Description of the new cloud model

2. Description of the new cloud model

2.1. Aerosol composition and the initial particle sizes for haze growth process Atmospheric observations have shown that in the northern hemisphere, sulfate Ž . aerosols now dominate at the sources Hobbs, 1993 . Observations in urban, remote continental, polar and maritime areas reveal the chemical composition of tropospheric aerosols in which the negative ions are largely populated by in mass concentration for both continental and maritime air masses, and the positive ions by NH q and Na q 4 Ž . Ž . Jaenicke, 1993 . These observations indicate that NH SO , ammonium sulfate, and 4 2 4 the derivatives represent the majority of CCN. It should then be reasonable to select ammonium sulfate as the solute of haze particles before being involved in cloud droplet formation. In the present cloud model to describe the haze process, we first obtain size distribution of dry particles out of the selected CCN activity spectrum, assuming the Ž . particles are all ammonium sulfate. Similar to the cloud model of Young 1993 , the mass of ammonium sulfate in the range 10 y1 9 –10 y1 1 g is logarithmically divided into 100 groups as m j s 10 w0 .08 Ž101yj.y19x , 1 Ž . Ž . N where j s 1, 2, . . . , 101. The dry nucleus particles thus divided are assumed to be perfectly spherical, so that they may be converted into the corresponding sizes using 1r3 3m j Ž . N r j s , 2 Ž . Ž . N ž 4pr N where r is the density of the dry nucleus compound. Then the 100 groups of dry N particles are allowed to grow at a prescribed atmospheric condition of pressure, temperature, relative humidity in the updraft. In this way we can follow the change of the size distribution of haze particles by growth in a pre-cloud parcel. 2.2. Condensation growth of haze particles and cloud droplets In the present cloud model, both haze particles and cloud droplets are allowed to Ž . Ž grow by condensation, all adopting thermodynamically improved L r R T Mason, c v ` . Ž . 1971 and including Kelvin surface tension as well as the solute effects under low concentration. The following growth formulas are selected: Maxwellian droplet growth equation is given as a b S y 1 y q 3 d r r r r s . 3 Ž . L L 1 d t c c r y 1 q L ž R T KT r D v ` ` s` Fukuta DK equation is described as a b S y 1 y q 3 d r r r r s , 4 Ž . L L 1 1 1 d t c c r y 1 P q P L ž R T KT f r D f v ` ` a s` b where 1r2 r K 2p R T Ž . a ` f s , l s , X a a 1 r q l a a p c q R v a ž 2 r D 2p f s , l s , X b b ž r q l b R T b a ` 2 a 2 b X X a s , b s . 2 y a 2 y b Ž . Eq. 4 is based on Knudsen treatment of accommodation phenomena right above the Ž . surface of the droplet Chapman and Cowling, 1970 and is basically the same as that of Ž . X X Fukuta and Walter 1970 , except that representative a and b of the latter are replaced Ž . Ž . Ž . with their true values Fukuta and Xu, 1996 . In Eqs. 3 and 4 , Ž y4 . 0 .167q3 .26=10 T ` 273.15 6 y1 L s 2.5 = 10 J kg , Ž . c ž T ` y3 y1 y1 y1 K s 4.18 = 10 5.69 q 0.017 T y 273.15 J m s K , Ž . Ž . ` 1 .94 1013.15 T ` y5 2 y1 D s 2.11 = 10 m s , Ž . ž ž p 273.15 Ž . Pruppacher and Klett, 1978 may be taken, where T is in Kelvin and p in hPa. ` Ž . Ž Eq. 4 is now to be compared with other DK droplet growth formulas Fuchs, 1959; . Fitzgerald, 1972 . Fuchs droplet growth equation a b S y 1 y q 3 d r r r r s , 5 Ž . L L 1 1 d t c c r y 1 q X L ž R T KT r D f v ` ` s` b where 1r2 X y1 r l D 2p b X X f s q , l s , b b ž ž r q D r b R T a ` and D is said to be the thickness of ‘collision free zone’ just above droplet surface equalling the mean free path of air molecules. Ž . Ž . Ž . Fitzgerald 1972 and Fitzgerald 1978 , along the treatments of Fuchs 1959 and Ž . Fukuta and Walter 1970 , obtained the following equation X X 2 d r e D K R T wp a b s` v ` r s y 1 q y , 6 Ž . X X 2 3 2 3 ž d t e w q ´ r r R K T q