Cloud Microphysics

Dr. Corey Potvin, CIMMS/NSSL

METR 5004 Lecture #1Oct 1, 2013

Preliminaries• Primarily following Wallace & Hobbs Chap. 6

• Rogers & Yau useful as parallel text (deeper explanations)

• Google is your friend!

• Will make these slides available

• Figures from W & H unless otherwise noted

• Leaving out some important details – read book!

• corey.potvin@ou.edu; office #4378 (once gov’t reopens)

Warm cloud microphysics

Two fundamental phenomena that warm cloud microphysics theory must explain:

• Formation of cloud droplets from supersaturated vapor

• Growth of cloud droplets to raindrops in O(10 min)

Lyndon State College

Saturation vapor pressure

• Increases with temperature T (Clausius-Clapeyron)

• By default, refers to vapor pressure e over planar water surface within sealed container at T at equilibrium (evaporation = condensation); denoted es

• BUT can also refer to equilibrium e over cloud particle surface (e.g., ei, e’)

Wikipedia

Saturation vapor pressure

• By default, supersaturation refers to e > es, as when rising parcel cools (es decreases)

• e > es does NOT guarantee net condensation onto cloud particles – not surprising given artificiality of es!

• But, es useful as reference point when describing (super-) saturation level of air relative to cloud droplet or ice particle Lyndon State College

Homogeneous nucleation

• Consider supersaturated air with no aerosols• Are chance collisions of vapor molecules likely

to produce droplets large enough to survive?– Consider change in energy of system ΔE due to

formation of droplet with radius R– Compute R for which ΔE ≤ 0 (lazy universe always

seeks equilibrium) – growth favored– Determine whether such R occur often enough

How big must embryonic droplets be for growth to be favored?

• Take droplet with volume V, surface area A, and n molecules per unit volume of liquid

• Consider surface energy of droplet and energy spent for condensation:

ΔE =Aσ −nV μv−μl( ) =4πR2σ −

43πR3nkT ln

ees

Gibbs free energies – roughly, microscopic energy of system

Net change in system energy

Work to create unit area of droplet surface

Critical radius r

• Subsaturation (e < es) dΔE/dR > 0 embryonic droplets generally evaporate (all sizes)

• Supersaturation (e > es) dΔE/dR < 0 for r > R sufficiently large droplets tend to grow (energy loss from condensation > energy gain from droplet surface)

• R = r: droplet in unstable equilibrium with environment – small change in size will perpetuate

dΔEdR

=8πσR−4πR2nkTlnees

Kelvin’s equation & curvature effect• Solve d(ΔE)/dR = 0 to relate r, e, es:

• In latter equation and hereafter, e = droplet saturation vapor pressure! Otherwise, net evaporation or condensation of droplet would occur (disequilibrium).

• “Kelvin” or “curvature” effect: e > es required for equilibrium since less energy needed for molecules to escape curved surface

• Smaller droplet larger RH (= 100 × e/es) needed

Critical radius for given ambient vapor pressure

Vapor pressure required for droplet of radius r to be in unstable equilibrium

Heterogeneous nucleation• Some aerosols dissolve when water condenses on them - cloud

condensation nuclei (CCN)

• Due to relatively low water vapor pressure of solute molecules in droplet surface, solution droplet saturation vapor pressure e’ < e

• Raoult’s law for ideal solution containing single solute: saturation vapor pressure of solution = saturation vapor pressure of pure solvent, reduced by mole fraction of solvent. Thus,

where f is mole faction of pure water. e' = fe

Computing f

• Vapor condenses on CCN of mass m, molecular mass Ms, forming droplet of size r

• Each CCN molecule dissociates into i ions• Solution density ρ’, water molecular mass Mw

• Effective # moles of dissolved CCN = im/Ms

• # moles pure water =

4

3πr3ρ' −m

⎛⎝⎜

⎞⎠⎟

/ Mw

Computing f (cont.)

f =#molespurewater

total #molesinsolution=

43πr3ρ' −m

⎛⎝⎜

⎞⎠⎟

/ Mw

43πr3ρ' −m

⎛⎝⎜

⎞⎠⎟

/ Mw+ im/ Ms

= 1+imMw

Ms43πr3ρ' −m

⎛⎝⎜

⎞⎠⎟

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

−1

Kohler curves• Combining curvature and solute effects, can model

equilibrium conditions for range of droplet sizes:

S ≡e'es

=e'e

ees

= f exp2σ '

n'kTr

⎛

⎝⎜

⎞

⎠⎟

(r*, S*) – activation radius, critical saturation ratio – spontaneous growth occurs for r > r*

Saturation ratio for solution droplet

Rogers & Yau

Stable vs. unstable equilibrium• A: r increase RH >

equilibrium RH r increases further; similarly for r decrease (unstable equilibrium)

• B: r increase RH < equilibrium RH r decreases; similarly for r increase (stable equilibrium)

• C: as in A, but droplet activated (grows spontaneously, i.e., without further RH increases)

Adapted from www.physics.nmt.edu/~raymond/classes/ph536/notes/microphys.pdf

Kohler curves (cont.)

(r*, S*)Left of peak Right of peak

Equilibrium Stable Unstable

Dominant effect Solute Curvature

Droplet type Haze particles Activated CCN

Kohler curves (cont.)

• Slightly different perspective – assume solution droplet inserted into air with ambient S

• Red curve – solution droplet grows indefinitely since droplet S < ambient S

• Green curve – droplet grows until stable equilibrium point “A”

Two fundamental phenomena that warm cloud microphysics theory must explain:

• Formation of cloud droplets from supersaturated vapor

• Growth of cloud droplets to raindrops in O(10 min)