1. 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

2. 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
Gamma(n) = (n-1)!

4. 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

5. Reminder: Warm cloud (supersaturation via adiabatic expansion)
What are these?

6. Diabatic condensation: entrainment effects
Exchange of energy with environment by virtue of a T difference
Diabatic condensation: entrainment effects
An air parcel may rise adiabatically in the core of a cloud, but turbulent motions eventually mix in dry environmental air
This mixing smooths out gradients  dry air moistens and cloudy air dries
Compare the time scales: τevap << τmixing
This means that the drops respond quickly to the drier air  some drops evaporate until the air is resaturated (according to the new conditions)
Overall, retain the original distribution but ND has decreased
BUT: now ND decreased  condensational sink is REDUCED!  supersaturation RISES faster than in neighboring cloud elements growth rates increase  those surviving drops get larger than neighbors who didn’t see an entrainment event
When the processed air gets mixed in with those neighboring drops later, the final drop distribution will be broader

7. Diabatic condensation: radiative effects
Consider droplets near the top of the cloud: if they “see” a cooler atmosphere above them, they can radiate away some of their energy  their temperature DROPS
So the vapor pressure at the drop surface is DECREASED  drops grow faster
Interestingly, the radiative cooling is proportional to the cross-sectional area of the droplet  so large drops cool more
Harrington et al. (2000) showed that in a marine Scu environment, when drops compete for a limited supply of water vapor, the larger drops grow so rapidly via this enhancement that drops with diameters < 20 µm evaporate  bimodal drop size spectrum!
Only effective in clouds where drops can reside near cloud top for 12 min or more
Cumulus clouds with vigorous overturning “expose” drops to space for too short a time
Hartman and Harrington, 2005

8. Arctic stratus cloud example
Drops smaller than ~10 µm are prevented from growing!
With radiative effects
Without

9. Autoconversion
“Autoconversion” is the process of collision-coalescence that leads to the formation of new small drizzle drops
In models, we generally cannot represent this process explicitly
The “drizzle drop threshold” is generally taken to be r = 20 µm (other choices exist)
Parameterizations express the autoconversion rate in terms of drop size distribution moments, such as liquid water content (LWC) [Kessler, 1969], cloud droplet number concentration, or spectral dispersion
The computed autoconversion rate is used to transfer water mass from “cloud drops” into “drizzle drops” (sets time scale) in order to initiate precipitation
When autoconversion is active, an average collision frequency is assumed for all cloud droplets, resulting in an autoconversion rate that scales with LWC7/3
On the representation of droplet coalescence and autoconversion: Evaluation using ambient cloud droplet size distributions, W. C. Hsieh et al., JGR, 2009
“In this study, we evaluate eight autoconversion parameterizations against integration
of the Kinetic Collection Equation (KCE) for cloud size distributions measured during the
NASA CRYSTAL-FACE and CSTRIPE campaigns. KCE calculations are done using
both the observed data and fits of these data to a gamma distribution function; it is found
that the fitted distributions provide a good approximation for calculations of total
coalescence but not for autoconversion because of fitting errors near the drop-drizzle
separation size.”

10. Measured DSD with gamma distribution fits

11. Generating supersaturations to create clouds
So far, we have focused on generation of supersaturation in an air parcel (really, COOLING that results in supersaturation generation) by adiabatic expansion. In general,
For vertical lifting,
And using mole fraction of total water that is liquid (yL):
But other cooling mechanisms exist, in addition to uplift:
Radiation
Conduction
Mixing
Updraft speed
Heating due to condensation
Cooling due to expansion

12. Isobaric, diabatic cooling Isobaric, adiabatic mixing
Typical case: Earth’s surface radiates energy to space under clear skies at night
Air in contact with surface loses thermal energy by conduction, and cools neighboring air parcels by mixing (dT/dt)
If moist air cools below its dew point, a radiation fog is created
Advection fog: moist warmer air flows over cooler surface (e.g., cold lake)
Isobaric, adiabatic mixing
Typical case: contrails (warm, moist engine exhaust + cold ambient air)
Notice both starting points were undersaturated
-- but mixing can produce supersaturations over wide range of mixing fraction

13. Cloud properties
Table 2. Observed typical values for the properties of clouds. The values are merely modal-means. The range of observed values is quite large.
The radius of cloud droplets is r (microns), the effective optical radius optically is r', N is the number of droplets per cubic centimetre, L is the liquid water content of the cloud (g/m3).
For all clouds, the level of observation is just below the freezing level, except for fog and cirrus. Cirrus consists entirely of ice crystals, and the values shown in this table are liquid equivalents (Source: (3,4)).

14. 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

15. Marine boundary layer Subsidence creates warming that caps the BL
Radiative cooling (creates negative buoyancy at cloud top) and entrainment (grows against subsidence) force the circulations
Middle: notice that strong drying means VPT in downdraft can be warmer than updraft – cloud has to try to compensate
Right: effect of drizzle is similar to that of strong entrainment; can stabilize BL which slow circulation
Left case: Air is cooled, and condensate lost to entrainment of dry air – downdraft cloud base slightly higher than updraft