1. Condensational Growth

2. Reading
Wallace & Hobbs
pp 221 – 224

3. Condensational Growth
Objectives
Be able to describe the factors that determine the condensational grow rate of a cloud droplet
Be able to state the relationship between droplet size and growth rate
Be able to describe how ventilation effects influence growing cloud droplets

4. Condensational Growth
Objectives
Be able to describe the initial growth of a cloud including typical supersaturation, height of maximum supersaturation, activated CCN and resulting cloud droplet spectrum

5. Condensational Growth
How do droplets grow? t1 t2 t3

6. Condensational Growth
Droplet gains water molecules

7. Condensational Growth
Flux of water molecules towards droplet

8. Condensational Growth
Equation of continuity
The net mass flux into the system equals the rate of increase of mass of the system
rw = density of water vapor molecules
rwV = mass flux of water molecules

9. Condensational Growth
How do the molecules move towards the droplet?
Kinetic Theory of Gases
Flux Density – the net rate of transport per unit area

10. Condensational Growth
Flux Density
lw = molecular mean free path
vw = mean molecular speed
rw = density of water molecules

11. Condensational Growth
Diffusional Coefficient
Flux Density

12. Condensational Growth
Flux density is the same as mass flux
Substitute into or

13. Condensational Growth
Diffusion Equation for Water Vapor
the change in water vapor density over time is a function of
Diffusion Coefficient
Distribution of water vapor

14. Condensational Growth
Let’s solve this equation physically
Imagine a sphere around a growing droplet r
Surface Area of Sphere = 4pr2

15. Condensational Growth
Rate of droplet growth r
m = mass of water

16. Condensational Growth
Why is the drop growing?
Environmental water vapor density is greater than that at droplet surface r
Water vapor gradient

17. Condensational Growth
New equation for a growing droplet
Integrate
vapor density adjacent to droplet surface r
vapor density a great distance away from droplet

18. Condensational Growth
Assuming the change in mass with time is independent of radius

19. Condensational Growth
Substitute for the mass of water (assuming a spherical droplet)
density of liquid water

20. Condensational Growth

21. Condensational Growth
Using the Ideal Gas Law
Temperature at droplet surface
Temperature far from droplet

22. Condensational Growth
Assume temperature at droplet surface is same as environment

23. Condensational Growth
Using the Ideal Gas Law again

24. Condensational Growth
Vapor pressure at droplet surface
Vapor pressure far from droplet

25. Condensational Growth
Vapor pressure at droplet surface depends on
Solute Effect
Surface Tension

26. Condensational Growth.3
Solute & Kelvin effects are small for droplets > 1mm
Pure
Water
Supersaturation (%) .2 .1
100 95
Condensational
Growth
Relative Humidity (%) 90
10-16 g NaCl
10-15 g NaCl
10-14 g NaCl
10-13 g NaCl 85 80
.01 .1 1 10
Droplet Radius (mm)

27. Condensational Growth
Vapor pressure at the droplet surface is approximately equal to that over a plane surface of water @ eo es

28. Condensational Growth
If the vapor pressure at the droplet surface is not too different from the vapor pressure away from the drop

29. Condensational Growth
Let’s review what’s happening
Environmental water vapor pressure is greater than that at droplet surface

30. Condensational Growth
Supersaturation
Substitute into
Supersaturation here is a fraction rather than a percentage

31. Condensational Growth
Rearranging and grouping terms
Gl can be considered constant for a given environment at a fixed temperature
where

32. Condensational Growth
All that just to say.....

33. Condensational Growth
Rate of Droplet Growth
Proportional to supersaturation
Bigger SS, grows faster

34. Condensational Growth
Rate of Droplet Growth
Inversely proportional to droplet radius
Smaller radius, grows faster

35. Condensational Growth
Ventilation Effects
Proportional to droplet terminal speed
Unimportant for growing droplets
Significant for falling raindrops

36. Written and Illustrated by Prof. Fred Remer
A Cloud Story
Written and Illustrated by
Prof. Fred Remer

37. Cloud Story Once upon a time, there was a rising parcel of air
It had aerosols

38. Cloud Story As the parcel rose, it cooled adiabatically
It reached saturation with respect to liquid water
RH = 100%

39. Cloud Story It kept rising! Soon it was supersaturated!
The supersaturation increased at a rate proportional to the updraft velocity SS

40. Cloud Story
The biggest (and most efficient) CCN were activated first

41. Cloud Story Maximum Supersaturation
Rate of condensation approaches rate of moisture supply
SSmax

42. Cloud Story Maximum Supersaturation
Smallest cloud droplets are activated
Determines cloud droplet concentration
SSmax

43. Cloud Story Maximum Supersaturation
Occurs within a few hundred meters of cloud base
SSmax

44. Cloud Story Supersaturation begins to decrease
Rate of condensation greater than rate of moisture supply

45. Cloud Story Haze droplet begin to evaporate Activated droplets grow
Metastable droplets
Did not activate
Activated droplets grow

46. Cloud Story Smallest droplets grow fastest Bigger droplets grow slower
Droplet spectrum becomes more uniform

48. Condensational Growth
Monodisperse spectrum
Droplets grow to 10 mm after 5 min.
Slower growth at larger sizes

49. Units: r [mm], n [liter-1], v [cm/s]

50. Condensational Growth
Precipitation sized particles
Large Cloud Droplet
(50 mm)
Small Raindrop
(100 mm)
Cloud Droplet
(10 mm)
Typical Raindrop
(1000 mm)

51. Condensational Growth
Condensational growth cannot account for precipitation sized particles
Cloud Droplet
(10 mm)
Typical Raindrop
(1000 mm)

52. The End