cloud droplet formation and growth - folk.uio.nofolk.uio.no/truds/droplet_formation_growth.pdf · i...

Nucleation Kelvin equation Raoult’s law Kohler curve Condensation Rain formation Collision efficiency Growth equation

Cloud droplet formation and growth

Trude Storelvmo Condensation and rain formation 1 / 26


Phase changes

I Phase changes from left to right: increasing molecular order:

I Vapor ↔ liquid (condensation, evaporation)

I Liquid ↔ solid (freezing, melting)

I Vapor ↔ solid (deposition, sublimation)

I Recall: Clausius-Clapeyron equation describes equilibrium

condition for bulk water and its vapor.

I Saturation: equilibrium situation with rates of evaporation and

condensation being equal.

I However, for small droplets, because of the energy barrier

(surface tension), phase transitions do not generally occur at

the equilibrium saturation of bulk water.



Nucleation

I Nucleation: Any process in which a free energy barrier must

be overcome, such as vapor to liquid or liquid to ice transitions.

I Homogeneous nucleation: Cloud droplets form directly from

the vapor phase

I Homogeneous nucleation requires several hundred percent

supersaturation

I Instead cloud droplet form when the ascending air just reaches

equilibrium saturation, because of the presence of CCN.

I Heterogeneous nucleation: Cloud droplets form on nuclei

from the vapor phase



Kelvin equation I

I Define saturation ratio S = es(r)es(∞) :

I The change in Gibbs free energy associated with the formation

of a droplet of radius r at saturation ratio S at temperature T:

∆G = −4πr3RvT

3αllnS + 4πr2σ (1)

I From ∂∆G∂r = 0. obtain Kelvin equation: vapor pressure in

equilibrium is larger over a droplet with radius r than over a

bulk surface:

es(r) = es(∞)exp(2σ

ρwRvTr) (2)

I where σ is the surface tension = 0.075 N/m.

Def: Surface tension: free energy per unit surface area of the

liquid. Work per unit area required to extend the surface of

liquid at constant temperature



Gibbs free energy (Fig. 9.10 Seinfeld&Pandis)



Kelvin equation II

Critical radii for droplet formation in clean air:

rc =2σ

RvρwTlnS; S =

e

esat(∞)(3)

Saturation ratio Critical radius number of moleculesS rc(µm) n1 ∞ ∞

1.01 0.12 2.47 x 108

1.1 0.0126 2.81 x 105

1.5 2.96 x 10−3 36452 1.73 x 10−3 7303 1.09 x 10−4 183

10 5.22 x 10−4 20



Raoult’s Law I

I Reduction in vapor pressure due to the presence of a

non-volatile solute over a plane surface of water:

e∗

es(∞)= 1− 3imMw

4πMsρw r3= 1− b

r3(4)

where e∗ is the equilibrium vapor pressure over a solution, i is

the degree of ionic dissociation in the solute, Ms is the

molecular weight of the solute, Mw is the molecular weight of

water, m is the mass of solute

I If the vapor pressure of the solute is less than that of the

solvent, the vapor pressure is reduced in proportion to the

amount of solute present.



Kohler curve I

I Combination (multiplication) of Kelvin and Raoult’s equation

(evaluating it for e∗(r)/es(r)) gives the Kohler curve:

Ie∗(r)

es(∞)= (1− b

r3) ∗ exp(

a

r). (5)

with

a =2σ

ρwRvT≈ 3.3 · 10−7

T[m] (6)

I For r not too small, a good approx. is (exp(ar ) ∼ 1 + ar )

e∗(r)

es(∞)= 1 +

a

r− b

r3(7)

I 1. term: surface molecules possess extra energy

2. term: solute molecules displacing surface water molecules



Kohler curve II



Kohler curve III

I The critical radius rc and critical supersaturation Sc are given by:

rc =

√3b

a; Sc =

√4a3

27b(8)

I Kohler curve represents equilibrium conditions

I Large particles have large equilibrium radii and may have insufficient

times to grow to their equilibrium size in clouds with strong updrafts.

I As the size of a droplet increases, the equilibrium vapor pressure

above its surface decreases (Kelvin’s equation).

I The curves for droplets containing fixed masses of salt approach the

Kelvin curve as they increase in size, since the droplets become

increasingly dilute solutions.



Kohler curve IV

I Solution effect dominates for small particles: Small solution

droplets are in equilibrium with the vapor for RH < 100%. If

RH increases, droplet will grow until it reaches equilibrium

again.

I If droplet grows beyond rc , its equilibrium saturation ratio falls

below Sc . Then vapor will diffuse to the droplet. It will now

grow even if RH decreases → activated drop.

I Drops experiencing S > Sc can thus be activated

(S = e∗(r)es(∞) − 1).

I If cloud droplet is not activated and grows slightly, then S of

air adjacent to drop needs to be higher than that of ambient

air to maintain that state. Since it is not, the drop will shrink

again and vice versa.

I The higher S , the more and the smaller aerosols are activated.



Droplet growth by condensation

I At the early stages of a cloud droplets development, it grows

by diffusion of water molecules from the vapor onto its surface.

I Droplet growth equation:

rdr

dt=

(S − 1)− ar + b

r3

Fk + Fd(9)

I Fk = thermodynamic term:(

LRvT− 1)

LρlKT ; “-1” can be

neglected

I Fd = vapor diffusion term: ρlRvTDes

.

I Equation (9) cannot be solved analytically. For sufficiently

large droplets it can be approximated by neglecting the solution

and curvature effects on the drop’s equilibrium pressure:



Droplet growth equation

rdr

dt=

S − 1

Fk + Fd(10)

Figure: Droplet growth equation for different r0 at 0.5% supersaturation



Droplet growth by condensation

I The cloud droplet radius increases with time according to:

r(t) =√r20 + 2ξt (11)

where ξ = (S − 1)/(Fk + Fd).