r e D L r Ž . s` L v ` L s` c where r K X K s K q , r q D rnr c a a p r D X D s D q , r q D rnb 1r2 1 8 R T a ` Õ s , ž 4 p where w is the mixing ratio of water vapor, e saturated vapor pressure at the s` Ž . temperature of cloud environment, and ´ s 0.622 is the ratio of molecular weight Ž . between water and air. Other parameters in Eq. 6 are the same as those used before Ž . Fukuta and Xu, 1996 . In these treatments, the origin of D traces to the concentration gradient at a distance Ž . away from the surface Fuchs, 1959 , and in this D-zone, transfer of the water molecules is assumed to take place without concentration gradient, an obvious contradiction since a finite gradient is the necessary condition for the transfer even in the free molecule regime. Ž . Ž . Note that in Eqs. 3 – 6 , a and b express Kelvin and solute effects on droplet growth, respectively, 2 s a s , 7 Ž . r R T L v ` 3im M N L b s , 8 Ž . X 3 y3 4p M r y r r r Ž . N L N N where r X is the density of solution droplet and surface tension and van’t Hoff factor are L Ž . Ž functions of molality X of solute Fitzgerald, 1972; Young, 1993; Young and Warren, . 1992 : y1 1000 m q m L N X s y 1 , ž M m N N s s 74.22 q 1.66 X , i s 1.9242 y 0.1844 ln X y 0.007931 X 2 , Ž . where m is the mass of water and M and m are the molecular weight and the mass L N N of ammonium sulfate in the solution droplet, respectively. Ž . Ž . Now, we try to convert Eq. 6 into the form comparable to Eq. 4 . Since w ´, wp e s s S, e w q ´ e Ž . s` ` Ž . where S is the saturation ratio. Then, Eq. 6 becomes a b S y 1 y q 3 ž d r r r r s , 9 Ž . 2 d t R T L v ` c r q X X L 2 ž e D R T K s` v ` a Maxwellian growth equation for dilute solution droplet. By making use of these Ž . conditions, Eq. 6 can be written as a b S y 1 y q 3 d r r r r s , 10 Ž . L L 1 1 1 d t c c r y 1 q X X L ž R T KT f r D f v ` ` a s` b where 1r2 X y1 r l K 2p R T Ž . a a ` X X f s q , l s , a a ž r q D r a p c q R Ž . v a 1r2 X y1 r l D 2p b X X f s q , l s . b b ž ž r q D r b R T a ` Ž . Ž . Eq. 10 is the droplet growth equation of Fitzgerald 1972, 1978 , which, together with Ž . Ž . Eqs. 3 – 5 , will be used for comparison in our computation for droplet growth. However, for haze particle growth, both Kelvin effect term and solute effect term are to be used in their original forms instead of their linear approximation to make the equation more accurate for high concentrations of solute: y1 2 s 3im M N L S y exp 1 q X 3 ž r R T r d r 4p M r r y m Ž . L v ` N L N r s . 11 Ž . L L 1 1 1 d t c c r y 1 q L ž R T KT f r D f v ` ` a s` b Ž . It is clear that Eq. 3 does not consider any DK effect on droplet growth, while Eq. Ž . 5 includes the DK effect but on water vapor transfer process only across a thickness of D without concentration gradient, an unrealistic physical process as pointed out above. Ž . On the other hand, Eq. 4 treats the DK effect on both water vapor transfer and heat conduction processes. The differences between Fukuta’s and Fitzgerald’s droplet growth equations are Ž found in three places. First of all, the former relies on Knudsen’s treatment Fukuta and . Xu, 1996 , while the latter follows Fuch’s unrealistic method involving D. Secondly, the former uses b X and a X following the Knudsen method, but the latter employs b and a , Ž . instead. The third difference is that the former takes c q 1r2 R in l , including the v a a Ž . contribution of hotter molecules, but the latter takes c q R s c , which finds no v a p place in the steady state heat flow. Only in the non-steady heat flow, which is not the present case, the c transfer is accompanied by gas expansion to bring the R term in, so v a that c becomes involved. These different treatments lead to a substantial discrepancy in p the predicted cloud supersaturation. 