Drop growth by condensation: Quick review Drop growth by condensation is modeled as a process of diffusion of water vapor to the surface The “driving force” is the difference between the vapor pressure of water over the droplet, and the vapor pressure of the environment A gradient (profile of concentration changes with distance) exists between the drop surface and the far-field environment Sign can be positive or negative – same equation is used for evaporation / condensation The rate of mass addition is largest for larger drops, BUT the rate of change of radius is largest for smaller drops (condensation narrows a drop size distribution) What is the water vapor pressure at the drop surface? May depend on curvature and on solutes in the drop If the drop is very dilute, and large enough to approximate a plane surface, it may simply equal the water saturation vapor pressure … BUT The drop temperature is not the same as the environmental T if the drop is gaining or losing mass, because of the release / absorption of latent heat during the phase change! So we need to find out what the T of the drop is, in order to look up the saturation vapor pressure at that temperature The latent heating effect will tend to REDUCE the driving force (slows evaporation or condensation)

Review, continued The drop T rises until the T gradient between the drop and the environment is large enough to make the rate of heat conduction from the drop, equal to the rate of heat released by condensation Can see from above the equations are coupled (can’t get Ta until you know growth rate, which in turn depends on Ta) Invoke Clausius-Clapeyron: plug into drop growth equation and use approx that δ << 1 End up with single drop growth equation with Ta not appearing explicitly anywhere – can solve numerically

How does nature create a raindrop this big?? Wallace & Hobbs (1977) How does nature create a raindrop this big?? We only need about 1 per liter… Growth mechanisms: Water vapor condensation Droplet coalescence Ice processes Simple calculation shows that for constant supersaturation of 1%, it takes an hour for a drop to grow to ~100 µm  Only 0.1% of the mass of an average raindrop! “indirect effects”

Collision-Coalescence Growth Since we know that condensation growth alone cannot produce precipitation-sized drops (> 100 µm) in reasonable cloud lifetimes, we must invoke other mechanisms to explain the production of precipitation in ‘warm’ clouds. Warm clouds contain no ice and their summit temperatures are greater than 0°C. Invoke the collision-coalescence mechanism Collisions between cloud droplets can occur to produce a larger droplet by coalescence with neighboring droplets These initial collisions are motivated by the presence of different-sized cloud droplets -- this size distribution can be caused by nucleation on both small (hygroscopic) and large (wettable) CCN. (Broadening of the cloud droplet size distribution is discussed later) Once a larger cloud droplet is formed, it will collide with other droplets and thus grow rapidly. CCN spectra (giant nuclei) Broadening via turbulence Inhomogeneous mixing Three mechanisms that act to broaden the droplet size distribution.

Droplet fall speeds As a droplet falls, the streamlines around it bend. The terminal fallspeed is determined by the balance between acceleration due to gravity (which depends on droplet MASS) and the drag on the particle (which depends on an “effective” CROSS SECTIONAL AREA – size and shape, and speed)

Regimes for droplets

Over certain ranges, power laws apply: vp a Dpn

Conceptual model A bimodal spectrum produced by some means r1 r1=radius of collector (drop) r2=radius of collected droplet Collision efficiency E drops making collisions not all drops in path of collector are expected to collide with the collector E= drops in path of collector Coalescence efficiency E’ sticking collisions E’= not all drops that collide are expected to coalesce total collisions Collection efficiency

Geometry (from Lamb & Verlinde) Imagine we have a bimodal spectrum, produced by some means: So drops are falling with DIFFERENT fallspeeds rL=radius of collector (drop) rs=radius of collected droplets Volume of cylinder that is swept out, per unit time: Area of cylinder that is swept out

Some of the small drops will follow the air streamlines around the drop, as shown. This means that not all the drops located within the cylinder that is swept out, will actually hit the collecting drop. To account for the “limited success” of potential collisions, we introduce an efficiency factor, E Here yc is the limiting impact parameter SO the remaining problems are: - Defining the limiting impact parameters for pairs of drop sizes - Deciding if each collision actually “sticks” the drops together (coalescence)

