A Comparison of Continuous and Stochastic Methods for Modeling Rain Drop Growth in Clouds

Two models for raindrop growth in clouds are developed and compared. A continuous accretion model is solved numerically for drop growth from 20-50 microns, using a polynomial approximation to the collection kernel, and is shown to underestimate growth rates. A Monte Carlo simulation for stochastic growth is also implemented to demonstrate discrete drop growth. The approach models the effect of decreased average time between captures as the drop size increases. It is found that the stochastic model yields a more realistic growth rate, especially for larger drop sizes. It is concluded that the stochastic model showed faster droplet accumulation and hence shorter times for drop growth.


Introduction
In a cloud, the development of a size distribution of rain drops with radius R, as they collect droplets of radius , is described by a nonlinear differential equation relating the mean number concentration of droplets N( ) to the rate at which drops and droplets collide and coalesce. The effect of mixing between upwards and downwards moving entities is to reduce the concentration of droplets in the ascending air. The super supersaturated created in the updraft is then distributed over fewer drops, permitting them to grow to larger sizes.
Rain drop collision does not guarantee coalescence. When a pair of drops collides they may subsequently (i) bounce apart, (ii) coalesce and remain so, (iii) coalesce temporarily but then break apart, retaining their initial identities (iv) coalescence temporarily but then break apart to a number of smaller drops. For sizes smaller than 100 microns in radius, the important interactions are (i) and (ii), described by Barnet (2011) and Rogers et al. (1989).
In stochastic raindrop growth, coalescence can broaden the droplet spectrum, but is hindered in the early growth stages by the fact the collection efficiencies between small droplets are extremely small. Coalescence is not sufficient to account for rain development over short periods as shown by an earlier study Robertson (1973). It is now recognized that statistical effects are crucial in the early stages of coalescence.
Consequently a stochastic coalescence model provides a convenient means to describe this process .
According to Rogers et al. (1989), as droplets grow, their collection efficiencies increase, increasing the probability of coalescence. Once it begins coalescence proceeds rapidly, as indicated by the fast decline in the number of drops. At the same time, super saturation increases sharply because the drops, now fewer in number, are no longer able to consume the excess vapor at the rate it is created.
In general the continuous and stochastic growth of rain drop are classify by the relative amount of water collected from the different sizes of small droplets to large droplets, which is mainly depends upon the mass and size of the droplets. Droplets growing according to the continuous model collect most of their water by capture of droplets while droplets growing by stochastic model collect water from droplets of all the small sizes. According to Berry (1967), the average rate of mass and size increase of n th droplet due to the capture of r th droplets is equal to the product of the collection kernel (volume swept out per unit time and the mass density function (mass per unit volume per unit size of interval).
The effects of turbulence in a cloud can be modeled by a probabilistic collection kernel where the magnitude of the collection kernel indicates, the importance of turbulence (Berry 1967).
In this work we developed and compared two models for raindrop growth in clouds based on continuous accretion and stochastic technique by using numerical solution and Monte Carlo simulation. It is found that the stochastic model yields a more realistic growth rate, especially for larger drop sizes.
We applied MAPLE 13 for supporting numerical techniques and programming features.

Theory
Consider a collector (larger) drop of radius R that is falling relative to a field of smaller droplets of radius r. The rate at which the collector collides with the smaller droplets is proportional to the shared collision volume, ( , ) c V R r , which is given by the cross-sectional areas of both the drop and the droplet and their vertical velocities ( ), u(r). Derivation and discussion of Equations can be found in Long (1973Long ( , 1974 and Robertson (1974).
The probability that a collision between a drop and a droplet results in an actual capture (coalescence) is described by the collection efficiency E(R,r). Given that the mean number of droplets within the collision volume is ( , ) ( ) is the mean number concentration of droplets, the probability per unit time that a drop captures a droplet is The realistic growth of a collector drop is discrete, where capture of each droplet increases the mass of the drop M( ) by the finite droplet mass m( ). The collector drop also grows stochastically, where each capture has a probability between 0 and 1. The mean growth rate of the collector drop is described by As a first approximation, we can consider the simplest type of model for collection growth, the continuous model, as Here we have two factors [2]: the droplet collection kernel , and the liquid water content of the droplets, A method for deriving an analytical solution for the droplet collection equation, using a polynomial approximation to the kernel, 2 ( , ) Here c is a scaling factor and ≡ (R) is the collector drop volume. Then the collection equation becomes, Here, v(r) is the droplet volume. An analytical solution for V(t) is found by integrating the above equation, to give where is the initial collector drop volume and = 1.1 x 10 10 cm -3 s -1 is the constant related to the polynomial kernel according to Long et al. (1974).

Method
For the continuous model of collection growth, equation (8) is numerically solved using an implicit or semi-implicit integration scheme. The implicit scheme is The semi-implicit equation is A Monte Carlo simulation of stochastic drop growth is also implemented. First we calculate the time interval ∆t to perform a discrete simulation step for which the probability of capture ( , )Δ q P R r t = , where q is chosen to be a small value such as 0.1 as suggested by Long (1973). If a uniformly distributed random number x between 0 and 1 is generated and x > q, then no capture occurs during the time interval Δ / ( , ) t q P R r = . If x ≤ q, a capture is deemed to have occurred and M( ) is increased by m( ). Before the next time step, P( , ) and ∆t are recalculated using the new value of according to Robertson (1974).

Droplet Terminal Velocity
One important factor in drop formation is the droplet terminal velocity. In general when downward net gravitational force is equal to upward drag force (i.e. F G = F drag ), the droplet reaches a steady fall speed, its terminal velocity. Terminal velocities depend mainly on the size of the droplet. Fig.1. shows the droplet terminal velocity as a function of its radius, with different droplet regimes showing different behaviors by Rinehart (1990). By Rogers et al. (1989), for small droplet sizes (r ≤ 30µm), flow is completely dominated by air viscosity, and the terminal velocity increases quadratically: = k 1 r 2 with k 1 = 1.19 x 10 8 s -1 m -1 . For larger sizes (30µm ≤ r ≤ 10 3 µm), flow is turbulent, and the velocity grows linearly: u = k 3 r with k 3 = 8 x 10 3 s -1 .

Collection Efficiency
The probability that a collision between a drop of radius and a droplet results in a capture is called efficiency and is given by E (  In the Monte Carlo technique, the average time between captures gets smaller as the drop grows. As expected, after a sufficiently large number of captures i.e. at a larger drop radius R, the growth curves stabilize, and increase in parallel to the continuous growth curve also explained by Robertson (1974).