Condensational Growth

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Condensational Growth
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Reading

• Wallace Hobbs
• pp 221 224

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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

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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

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Condensational Growth

• How do droplets grow?

t2
t1
t3
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Condensational Growth

• Droplet gains water molecules

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Condensational Growth

• Flux of water molecules towards droplet

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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
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Condensational Growth

• How do the molecules move towards the droplet?

Kinetic Theory of Gases
Flux Density the net rate of transport per unit
area
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Condensational Growth

• Flux Density

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

• Diffusional Coefficient

Flux Density
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Condensational Growth

• Flux density is the same as mass flux

Substitute into
or
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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

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Condensational Growth

• Lets solve this equation physically
• Imagine a sphere around a growing droplet

r
Surface Area of Sphere 4pr2
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Condensational Growth

• Rate of droplet growth

r
m mass of water
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Condensational Growth

• Why is the drop growing?
• Environmental water vapor density is greater than
  that at droplet surface

r
Water vapor gradient
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Condensational Growth

• New equation for a growing droplet

• Integrate

vapor density adjacent to droplet surface
r
vapor density a great distance away from droplet
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Condensational Growth

• Assuming the change in mass with time is
  independent of radius

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Condensational Growth

• Substitute for the mass of water (assuming a
  spherical droplet)

density of liquid water
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Condensational Growth
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Condensational Growth

• Using the Ideal Gas Law

Temperature at droplet surface
Temperature far from droplet
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Condensational Growth

• Assume temperature at droplet surface is same as
  environment

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Condensational Growth

• Using the Ideal Gas Law again

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

• Vapor pressure at droplet surface depends on
• Solute Effect
• Surface Tension

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Condensational Growth
.3

• Solute Kelvin effects are small for droplets gt
  1mm

Pure Water
Supersaturation ()
.2
.1
100
95
Condensational Growth
Relative Humidity ()
90
10-15 g NaCl
10-14 g NaCl
10-13 g NaCl
10-16 g NaCl
85
80
10
.1
1
.01
Droplet Radius (mm)
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Condensational Growth

• Vapor pressure at the droplet surface is
  approximately equal to that over a plane surface
  of water

eo
_at_
es
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Condensational Growth

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

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Condensational Growth

• Lets review whats happening
• Environmental water vapor pressure is greater
  than that at droplet surface

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Condensational Growth

• Supersaturation

• Substitute into

• Supersaturation here is a fraction rather than a
  percentage

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Condensational Growth

• Rearranging and grouping terms

Gl can be considered constant for a given
environment at a fixed temperature
where
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Condensational Growth

• All that just to say.....

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Condensational Growth

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

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Condensational Growth

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

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Condensational Growth

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

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A Cloud Story

• Written and Illustrated by
• Prof. Fred Remer

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Cloud Story

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

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Cloud Story

• As the parcel rose, it cooled adiabatically
• It reached saturation with respect to liquid water

RH 100
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Cloud Story

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

SS
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Cloud Story

• The biggest (and most efficient) CCN were
  activated first

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Cloud Story

• Maximum Supersaturation
• Rate of condensation approaches rate of moisture
  supply

SSmax
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Cloud Story

• Maximum Supersaturation
• Smallest cloud droplets are activated
• Determines cloud droplet concentration

SSmax
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Cloud Story

• Maximum Supersaturation
• Occurs within a few hundred meters of cloud base

SSmax
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Cloud Story

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

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Cloud Story

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

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Cloud Story

• Smallest droplets grow fastest
• Bigger droplets grow slower
• Droplet spectrum becomes more uniform

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Condensational Growth

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

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Units r mm, n liter-1, v cm/s
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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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Condensational Growth

• Condensational growth cannot account for
  precipitation sized particles

Cloud Droplet (10 mm)
Typical Raindrop (1000 mm)
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The End