulence on the warm rain initiation [6]. The above turbulent collection kernel is then used in the kinetic collection equation to solve for droplet size distributions FIG. 3: The ratio of a typical turbulent collection kernel to the Hall kernel. The Hall kernel [36] is a hydrodynamical gravitational kernel in still air. The ratio on the 45 o degree line is undefined due to the zero value of the Hall kernel. The ratio is essentially one when droplets are above 100 µm. The flow dissipation rate is 400 cm 2 /s 3 and rms velocity is 202 cm/s. at different times, starting from the initial number density distribution n(x,t = 0) = L ) 0 X0 2 exp ( xx0, (2) where x is the droplet mass, L 0 is the liquid water content and is set to 1 g/m 3, and X 0 is the initial average mass of the cloud droplets and is assumed to be g which corresponds to a mean radius of 9.3 µm. The kinetic collection equation is solved by an accurate method [37] which combines the advantages of flux-based methods [38] and spectral moment-based methods [39]. A small bin mass ratio of ensures that the numerical solutions are free from numerical diffusion and dispersion errors. The mass distribution g(lnr,t) 3x 2 n(x,t) is usually plotted at different times in order to examine growth processes [40]. In Fig. 4, we instead plot the local rate of change, g/ t, as a function of radius for times from 0 to 60 min every 1 min. We compare the results using the turbulent kernel to those using the Hall kernel. The plots naturally reveal the three growth phases first described qualitatively in [40]: (1) the autoconversion phase in which self-collections of small cloud droplets near the peak of the initial size distribution slowly shift the initial peak of the distribution toward larger sizes; (2) the accretion phase in which the accretion mode dominates over the autoconversion mode and serves to quickly transfer mass from the initial peak to the newly formed secondary peak at drizzle sizes; and (3) the large hydrometeor self-collection phase in which the

4 FIG. 4: The rate of change ( g/ t, g m 3 s 1 ) of droplet mass density in each numerical bin as a function of droplet radius: solutions using the Hall kernel; solutions using a turbulent kernel at flow dissipation rate of 400 cm 2 /s 3 and rms fluctuation velocity of 2.0 m/s. There are 61 curves in each plot, representing t = 0 to t = 60 min with a time increment of 1 min. The curves for t > 0 is shifted upwards by a constant in order to distinguish them. The value of g/ t can be either positive or negative, with the total integral over the whole size range equal to zero due to the mass conservation. At any given time, a positive g/ t for a given size bin implies that the mass density for that size bin is increasing. The two red lines mark the begin and the end of the accretion phase. The figure shows that air turbulence can shorten the growth time by a factor of about two. self-collections of drizzle droplets near the second peak dominate over the accretion mode; the initial peak diminishes and the second peak gains strength and shifts toward the raindrop sizes (a few millimeters). By examining the locations corresponding to the maximum and minimum g/ t, one can unambiguously identify the time intervals of the three phases [6]. In Fig. 4, we indicate in each plot the beginning and end of the accretion phase by two red horizontal lines. Figure 4 highlights striking differences between the two collection kernels. The intensity of the autoconversion is significantly increased by the turbulent effects as shown by the magnitude of g/ t at early times (Fig. 4b) when compared to the base case (Fig. 4a). The time interval for the autoconversion phase is reduced from about 32.5 min (Hall kernel) to only 10.5 min (the turbulent kernel). This demonstrates that turbulence has a strong impact on the autoconversion phase, which is typically the longest phase of warm rain initiation. The time interval for the accretion phase is also significantly reduced and smaller drizzle drops ( 100 to 300 µm) are produced during this phase. If a radar reflectivity factor of 20 dbz (or the massweighted mean droplet radius of 200 µm) is used as an indicator for the drizzle precipitation, the time needed to reach such a reflectivity (or mean droplet radius) changes from about 2450 s (or 2470 s) for the Hall kernel to 1230 s (or 1250 s) for the turbulent kernel. This two-fold reduction factor increases with either the dissipation rate or rms turbulent fluctuation velocity [6]. 5. SUMMARY AND ON-GOING WORK Studies during the last 10 years have significantly advanced our understanding of the effects of cloud turbulence on the collision-coalescence of cloud droplets. Since the moderate enhancements by air turbulence occur within the bottleneck range and since the autoconversion is the longest phase of the initiation process, there is now convincing evidence showing that turbulence plays a definite role in promoting rain formation. The level of reduction in rain initiation time by turbulence appears to be at the level needed to resolve the discrepancy between the observed time and the time predicted based on the gravitational mechanism alone.