I ξ depends on temperature and pressure. The larger T , the lower p,

the higher ξ

I The parabolic form of (11) leads to a narrowing of the drop-size

distribution as growth proceeds.

I Consider 2 cloud droplets with initial radii of r1(0) and r2(0) with

r2 > r1. From (11) it follows:

r2(t)− r1(t) =r22 (0)− r2

1 (0)

r2(t) + r1(t)(12)

because the difference between the squares of the initial radii remains

constant, at any time t, the difference in radii becomes smaller.



Growth of droplet population

I Condensation growth is controlled by ambient S , thus we need

to examine the vapor budget of a developing cloud

I Assume that the vapor is provided by saturated air that is

cooled in ascent and that it is lost by condensation on the

growing cloud droplet, so the rate of change of the saturation

ratio S is given by:

dS

dt= P − C = Q1

dz

dt− Q2

dχ

dt(13)

where:

P = production, C = condensationdzdt = w = vertical air velocitydχdt : condensation rate [kg/kgs]

Q1 = 1T

[εLg

RdcpT− g

Rd

], and Q2 = ρ

[RdTεes

+ εL2

pTcp

]Trude Storelvmo Condensation and rain formation 16 / 26


Initiation of rain in nonfreezing (=warm) clouds

I Rain formation by coalescence is the dominant precipitation

formation process in warm clouds in the tropics and in

midlatitude cumuli whose tops may extend to subfreezing T .

I Need to explain how raindrops can be created by condensation

and coalescence in 20 min (∼ time between first appearance of

Cu and appearance of rain).

I 50-fold increase in radius (from 10 µm cloud drop with 105 l−1

to a 0.5 mm rain drop with 1 l−1) is achieved by collision and

coalescence if some drops exceed 20µm.

I Smaller drops have small collision cross section and slow

settling speeds → small chance to collide (collision rate ∼ r4)

I A raindrop of 1 mm results from ∼ 105 collisions

I Cloud droplets can grow by condensation to r = 20µm in 10

min if S ≥ 0.5% or u ≥ 5m/s (developing Cu).



Droplet growth by collision, coalescence

I Collision may occur due to gravitational effects once some

drops are larger than others

I Define the coalescence efficiency as number of coalescence

events / number of collisions

I The growth of droplets by the collision-coalescence process is

governed by the collection efficiency = collision efficiency *

coalescence efficiency

I Assume a coalescence efficiency=1, so that collision efficiency

= collection efficiency

I The problem then is to determine collision rates among a

droplet population



Droplet terminal velocity (u)

I Assuming equilibrium between the drag force of a sphere in a

viscous fluid and the gravitational force:

FD =π

2r2u2ρCD = 6πµru

(CDRe

24

)=

4

3πr3gρl = Fg (14)

where CD is the drag coefficient, µ the dynamic viscosity, r the

droplet radius and Re = 2ρurµ the Reynolds number. This gives:

u =2

9

r2gρl

µCDRe

24

(15)

I For r < 40µm, CDRe

24 ≈ 1, so that

u =2

9

r2gρlµ

= k1r2 (16)

where k1 ∼ 1.19 x 106 cm−1 s−1 (Stokes law)



RaindropsI For spherical raindrops 0.6 mm < r < 2 mm.

u = k2

√r (17)

where k2 = 2200√

ρoρ [√cm/s] and ρo = 1.2 kg/m3.

I In the intermediate range (40 µ m< r < 0.6 mm), an approx.

formula for the fall speed with k3 = 8000 s−1 is given by:

u = k3r (18)



Collision efficiency

E (R, r) =x2o

(R + r)2(19)

I xo = impact parameter within which a

collision is certain to occur. Outside xodrops will be deflected out of the path.

I E = fraction of drops with r colliding

with collector drops over the path swept

out by the collector drop

I or: E = probability that a collision will

occur with a drop located at random in

the swept volume.



Collision efficiency

Collision efficien-

cies for small col-

lector drops from

3 sets of theoret-

ical calculations.

E vs. r/R.



Growth equation

I Suppose a drop with radius R falls with terminal velocity u(R)

through a population of smaller droplets. During unit time it

sweeps out droplets of radius r from a volume given by:

π(R + r)2[u(R)− u(r)] (20)

I The number of drops with radii between r and r+dr then is:

π(R + r)2[u(R)− u(r)]n(r)E (R, r)dr (21)

where E(R,r) is the collection efficiency and n(r) is the number

concentration of droplets with radius r.

I If r and R < 100µm → coalescence efficiency = 1 and the

collection efficiency = collision efficiency.



Growth equation

I The increase in the droplet radius is then given as:

dR

dt=π

3

∫ R

o(R + r

R)2n(r)[u(R)− u(r)]r3E (R, r)dr (22)

I If droplets are much smaller than the collector drop u(r) ∼ 0

and R + r ∼ R, so that

dR

dt=π

3

∫ R

on(r) u(R) r3 E (R, r)dr =

EMu(R)

4ρl(23)

where E is the average collection efficiency for the droplet

population and M is the cloud liquid water content.

I These equations describe drop growth as a continuous

collection process, regarding the cloud as a continuum. Growth

actually occurs by the discrete events of droplet capture.



Marine case: Onset

of coalescence reduces

drop number. S in-

creases because the few

drops cannot consume

the excess vapor. New

CCN are activated, but

quickly consumed by

coalescence.

Continental case: Coa-

lescence causes a slight

reduction in drop num-

ber, but not enough to

cause a sudden rise in S .


cloud droplet formation and growth - folk.uio.nofolk.uio.no/truds/droplet_formation_growth.pdf · i...

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