# ESCI 340 Physical Meteorology Cloud Physics Lesson 2 Formation of Cloud Droplets

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1 ESCI 40 Phyical Meteorology Cloud Phyic Leon 2 Formation of Cloud Droplet Reference: A Short Coure in Cloud Phyic, Roger and Yau Reading: Roger and Yau, Chapter 6 The objective of thi leon are: 1) Undertand the effect curvature and olute on aturation vapor preure, and ho thee effect are quantified. 2) Decribe and undertand ho Kohler curve are created. ) Decribe the ignificance of the critical radiu, r *, and the different tability regime for r < r * and r > r *. 4) Decribe the different type of atmopheric aerool, and hich one are important a cloud condenation nuclei (CCN). SATURATION VAPOR PRESSURE OVER A CURVED DROPLET (THE CURVATURE EFFECT) The aturation vapor preure over a curved ater urface, e ( r ), i greater than that over a flat urface, e ( ). Thi i expreed mathematically a 2γ 1 e ( r) = e ( )exp = e exp a r Rv ρlt r ( ) ( ) here γ i the urface tenion of the ater-air interface (~0.075 N/m), ρ L i the denity of liquid ater (~1000 kg/m ), and r i the radiu of curvature (or radiu of the droplet). For a droplet to be in uilibrium ith the environment (meaning the droplet ill neither gro nor evaporate), then the environmental vapor preure e mut be ual to e ( r ). If thi i not true, the droplet ill either gro or evaporate. Thi i ummarized a: Condition The droplet ill e < e ( r) evaporate e = e ( r) remain the ame ize e > e ( r) gro via condenation (1)

2 For a droplet to be in uilibrium then Dividing both ide of (2) by e ( ) e get here S i the uilibrium aturation ratio. e = e ( )exp( a r). (2) S = e e ( ) = exp( a r) () ο The aturation ratio i jut relative humidity expreed a a ratio rather than a percent. The uilibrium aturation ratio i the aturation ratio ruired for the droplet to be in uilibrium. ο If the environmental aturation ratio i le than the uilibrium aturation ratio the droplet ill evaporate. ο If the environmental aturation ratio i greater than the uilibrium aturation ratio the droplet ill gro. For very mall drop the uilibrium aturation ratio i extremely large. The increae of uilibrium aturation ratio ith decreaing radiu i knon a the curvature effect. Homogeneou nucleation (condenation of pure ater ith no dut or aerool preent) ruire a relative humidity of %! Though thi can be achieved in a laboratory, uch high relative humidity doe not occur in the atmophere. Therefore, homogeneou nucleation cannot explain the initial formation of cloud droplet. SATURATION VAPOR PRESSURE OVER A SOLUTION (THE SOLUTE EFFECT) A diolved ubtance (olute) loer the aturation vapor preure of ater. A formula expreing thi for dilute olution i given by Raoult La e = χ e (4) here χ i the mole fraction of the ater n χ = n + n (n i the number of mole of olute and n i the number of mole of ater). For very dilute olution e can approximate χ a 2 (5)

3 The number of mole of olute and ater are given by χ 1 n n. (6) n = im / M olute n = m / M here m i ma, M i molecular eight, and i i the ion factor (the number of ion that one molecule of ubtance diociate into). Therefore, e get The ma of the ater in the droplet i o that χ m M χ =1 i. (8) m M m 4 (7) = πρlr (9) im M 1 = 1 1 b r 4πρ L M r =. (10) The ratio of the aturation vapor preure of the mixture over that of pure ater i then e e = 1 b r. (11) The reduction of aturation vapor preure by introducing a olute i knon a the olute effect. For droplet of mall radiu the aturation vapor preure over the drop i much le than that over pure ater. A radiu increae the aturation vapor preure of the olution approache that of pure ater. COMBINING THE CURVATURE AND SOLUTE EFFECTS The curvature effect increae the aturation vapor preure and ha the greatet impact for mall droplet. The olute effect decreae the aturation vapor preure and alo ha the greatet impact for mall droplet. Which effect in depend on the droplet ize and on the amount of olute preent.