Collection efficiencies near unity only for Points to note: – E is small if rs < 5 µm, independent of rL (so, for E > 0.1, rs > 5 µm – ~minimum threshold in cloud) – E increases rapidly for rs > 5 µm (don’t follow streamlines as well) – but also need rL > 20 µm for E > 0.1 – Decrease in E after maximum is due to increasing similarity in fallspeeds – when E increases (upturn) near end of curves, this is due to wake effect (lower drop distorts flow field, lets larger drop fall faster) Displayed as family of constant-rL curves Collection efficiencies near unity only for rL > 50 µm and rs > 10 µm

For fixed , E increases with a1. Larger drop has more inertia. Pruppacher & Klett (1978) For fixed , E increases with a1. Larger drop has more inertia. E is small for , small. Small droplets tend to move around collector in streamline flow. When , increases, E increases. Collected droplets tend to move more in straight line paths and do not flow around collection as readily. When , E decreases. Relative velocity between the droplets of similar size is small allowing deflective forces to inhibit collection For large a, and WAKE CAPTURE

As drop sizes approach each other, coll eff increases (but prob in atmosphere is small…) Drops with diameters < 50 µm have small collision efficiency  growth to sizes larger than this is required to accelerate accretional growth Need to multiply these numbers by coalescence efficiencies

COALESCENCE Ec = E × ε water surface water surface Grazing Collision Ec = E × ε water surface More direct collision water surface results in coalescence event Critical angle is but relative drop sizes is also an important factor.

The collision Weber number expresses the ratio of external pressure force to surface tension force. As Weber number increases, drop deforms more easily. The figure, from experimental data, shows that the ability of the large drop to deform as the small drop approaches, results in cushion of air that impedes coalescence.

If collision occurs, coalescence will result. Not all collisions will result in coalescence. For drops rebound due to cushion of air that exists between them. Air cushion most likely squeezed out for dissimilar sized drops.

Net effects Ec ~ 50% over a broad range Rapid increase as rs > 5 µm Strong decrease as rs increases Net effect: Ec maximizes at intermediate small-drop sizes Ec ~ 50% over a broad range

Note about breakup (large drops)

Continuous Collection Equation GOAL: Derive an expression for the growth rate of a droplet by collection of small droplets terminal fall speed r2 with fall speed V2 (depends on r) Assume small collected droplets are uniformly distributed in the cloud. Collector drops of the same size collect droplets at the same rate, that is, collector drops of a given size all grow at the same rate. Drops of size R have the same probability for undergoing collisions with small droplets in given time We need to consider the portion of the cloud liquid water content that is collected by the drop. The mass to be collected resides within an imaginary cylinder swept out by the collector drop.

wl Ec Liquid water content Assuming cloud drafts are small Volume swept out per unit time wl Ec collection efficiency Mass of water in volume collected This is the Continuous Collection Equation

For spherical water droplets, Since Equation (1) becomes For spherical water droplets, Therefore, Since increases with , and increases with , Coalescence growth is an accelerating process condensation collection time

Consider growth of a droplet in a modest cumulus cloud - collisional growth Drop fall speed exceeds updraft speed – drop falls back through the cloud w, updraft speed CCN activated Derive equation for drop size as a function of position in the cloud Motion of collector drop updraft speed drop fall speed (function of ) From chain rule, from continuous collection equation

drop size as a function of drop position . or Consider drop having size at [cloud base]. Let at some height in the cloud. Drop grows as it moves upward in the cloud to height . Now integrate above equation; For constant with height, On drops downward path, assume Therefore (- sign needed since increases with decreasing )

Continuous growth through a layer with liquid water content 1 EXAMPLE Continuous growth through a layer with liquid water content 1 Uniform liquid water content . Assume Then Therefore,

Calculations with this equation indicate the following, as an example, 1; 45mins, ascent height 2.2km, r=.15 2; 15mins, cloud base, r=0.7mm 2 1 Examples: Produce small drops since drops begin to fall back to cloud base with relatively small sizes weak updraft case Drops carried to greater heights since path length through liquid water content is much larger compared to clouds with weak updrafts – however clouds with strong updrafts must also be deeper – which they normally are – and they contain larger strong updraft case

Updraft velocity (m s-1) Maximum supersaturation (%) Some Useful (Ballpark) Values (Table 15.3, Seinfeld & Pandis) Cloud Type Updraft velocity (m s-1) Maximum supersaturation (%) Continental cumulus ~1 – 17 0.25 – 0.7 Maritime cumulus ~1 – 2.5 0.3 – 0.8 Stratiform ~0 – 1 ~0.05 Fog -- ~0.1