1. Putting it together: Thermo + microphysics in warm clouds
Condensation “activates” aerosol particles into drops  condensation grows liquid droplets (initial stages)  drop-drop collisions / coalescence create larger drops  formation of rain in mature clouds (removes water from parcel, disrupts the process)

2. Putting it together: Thermo + microphysics in warm clouds
Condensation “activates” aerosol particles into drops  condensation grows liquid droplets (initial stages)  drop-drop collisions / coalescence create larger drops  formation of rain in mature clouds (disrupts the process)
Assume for now that the air parcel is condensing the adiabatic liquid water content at any height z in the updraft (given by the difference between the total initial water content, and the water contents along the moist adiabat – recall the skew-T diagram)
The liquid water content (LWC) can also be calculated from knowledge of the drop size distribution:
Ratio of concentrations is the inverse of the ratio of sizes: LWC is ~ conserved, but mass is shifted from smaller  larger drops as the cloud evolves

3. Time history of supersaturation: Initial stages
Near cloud base: aerosol particles activate according to their critical supersaturations
As they deplete water vapor via their growth, supersaturation reaches a maximum, then declines
Note: for same updraft, peak s in low-CCN cases is higher than in high-CCN cases: why?
driving force gets small  condensational growth slows down / stops
“colloidal stability”: the mixture of small liquid droplets in air (following air motions, separated from each other) can persist for long periods
…. until broken up by collection

4. Switching over to collection as dominant process
Recall that the collision efficiency approaches unity only when the “large” (collector) drop radius is > 50 µm, and the “small” drop radii are > 10 µm
~20 µm radius limit
Notice curvature: condensation slows down, but collection speeds up as radii increase
Uncertain mechanisms by which drops grow to sizes required to initiate collision-coalescence
New drops form (formerly haze), but are then collected rapidly  supersaturation can rise rapidly
total mass condensation rate a ND × rD × (S-1)

5. Contrasting maritime and continental clouds (model result)
dS/dt>0
Supersaturation rises as drop number falls due to coalescence
Onset of coalescence
Supersaturation does not show increase as it does in maritime cloud since drop concentration remains fairly high
(Rogers and Yau, 1989)

6. Recall features of the continuous collection equation
We ignore that the actual drop population is a continuum of sizes, and simply apply averaged collection efficiencies between “large” and “small” drops, to the entire “collectible” liquid water content
When rL >> rS, vL – vS ≈ vL
vL ≈ k1 rL (recall how fall speeds depend on r (regime II)
So K ≈ p k1 rL3
Mass growth rate by collection is proportional to drop mass  grows exponentially!
Comparing relative sizes (and number concentrations), each mm-sized rain drop must collect about a million cloud droplets.
Let’s relax the approximations in the continuous collection equation, to look more closely at how collection is initiated and proceeds inside the cloud
“collection kernel”

7. Stochastic collection
“Stochastic”: occurs with some degree of chance  initiation of coalescence (vs. deterministic: happens same way every time, described by an equation that can be solved with initial and/or boundary conditions)
Why is drop collection stochastic?
Drops are distributed randomly in space, so depending on nearest neighbors, a particular drop may or may not have a collision in a time interval
Each coalescence event is discrete – in continuous collection, fractions of mass can be collected – not so with individual drop-drop events
Mean growth calcs usually longer than observed time scales -- so a few drops on the (large) tail probably are favored and go on to form rain (only need 1 in 106)

8. Stochastic collection equation (SCE)
Drops grow by (potentially) accumulating water from all drops of smaller sizes
For a small time interval, δt, over which the collector drop can collect at most one other drop, the probability P(mL,mS)
The number of collection events (per unit volume of cloud, per unit time) is
Discrete event that ADDS drops to the “m” bin
Discrete event that REMOVES drops from the “m” bin

9. Stochastic collection equation (SCE)
Put into full, general equation:
An “integro-differential” equation (have to deal with numerically)
Avoids double counting
Twomey (1964): have to look closely at the tail: very small number of large drops even after 20 s
Berry & Reinhardt (1974): famous result
Takes ~ 15 min to generate the rL mode

10. Stochastic collection equation (SCE)
But this is not really “stochastic” – no probabilities in here (can solve + get same answer every time)
We can add the probabilities back in, by considering the time between collection events as a random variable, and using statistics for unlikely events (Poisson counting stats)
Let the MEAN time to collection be
The times between collection event decrease as collector grow larger  but small times matter the most (control the overall timescale)  thus the time to grow cloud drops to drizzle drops depends strongly on first few events

11. Some considerations of real (non-idealized) behavior
that can lead to drop distribution broadening
Turbulence can alter S in “parcel”, enhance collision-coalescence
S increases  new drops?
S decreases, coll / coal enhanced?

12. Some considerations of real (non-idealized) behavior
that can lead to drop distribution broadening
Entrainment can alter drop sizes, number concentrations  affects S
Entrainment of dry air causes small drops to evaporate  reduces competition for vapor  higher S  large ones have enhanced condensation  broadening

13. Warm rain
We consider the simple (parcel-type) model due to Bowen (1950) to understand rain generation
Air is rising in an updraft (speed w), while a “representative” rain drop grows (by continuous collection) and falls against this updraft
The mass of the collector drop increases at a rate
The radius of the drop changes as
Solving, (exponential in time)
The altitude of the drop is
Plug in rd(t) and solve: drop stays with air motion initially, then deviates and eventually falls (accelerating in downward motion)

14. Warm rain
Using the chain rule to convert change in radius with height, from change in radius with time, we get the approximate rate of change of radius with change in altitude:
If w and other parameters are constant, then
Approximate treatment but gives general behavior:
The larger the updraft, the higher the drop can travel 
the more liquid water it can sweep out  and the larger the raindrop that falls out of the cloud

15. Rain rate
If the drop size distribution is n(D), and fall speeds v(D), net vertical flux of drops (m-2 s-1)
The “threshold diameter” has v(Dth) = w. Smaller drops move up, larger ones move down.
The larger that w is, the larger Dth must be  Large rain rates tend to have large drops
The rain rate at the surface is this flux computed at the ground (w=0)
Mass flux [kg m-2 s-1] of rain hitting the ground:
rain

16. Size distributions of rain drops
Modified gamma distribution
Alternative form of gamma distribution:
Setting b=0 yields an exponential –
with n0 = 8×103 m-3 mm-1 and
Λ = 41 R-0.21 (R in mm h-1), this is the Marshall-Palmer distribution
 only captures large tail

17. Evolution of drop size distribution (starting with Marshall-Palmer)1 2
Largest break up
And fragments end up here 3
Collection and disruptions oscillate, but after 30 min get trimodal
Smallest get collected quickly – but some replenished as smaller “grow in” by condensation
Important to note that very large drops not allowed to survive by physics – 8 mm diam sometimes observed, but GCCN origin invoked