5 FIG. 5: Dependence of pair density on the orientation angle as shown by the angle-depdnent radial distribution function. The horizontal line is the usual orientation-averaged radial distribution function. Droplets are assumed to be ghost particles. The data were from simualtions at R λ = µm 50 µm cross-size pairs, 40 µm 40 µm same-size pairs. (c) The parameterization of the radial distribution function discussed in [33] is far from satisfactory. To obtain a better understanding, we are investigating if the droplet sedimentation introduces an orientation dependence. Prelimary results are shown in Fig. 5. For cross-size geometric collision between 10 µm and 50 µm droplets, there is essentially no orientation dependence. However, for the same-size collision between 40 µm droplets, it is more likely to find pairs approaching each other in the vertical direction than in the horizontal direction. The hybrid DNS results on dropelt collision statistics we have obtained to date were based on OpenMP implemtation at grid resolution or less. The relatively low flow Reynolds number in DNS implies that only a very limited range of scales of flow motion was explicitly represented in DNS. The relative motion and pair statistics of larger cloud droplets may be affected by larger-scale fluid motion not included in our DNS flow. Currently, we are implementing MPI into our simulation code in order to embark on simulations containing a large cloud volume. This allows us to increase the DNS flow FIG. 6: Vorticity contours from flow simulations at three higher grid resolutions: and R λ = 145, and R λ = 168, (c) and R λ = 241. The contour level for all visualizations was set to twice the field-mean vorticity magnitude. Reynolds number and the number of droplets followed in the simulation. Visualizations from preliminary flow simulations up to are shown in Figure 4. These high-resolution simulations allows a direct representation of the interaction of large cloud droplets with air turbulence. Short-range scaling law of the radial distribution function is also being studied to extract the scaling exponent, which may be important for modeling the at-contact radial distribution function. For the case of same-size (i.e., monodisperse) interactions, the power-law scaling appears to remain valid for sedimenting droplets (Fig. 7). The power-law exponent for g 11 is plotted as a function of droplet radius in Fig. 8 for two different grid resolutions, with ε = 400 cm 2 /s 3. A Stokes

6 number of one corresponds to a = 40 µm [25]. The powerlaw exponent increases quickly with droplet radius and then levels off at around a = 30 µm. The similar results between the two grid resolutions implies that most of the interactions of droplets with the air turbulence are captured in our simulations, for the size range considered in Fig. 8. The initial impact study discussed above and in [6] is also being extended to a detailed microphysics parcel model where condensational growth and collision-coalescence growth are considered together. In the parcel model, the changing environment (e.g., pressure and temperature) and parcel rising velocity can also be incorporated. Acknowledgments This study has been supported by the National Science Foundation through grants ATM and ATM , and by the National Center for Atmospheric Research (NCAR). FIG. 7: The short-range power-law scaling of monodisperse radial distribution function for sedimenting droplets. FIG. 8: The power-law exponent for g 11 as a function of droplet radius. [1] M. Pinsky, A. Khain, M. Shapiro, J. Aero. Sci. 28, 1177 (1997). [2] G. Falkovich, Nature 419, 151 (2002). [3] N. Riemer, A.S. Wexler, J. Atmos. Sci. 62, 1962 (2005). [4] S. Ghosh, J. Davila, J.C.R. Hunt, A. Srdic, H.J.S. Fernando, P. Jonas, Proc. Roy. Soc. A 461, 3059 (2005). [5] L. P. Wang, Y. Xue, O. Ayala, W. W. Grabowski, Atmos. Res. 82, 416 (2006). [6] Y. Xue, L.-P. Wang, W.W. Grabowski, J. Atmos. Sci., 65, (2008). [7] H.R. Pruppacher, J.D. Klett, Microphysics of clouds and pre-

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