Initiation of rain in nonfreezing clouds

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1 Collision-coalescence Topics: Initiation of rain in nonfreezing clouds ( warm rain process) Droplet terminal fall speed Collision efficiency Growth equations Initiation of rain in nonfreezing clouds We re well on our way exploring how precipitation forms. One important issue in cloud physics is to explain the short amount of time that elapses (~20 min) between first seeing a cumulus cloud until the development of precipitation. During this time, a population of cloud droplets, say of 10 µm radius and concentration 100 cm -3, transforms to a population of rain drops with typical diameters of 1 mm and concentration of 1000 m -3. That s 50 times the original cloud drop size and a five orders of magnitude change in concentration! A process called collision-coalescence is considered to be responsible for this behavior, but it requires at least some of the droplets be about 20 µm in size. There need be only a few of the 20 µm droplets, say 1 in 10 5 droplets or 1 L -1. This is as ongoing controversial topic in cloud physics how, from a population of 10 µm cloud droplets, do we get a few larger in size necessary for the coalescence process to take root? By the time a droplet is 30 µm or so in radius, coalescence is the dominant process. A typical 1 mm raindrop (diameter) may be the result of 10 5 discrete collisions! Gravitational, electrical, or aerodynamic effects may bring about collisions. Gravitational effects are the most important in clouds, with larger drops falling faster than smaller ones, overtaking and sweeping them out. Usually electrical and turbulent effects are too weak in clouds to influence the collision-coalescence process. Aerodynamic forces are in play, however, when evaluating the flow of small droplets past a larger, falling raindrop. The number of small cloud droplets collected by a falling (growing) rain drop is evaluated by applying the concept of geometric sweep-out. The ratio of the number of actual collisions to the number for geometric sweep-out is called the collision efficiency. This efficiency is dependent on the size of the respective drops.

2 When drops collide, they may either 1. bounce apart, 2. coalesce and remain permanently united, 3. coalesce temporarily and break apart, while retaining most of their initial identities, or 4. coalesce temporarily and break apart into many small drops. The result depends on drop size and collision energy. Sizes smaller than 100 µm generally follow interactions 1 and 2. Coalescence efficiency is the ratio of the number of permanent coalescence events to the number of collisions. The collection efficiency is then the product of the collision and coalescence efficiencies. Now we ll explore three of the component pieces of collision-coalescence theory. Droplet terminal fall speed Drag force on an object is 1 F d = --ρac, 2 d v 2 where A is cross sectional area, ρ is air density, C d is a drag coefficient characterizing the flow, and v is the velocity of the object relative to the flow. For a spherical droplet, this becomes π F d = --r 2 ρc. (1) 2 d v 2 C d is dependent on the Reynolds number of the flow, Re = 2ρvr µ, where µ is the dynamic viscosity of the fluid. The Reynolds number is a nondimensional parameter that gauges the relative importance of inertial (numerator) and viscous (denominator) forces. Using the definition of the Reynolds number, Equation 1 can be written as F d = 6πrvµ ( C d Re 24). Flow around droplets up to about 30 µm is characterized by C d Re 24 1, which gives the Stokes drag F d = 6πrvµ. At terminal velocity, the drag on the falling droplet is in balance with the gravitational force, 4 6πrvµ = --πr 3 ρ, 3 l g which we may solve for velocity to obtain 2 v t -- r2 gρ l = = k, 9 µ 1 r 2 where k 1 = cm -1 s -1.

3 For larger drops, C d becomes independent of Reynolds number and has a value of ~0.45. Applying the force balance as above gives a terminal velocity of v t = k 2 r 1 2, where k [cm 1/2 s ρ = ]. ρ The definitions of k 1 and k 2 are as in Rogers and Yau. k 2 contains a dependence on air density. Raindrops of the largest possible size have a terminal velocity of about 9 m s -1 at the surface and 13 m s -1 at 500 mb. In the intermediate range between the Stokes Law region and the square root law, the fall speed is approximately linear with radius, for 40 µm < r < 0.6 mm, with k 3 = s -1. v t = k 3 r, Collision efficiency The following figure from Rogers and Yau shows illustrates the sweep-out geometry. If the small droplet had zero inertia, aerodynamic forces around the larger drop would sweep it aside and no collision would take place. If the small droplet had infinite inertia, a collision would always take place. In reality, the probability of a collision depends on the relationship between the inertial and aerodynamic forces, and the separation x between drop centers. For any given r and R, a critical value x 0 exists within which a collision is certain to occur, and outside of which the small droplet will be deflected out of the path of the large droplet. A collision efficiency is defined by E( Rr, ) x 0 2 =. ( R+ r) 2

4 Collision efficiency is thus defined to equal the fraction of droplets in the sweep-out path that actually collide with the falling collector drop. Collection efficiencies can be plotted as a function of r/r. The following figure shows collection efficiencies from three sets of theoretical calculations. E is small for small values of r/r, no matter the size of the collector drop. The collected droplets are small and have little inertia, so they are easily deflected by the aerodynamic flow around the collector drop. The inertia of small droplets increase with r/r, which explains the increase in efficiency up to ~0.6. Beyond this point, the efficiencies can actually decrease. Because the droplets are becoming similarly sized, the relative difference in fall speed tends to be small, enabling a longer interaction time between droplets. This longer time tends to enhance the possibility of deflecting the droplet around the collector drop without an actual collision.

5 The following figure contours the collection efficiency E as a function of r and R. Growth equations Consider a collector drop of radius R falling at terminal speed v(r), through a population of droplets characterized by radius r and fall speed v(r). The volume swept out is π( R+ r) 2 [ ur ( ) ur ( )]. For a spectrum of droplet sizes, the number of droplets per unit volume is nr ( )dr, so the number of droplets with radii between r and r+dr is given by π( R+ r) 2 [ ur ( ) ur ( )]nr ( )ERr (, )dr, (2) where we have included E(R,r), the collection efficiency, which is equal to the product of collision and coalescence efficiencies. We are interested in how fast the drop grows from collision-coalescence. Equation 2 represents the number of droplets in the sweepout volume, and multiplying it by volume of the individual collected droplets gives the volume of water in the collected droplets that is transferred to the collector drop. This gives the volume growth rate

6 R dv =, dt π( R+ r) 2 [ ur ( ) ur ( )]nr ( ) πr3 ERr (, ) dr 0 which can be recast in terms of radius, R π = R+ r. dt 2 [ ur ( ) ur ( )]nr ( )r 3 ERr (, ) dr 3 R 0 Assuming R» r and ur ( ) 0, enables us to write 1 R -- ur ( ) =. dt ρ l 3 --πr3 ρ l nr ( ) ERr (, ) dr 0 The expression in the brackets is the liquid water content of the droplets between r and r+dr. Assuming a bulk collection efficiency E, = --u( R), dt 4ρ l where M is the liquid water content [kg m -3 ]. This shows that the growth is proportional to the collector drop fall speed, which ultimately determines the sweep-out volume, and to the liquid water content of the collected droplets. Consider instead of growth as a function of time, growth as a function of height in a cumulus (for example) updraft. The chain rule gives, dz ---- = --u( R). dz dt 4ρ l The expression dz dt represents the velocity difference between the updraft speed and large drop fallspeed. Rearranging, -- ur ( ) = ---, dz 4ρ l U u( R) where U is the updraft speed. If the updraft is small relative to the fallspeed (not necessarily a good assumption in convection!), the relation becomes = --. dz 4ρ l