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A warm rain parameterization has been developed by solving the stochastic collection equation with the use of turbulent collision kernels. The resulting parameterizations for the processes of autoconversion, accretion, and self-collection are functions of the turbulent intensity of the flow and are applicable to turbulent cloud conditions ranging in dissipation rates of turbulent kinetic energy from 100 to 1500 [cm.sup.2] [s.sup-3]. Turbulence has a significant effect on the acceleration of the drop size distribution and can reduce the time to the formation of raindrops. When the stochastic collection equation is solved with the gravitational collision kernel for an initial distribution with a liquid water content of 1 g [m.sup.-3] and 240 drops [cm.sup.-3] with a mean volume radius of 10 [micro]m, the amount of mass that is transferred to drop sizes greater than 40 [micro]m in radius after 20 min is 0.9% of the total mass. When the stochastic collection equation is solved with a turbulent collision kernel for collector drops in the range of 10-30 [micro]m with a dissipation rate of turbulent kinetic energy equal to 100 [cm.sup.2][s.sup.-3], this percentage increases to 21.4. Increasing the dissipation rate of turbulent kinetic energy to 500, 1000, and 1500 [cm.sup.2][s.sup.-3] further increases the percentage of mass transferred to radii greater than 40 [micro]m after 20 min to 41%, 52%, and 58%, respectively, showing a substantial acceleration of the drop size distribution when a turbulent collision kernel that includes both turbulent and gravitational forcing replaces the purely gravitational kernel. The warm rain microphysics parameterization has been developed from direct numerical simulation (DNS) results that are characterized by Reynolds numbers that are orders of magnitude smaller than those of atmospheric turbulence. The uncertainty involved with the extrapolation of the results to high Reynolds numbers, the use of gravitational collision efficiencies, and the range of the droplets for which the effect of turbulence has been included should all be considered when interpreting results based on these new microphysics parameterizations.

Collision statistics of cloud droplets in turbulent flow have been calculated for 12 droplet size combinations in four flow fields with levels of the eddy dissipation rate of turbulent kinetic energy ranging from 95 to 1535 [cm.sup.2] [s.sup.-3]. The flow fields were generated by using a direct numerical simulation technique and large numbers of droplets were explicitly tracked through the flow field for each experiment. The effect of turbulence on the collision kernel increases with both increasing radius ratio and eddy dissipation rate. These increases range from fairly modest values to almost 10 times the gravitational geometric collision Kernel. The two physical processes responsible for these increases are the radial relative velocities and the preferential concentration or clustering of the droplets. The radial relative velocities increased by up to 3 times the corresponding gravitational value and the greatest increase in the clustering, as measured by the radial distribution function, is 4.5 times the value for a random distribution as for the gravitational case. Parameterizations have been developed for the effect of turbulence on the radial relative velocities and the clustering of the droplets. These models reduce the average root-mean-squared errors in the existing velocity parameterization of Saffman and Turner and Wang et al. by 32% and the clustering parameterization of Zhou et al. by up to 58%.

Prediction of the effect of air turbulence on statistics relevant to a collision-coalescence process represents a key challenge in the modeling of cloud microphysics. In this paper, collision-related statistics for gravity-driven motion of droplets are considered and various probability distributions associated with geometric configuration and relative motion of colliding droplets are theoretically derived. The theoretical results agree well with numerical results obtained from direct numerical simulations (DNSs). In the absence of air turbulence, the probability distributions, calculated at the beginning of the time steps used for collision detection, nontrivially depend on the time step size. Next, a novel theory is developed to quantify the effect of turbulence on the angle-of-approach [theta] and radial relative velocity [absolute value of [w.sub.r,c]] for colliding pairs. A logical decomposition is used to construct extended collision volumes for a specific level of radial motion caused by air turbulence. It is shown that the inward relative motion due to turbulent fluctuations dominates the effect of turbulence in modifying the probability distributions of [theta] and [absolute value of [w.sub.r,c]]. Two key dimensionless parameters are identified in the theory: one measures the effect of finite time step size in numerical collision detection and the second measures the relative magnitude of air turbulence. The theory is compared with 11 numerical experiments from DNS. It is shown that the theory captures the essential physics of the effect of air turbulence and provides a quantitatively good representation of the statistics for 0. For most numerical experiments, the theory predicts ([theta]) to within 5%. The probability distribution of [absolute value of [w.sub.r,c]] is more sensitive to the influence of air turbulence and shows larger intermittency at large [absolute value of [w.sub.r,c]] than what is assumed in the theory. The theoretical framework developed here may be of value to other problems involving gravitational settling and weak turbulence, such as parameterization of collision kernel and hydrodynamic interactions of droplets in warm rain processes.

Direct numerical simulations of an evolving turbulent flow field have been performed to explore how turbulence affects the motion and collisions of cloud droplets. Large numbers of droplets are tracked through the flow field and their positions, velocities, and collision rates have been found to depend on the eddy dissipation rate of turbulent kinetic energy. The radial distribution function, which is a measure of the preferential concentration of droplets, increases with eddy dissipation rate. When droplets are clustered there is an increased probability of finding two droplets closely separated; thus. there is an increase in the collision kernel. For the flow fields explored in this study, the clustering effect accounts for an increase in the collision kernel of 8%-42%, as compared to the gravitational collision kernel. The spherical collision kernel is also a function of the radial relative velocities among droplets and these velocities increase from 1.008 to 1.488 times the corresponding gravitational value. For an eddy dissipation rate of about 100 [cm.sup.2] [s.sup.-3], the turbulent collision kernel is 1.06 times the magnitude of the gravitational value, while for an eddy dissipation rate of 1500 [cm.sup.2][s.sup.-3], this increases to 2.08 times. Therefore, these results demonstrate that turbulence could play an important role in the broadening and evolution of the droplet size distribution and the onset of precipitation.