4 The combined effect are expreed by applying the correction from (11) to the uilibrium aturation ratio uation () to get S b = 1 exp r ( a r ) A plot of S for a olution containing three different amount of olute (each curve differ in olute ma by a factor of 10) i hon belo. The tallet curve i for the leat amount of olute.. (12) A plot like that hon above i referred to a a Kohler curve. Making ue of the approximation that for mall x, e x 1 + x,, e can rite S a b 1 + r r. (1) The radiu at hich the Kohler curve i a maximum can be found by taking S / r and etting it ual to zero. Thi radiu i called the critical radiu, r*, and the aturation ratio at thi point i called the critical aturation ratio,, S*. They have value of r* = b a S* = 1+ 4a 27b The critical radiu i of fundamental importance for cloud droplet groth. 4

5 At radii belo the critical radiu (r < r*) the droplet are in table uilibrium. If S increae the droplet ill gro to a larger ize and then top. If S decreae the droplet ill hrink to a maller ize and then top. ο Droplet at radii belo the critical radiu are called haze particle. At radii above the critical radiu (r > r*) the uilibrium i untable, and the droplet ill pontaneouly gro larger, even though S i not increaing. ο Droplet hoe radiu ual the critical radiu (r = r*) are aid to be activated. ATMOSPHERIC AEROSOLS Aerool are formed either directly by diintegration of liquid or olid (knon a primary ource) or indirectly by condenation of gae (knon a ga-toparticle converion). Indirect ource are knon a econdary ource Example of primary ource are ο ind-generated dut ο ea pray ο foret fire ο combution The gae reponible for ga-to-particle converion are ο Sulfur dioxide (SO 2 ) ο Nitrogen dioxide (NO 2 ) ο NH ο certain hydrocarbon Aerool are broken into three different group baed on ize. Thee group are ο Aitkin nuclei particle ith r < 0.1µm ο Large particle 0.1µm < r < 1.0µm ο Giant particle r > 1.0µm The ize ditribution of aerool population can be pecified by a ditribution function n d (D), here D i the uivalent diameter (the diameter of a pherical particle ith the ame volume a the actual particle). In many intance the aerool ize ditribution i given a n d D) β ( = cd. 5

6 ο Thi ditribution i knon a the Junge ditribution. CLOUD CONDENSATION NUCLEI Homogeneou nucleation doe not occur in the atmophere, ince aturation ratio rarely exceed Therefore, the olute effect i extremely important! The olute for the olute effect come from aerool particle in the air. There are to type of aerool ο Hygrocopic nuclei nuclei that are attractive to ater vapor molecule, and act a collection ite for condenation ο Hydrophobic nuclei nuclei that are repellent to ater and therefore cannot act a ite for condenation Not all hygrocopic nuclei are important for cloud droplet formation. Since aturation ratio rarely exceed 1.01, only thoe nuclei that activate at S < 1.02 are available for forming cloud droplet. It i thee nuclei that are knon a cloud condenation nuclei (CCN). EXERCISES 1. What i the relative humidity ruired to upport pure ater droplet of radiu 8µm at a temperature of 0 C? I a relative humidity of thi magnitude ever achieved in the atmophere? 2. Uing the approximate expreion for a Kohler curve ho that S a b 1+ r r r* = b a S* = 1+ 4a 27b. Uing a graphing calculator or computer program, plot the Kohler curve for a droplet containing gram of odium chloride (NaCl). Alo, plot the curve for a droplet containing one-tenth a much odium chloride and compare 6

7 the curve. Sodium chloride ha a molecular eight of g/mol, and ha an ion factor of Sho that for the Junge ditribution that a. The total area of particle having diameter beteen D 1 and D 2 i π c A = D D 1 2 β β β ( ) D ; D 2 1 b. The total ma of particle having diameter beteen D 1 and D 2 i π cρ M = D D ( ) ( 4 β 4 β ) β D ; D 2 1 7

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