2.3. Nucleation of haze particles In the present cloud model, particles in each of the total 100 groups contain different amounts of ammonium sulfate, but within a given bin, the amount is equally divided among them. Therefore, we consider the heterogeneous nucleation process of haze particles of different groups. When an air parcel reaches the level of cloud base, the supersaturation becomes positive and the nucleation of haze particles becomes possible. However, regardless of whether a haze particle has nucleated or not, the equilibrium Ž . radius r may be determined by Kohler, 1921 ¨ a b S y 1 y q s 0, 12 Ž . 3 r r where S-1 is the supersaturation. The critical supersaturation of this Kohler equation or ¨ the maximum can be obtained by differentiation as 1r2 3 4 a y3 r2 S y 1 s s C r , 13 Ž . Ž . critical 1 N ž 27b where r is the radius of dry nucleus, and N 1r2 3 4 2 s M N C s . 1 X 2 2 3 ž 3 3i r R T r M L v ` N L Strictly speaking, s and i are a function of r since they depend on the droplet molality. However, the molality of the ammonium sulfate solution droplet is below 0.3 when the Ž . critical supersaturation is less than 0.5 Pruppacher and Klett, 1978 , and the errors arising from using constant dilute solution values for s and i are below 1 and 6, Ž . respectively Low, 1969 . Under this assumption of dilute solution, the critical radius, for which the critical supersaturation corresponds, is 1r2 3b 3r2 r s s C r , 14 Ž . critical 2 N ž a where 1r2 3i r R T r M L v ` N L C s . X 2 ž 2 s M r N L Ž . At or after S-1 , haze particles in the group are considered to have nucleated and critical grow without restriction towards the cloud droplets. Smaller haze particles, whose critical supersaturations are greater than environmental or cloud supersaturation, remain unactivated, although they swell. The treatment of CCN in our cloud model is quite different from those of some other Ž cloud models in which aerosol size distribution e.g., Junges distribution, gamma . distribution, or log-normal distribution characterizes the CCN activities. It should be pointed out that aerosol particles are not necessarily CCN, and use of the former in place of the latter, merely based on size, may cause a serious error. We start with dried particle size distribution derived from the CCN activity spectrum, which is experimen- Ž . tally verified Twomey, 1959 as k N s C S y 1 , 15 Ž . Ž . where N is the total number concentration of CCN, C the constant for the number Ž . concentration of CCN particles, and k the slope index. N in Eq. 15 corresponds to activated CCN particles: N s n r , 16 Ž . Ž . Ý j j j Ž . where n r is the CCN number concentration of bin j. j Ž . The CCN number concentration in each bin n r is determined by first using Eqs. j Ž . Ž . Ž . Ž . Ž . 1 , 7 , 8 , 13 and 14 to evaluate the critical supersaturation and then to obtain the critical radius. After the cloud base, supersaturation exceeds 0, and CCN activation process becomes possible. As the cloud parcel rises, the supersaturation increases and more and more particles nucleate. The group of aerosols with the largest dry size as well as the critical radius is first activated in the cloud. The CCN number concentration in Ž . group j is calculated by Eq. 15 as n r s N y N . 17 Ž . Ž . j j jy1 Ž . Ž . This method satisfies both Eqs. 15 and 16 and helps us determine the CCN number Ž . concentration in each size group. The droplet radius corresponding to n r is deter- j mined by r q r jq 1 jy1 r s . 