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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

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2 Kelvin Curve Köhler Curve

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3 Koehler Curve Plus. Impact of 1 ppb HNO 3 vapor (curve 3). PSCs often form in HNO 3 /H 2 O mixtures. From Finlayson and Pitts, page 803.

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4 CCN spectra from Hudson and Yum (JGR 2002) and in Wallace and Hobbs (sp). page 214.

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

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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.

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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.

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8 Ficks 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 F w has units of mass/(unit areaunit time)

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9 Mass Transport Rate of mass transfer of water vapor toward the drop through any radius R is denoted T w (italics to distinguish from R & Temp) and Note that T w = A 1 (a constant) because we assumed a steady state mass transfer.

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10 Mass Transport - continued Integrate the equation from the surface of the droplet where the vapor density is vr to where it is v How far away is ? See below. (1)

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11 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 (A 1 ), then the latent heat transport is a constant (A 2 ). The equation for conduction of heat away from the droplet may be written as K is the thermal conductivity of air

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12 Conduction of Latent Heat - continued Integrate the equation from the droplet surface to several radii away which is effectively

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13 Radial Growth Equations

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15 Molecular diffusion to a droplet at 1.00 atm. How far is infinity? t = x 2 /D x = 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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16 Radial Growth - continued Note that, the radius of a smaller droplet will increase faster than a larger droplet..

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17 Important Variables e Ambient water vapor pressure e s Equilibrium (sat.) water vapor pressure at ambient temperature e s = CC(T ) : e r: Equilibrium water vapor pressure for a droplet e r: =e hr =CC (T r ) f(r) f(r) = e sr: Equilibrium water vapor pressure for plane water at the same temperature as the droplet e sr = CC(T r ) :

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18 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 e s (T ), denoted as e s, to the saturation vapor pressure at the droplet surface e s (T r ), denoted e sr to obtain

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19 Final Set of Growth Equations Clausius-Clapeyron equation Combined curvature and solute effects Mass diffusion to the droplet Conduction of latent heat away

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20 Summary The four equations are a set of simultaneous equations for e r, e sr, T r, 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.

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21 Derivation of Droplet Radius Dependence on Time Steps to solve the Problem Expand Clausius-Clapeyron Equation Substitute for T r - T in (2) using the expansion Express the ratio (e sr /e s in terms of radial growth rate from (1) Solve resulting equation for r (dr/dt)

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22 Derivation

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23 Derivation - continued Solving for (e sr /e s ) one obtains But from the Clausius-Clapeyron equation because the argument of the exponent <<1 for most problems of interest

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24 Derivation - continued But, from Eq. (2) we can write

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25 Derivation - continued Note that some quantities always appear together. Lets define:

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26 Derivation - continued or

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27 Derivation - continued where

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28 Radius as a Function of Time Note that, in general, this requires a numerical integration

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29 Analytic Approximation Since (e r /e sr ) 1 after nucleation Consider the case where S, C 1, and C 2 are constant. Separate variables and integrate as:

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

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32 ξ 1 is normalized growth parameter where ξ = (S-1)/(F k + F d )

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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 (cms) 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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