Condensational Growth

Reading Wallace & Hobbs pp 221 – 224

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

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

Condensational Growth How do droplets grow? t1 t2 t3

Condensational Growth Droplet gains water molecules

Condensational Growth Flux of water molecules towards droplet

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

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

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

Condensational Growth Diffusional Coefficient Flux Density

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

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

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

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

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

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

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

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

Condensational Growth

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

Condensational Growth Assume temperature at droplet surface is same as environment

Condensational Growth Using the Ideal Gas Law again

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

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

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)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

The End