18 Ž . j 2 2.4. Microphysics-dynamics interaction inside of cloud In the present cloud model, our interest is to highlight the DK effect in the initial cloud formation process. So entrainment and turbulent mixing processes, as well as evaporation and precipitation processes are excluded from the model treatment. We also neglect droplet collision–coalescence processes. This is to say that our cloud model is a closed adiabatic one with haze particle growth, nucleation, and cloud droplet growth processes. Then, we have du L c s C, 19 Ž . d t c p p d q v s yC, 20 Ž . d t d q c s C, 21 Ž . d t dw u y u u s g q 0.608 q y q y q , 22 Ž . Ž . v v c d t u where T s pu , 23 Ž . R rc d p p p s , p in hPa 24 Ž . Ž . ž 1013.15 dp g s y , 25 Ž . d z c u p v 100 d q d 1 1 d m c j C s s n r d m s n , 26 Ž . Ž . Ý H j ž d t d t r r d t a a js1 Ž . is the condensation rate where n is given by Eq. 17 , and j a b 4p r S y 1 y q j 3 ž r r d m j j j s , L L 1 d t c c y 1 q ž R T KT f r Df v ` ` a s` b q e v S y 1 s y 1 s y 1, Ž . q e vs s` e s` q s 0.622 , vs p y e s` 17.67 T y 273.15 Ž . e s 6.112 exp . s` T y 29.65 The equations of state for air and water vapor are given, respectively, by p s r R T , 27 Ž . a d ` e s r R T . 28 Ž . v ` For the mass conservation of water substance in cloud, q s q q q . 29 Ž . v0 c v Ž . Ž . Ž . Ž . Eqs. 19 – 22 together with Eqs. 23 – 28 are to be solved in the present cloud model. 2.5. Physical processes outside of cloud In the present parcel model, although temperature, humidity and other physical quantities inside and outside of cloud are different, atmospheric pressure p is assumed to be the same at the given height, the hydrostatic approximation. In order to calculate Ž . p, we use the dry adiabatic or US standard atmosphere. Assuming temperature T and pressure p are given, then the height of cloud base above the sea level, z , can be calculated as b R rc d p T p b z s 1 y , 30 Ž . b ž g p where g s 0.988Cr100 m, and T and p are temperature and pressure at sea level, respectively, p s 1013.15 hPa R rc d p p T s T . 31 Ž . b ž p b The height of the parcel above sea level is t 2 z s z q w d t. 32 Ž . H b u t 1 If updraft velocity w s constant, then u z s z q w D t , 33 Ž . b u where D t s t y t . Then the temperature and pressure outside cloud at the height of 2 1 parcel are T z s T y g z , 34 Ž . Ž . and R rc d p T z Ž . p z s p , 35 Ž . Ž . ž T Ž . where p z is the pressure both inside and outside of cloud at the height of the parcel. 2.6. Computation procedure The procedure of our computation is as follows. First, when cloud base conditions, Ž . e.g., p , T , S-1 s 0 are decided, conditions outside of cloud are computed, such as b b b Ž . Ž . Ž . pressure p z , temperature T z , relative humidity RH below and above cloud base, which gives the vertical pressure profile inside of cloud, and for the start of haze particle growth. For comparison, the cloud base conditions are taken the same as that used in Ž . Ž . Twomey 1959, 1977 ; p s 800 hPa, T s 108C, and RH s 100. This leads to the b b b starting conditions of haze growth process below the cloud base: p s 809.21 hPa, Ž . T s 10.938C, and RH s 95. Then for prescribed size groups of dry solute and properties of the corresponding solution droplets considering van’t Hoff factor as a function of molality, the critical radius and the critical supersaturation are evaluated to obtain supersaturation change and haze size evolution below the cloud, which will provide the initial condition for nucleation and growth of droplets above the base. The nucleation of droplets takes place and droplet size distribution forms as n y n jq 1 jy1 f r s . 36 Ž . Ž . j 2 r y r Ž . j jy1 The subsequent estimation of droplet growth process involves estimation of temperature, supersaturation and mixing ratios of water vapor and liquid water which leads to the maximum supersaturation. For the Maxwellian droplet growth system, our model results for cloud supersatura- Ž . tion are checked with that of Rogers and Yau 1989 . In the DK droplet growth system, Ž . the predictions of our model are compared to those of the model by Young 1993 which uses the droplet growth equation of Fuchs. The output of the present model agrees with that of others when compared under the same condition.

3. Results and discussion