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Condensational Growth

Explore the factors influencing cloud droplet growth rate, relationship between size and growth rate, and ventilation effects. Understand how droplets gain water molecules and the equations governing condensational growth.

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Condensational Growth

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  1. r 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 10-14 g NaCl 10-13 g NaCl 10-15 g NaCl 85 80 10 .1 1 .01 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. 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 • Metastable droplets • Did not activate • Activated droplets grow

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

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

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

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

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