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1

ND

SD

North Dakota Thunderstorm Experiment

AOSC 620: Lecture 22 Cloud Droplet Growth

Growth by condensation in warm clouds

R. Dickerson and Z. Li

2

Kelvin CurveKöhler Curve

3

Koehler Curve Plus.Impact of 1 ppb HNO3 vapor (curve 3). PSC’s

often form in HNO3/H2O mixtures. From Finlayson

and Pitts, page 803.

4

CCN spectrafrom Hudson

and Yum (JGR 2002) and in Wallace and Hobbs (sp). page 214.

5

CCN measured in the marine boundary layer during INDOEX. Hudson and Yum (JGR, 2002).

↓ITCZ

6

Growth of Individual Cloud Droplet

Depends upon

• Type and mass of hygroscopic nuclei.

• surface tension.

• humidity of the surrounding air.

• rate of transfer of water vapor to the droplet.

• rate of transfer of latent heat of condensation away from the droplet.

7

Assumptions

• Isolated, spherical water droplet of mass M, radius r and density w

• Droplet is growing by the diffusion of water vapor to the surface.

• The temperature T and water vapor density v of the remote environment remain constant.

• A steady state diffusion field is established around the droplet so that the mass of water vapor diffusing across any spherical surface of radius R centered on the droplet will be independent of R and time t.

8

Fick’s Law of Diffusion

The flux of water vapor toward a droplet through any spherical surface is given as

where

D - diffusion coefficient of water vapor in air

v - density of water vapor

Note that Fw has units of mass/(unit area•unit time)

dR

dDDF v

vw

9

Mass Transport

Rate of mass transfer of water vapor toward the drop through any radius R is denoted Tw (italics to distinguish from R & Temp) and

122 4 4 A

dR

dDRFR

dt

dMT v

ww

Note that Tw = A1(a constant) because we assumed a steady state mass transfer.

10

Mass Transport - continued

) (4

but

) (4

1

1

vrv

vrv

Drdt

dMdt

dMA

r

AD

Integrate the equation from the surface of the droplet where the vapor density is vr to where it is vHow far away is ? See below.

r

RdR

vd A

v

vr

D21 4

(1)

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Conduction of Latent Heat

Assume that the latent heat released is dissipated primarily by conduction to the surrounding air. Since we assume that the mass growth is constant (A1), then the latent heat transport is a constant (A2).

The equation for conduction of heat away from the droplet may be written as

22 4 A

dR

dTKR

dt

dMLv

K is the thermal conductivity of air

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Conduction of Latent Heat - continued

Integrate the equation from the droplet surface to several radii away which is effectively

rr

RdRdT A

T

T

K22 4

)2( ) (4

r

) (4 2

TTL

rK

dt

dMdt

dM

r

LATTK

rv

vr

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Radial Growth Equations

14

15

Molecular diffusion to a droplet at 1.00 atm.How far is infinity?

t = x2/Dx = 1.0 cm → t ≈ 4 s

x = 0.32 m → t ≈ 4,000 s x = 1.0 m → t ≈ 40,000 s

Repeat at 0.10 atm.The lifetime of a Cb is only a few hours.

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Radial Growth - continued

Note that, the radius of a smaller droplet will increase faster than a larger droplet..

)2( ) (

)1( ) (

aTTL

K

dt

drr

aeeTR

D

dt

drr

or

TR

e

TR

eand

TR

e

rwv

rvw

rvvv

rv

rvr

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Important Variables

e Ambient water vapor pressure

es Equilibrium (sat.) water vapor pressure at ambient temperature

es= CC(T):

er: Equilibrium water vapor pressure for a droplet

er: =ehr=CC(Tr) f(r)

f(r) =

esr: Equilibrium water vapor pressure for plane water at the same temperature as the droplet

esr= CC(Tr):

) 1(3r

b

r

a

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Additional Equations

• Clausius-Clapeyron equation

• Combined curvature and solute effects

Integrate the CC equation from the saturation vapor pressure at the temperature of the environment es(T), denoted as es , to the saturation vapor pressure at the droplet surface es(Tr), denoted esr to obtain

)( )( – 1

– 1

ln r2

TTTR

L

TTR

L

e

e

v

v

rv

v

s

sr

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Final Set of Growth Equations

• Clausius-Clapeyron equation } )( – exp r2{

TTTR

L

e

e

v

v

s

sr

• Combined curvature and solute effects 3 1 r

b

r

a

e

e

sr

r

• Mass diffusion to the droplet

• Conduction of latent heat away

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Summary

The four equations are a set of simultaneous equations for er, esr , Tr , and r.

If we know the vapor pressure and temperature of the environment and the mass of solute, the four unknowns may be calculated for any value of r. Then, r may be calculated by numerical integration.

21

Derivation of Droplet Radius Dependence on Time

Steps to solve the Problem

• Expand Clausius-Clapeyron Equation

• Substitute for Tr - T in (2) using the expansion

• Express the ratio (esr/esin terms of radial growth rate from (1)

• Solve resulting equation for r (dr/dt)

22

Derivation

rs

r

s

sr

s

r

s

r

vw

srs

vw

s

e

e

e

e

e

e

thatNote

e

eS

TR

DeeeS

TR

D

dt

drr

asrewritenbemayEq

e

eS

)() (

1 .

astenvironmentheofratiosaturationtheSDefine , ,

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Derivation - continued

Solving for (esr /es) one obtains

But from the Clausius-Clapeyron equation

because the argument of the exponent <<1 formost problems of interest

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Derivation - continued

But, from Eq. (2) we can write

1

.

) (

2

2

dt

drr

KTR

L

e

e

yieldsEqClapeyronClausiusthewithCombining

dt

drr

K

LTT

v

wv

s

sr

wvr

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Derivation - continued

Note that some quantities always appear together. Lets define:

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Derivation - continued

dt

drrCS

e

e

dt

drrC

r

sr21 1

or

sr

r

sr

r

ee

CC

ee

S

dt

drr

12

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Derivation - continued

where

sr

r

sr

r

ee

CC

ee

S

dt

drr

12

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Radius as a Function of Time

Note that, in general, this requires a numerical integration

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Analytic Approximation

Since (er /esr )1 after nucleation

Consider the case where S, C1, and C2 are constant. Separate variables and integrate as:

30

31Lifetime of a Cb ~ 1 hr. Why are cloud droplets fairly uniform in size?

32ξ1 is normalized growth parameter where ξ = (S-1)/(Fk + Fd)

33

34

Summary for Cloud Droplet Growth by Condensation

1. Condensation depends on a seed or CCN.

2. Initial growth is a balance between the surface tension and energy of condensation.

3. Rate of growth depends on rate of vapor transfer and rate of latent heat dissipation.

4. Droplets formed on large CCN grow faster, but only at first.

5. Droplet growth slows after r ~ 20 m.

6. Diffusion is a near field (cm’s) phenomenon.

7. Cloud droplets that fall out of a cloud evaporate before they hit the ground.

8. Why is there ever rain?

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