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Constraints on droplet growth in radiatively cooled stratocumulus clouds Austin, Philip H. 2011

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JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 100, NO. D7, PAGES 14,231-14,242, JULY 20, 1995 Constraints on droplet growth in radiatively cooled stratocumulus clouds P. H. Austin Atmospheric Sciences Programme, Department of Geography, University of British Columbia, Vancouver, Canada S. Siems Depaxtment of Mathematics, Monash University, Clayton, Victoria, Australia Y. Wang Atmospheric Sciences Programme, Department of Geography, University of British Columbia, Vancouver, Canada Abstract. Radiative cooling near the top of a layer cloud plays a dominant role in droplet condensation growth. The impact of this cooling on the evolution of small droplets and the formation of precipitation-sized drops is calculated using a microphysical model that includes radiatively driven condensation and coalescence. The cloud top radiative environment used for these calculations is determined using a mixed- layer model of a marine stratocumulus cloud with a subsiding, radiatively cooled inversion. Calculations of the radiatively driven equilibrium supersaturation show that net long wave emission by cloud droplets produces supersaturations below 0.04% for typical nocturnal conditions. While supersaturations as low as this will force evaporation for droplets smaller than • 5 pm, radiatively enhanced growth for larger droplets can reduce the time required to produce precipitation-sized particles by a factor of 2-4, compared with droplets in a quiescent cloud without flux divergence. The impact of this radiative enhancement on the acceleration of coalescence is quivalent to that produced in updrafts of 0.1 - 0.5 m s -1, and varies linearly with the total emitted flux (the "radiative exchange"). 1. Introduction Long wave cooling at the top of the cloud-capped boundary layer can exert a controlling influence on layer dynamics and cloud microphysics. Observations and models show that stratocumulus clouds beneath a dry inversion are subjected to cooling rates of 7-10 K hr -1 [Caughey and Kitchen, 1984], with substantially larger flux divergences possible close to the top of thick clouds [Davies and Alves, 1989]. The cooling is due primar- ily to emission from droplets, and this emission allows larger cloud droplets to shed heat efficiently and grow at rates more than an order of magnitude greater than those experienced in the center of a quiescent cloud [Roach, 1976; Barkstrom, 1978]. The growing droplets deplete the available vapor and reduce the ambient su- persaturation as they cool the air; these two opposing influences force the cloud parcel to its equilibrium su- Copyright 1995 by the American Geophysical Union. Paper number 95JD01268. 0148-0227 / 95 / 95JD-01268505. O0 persaturation, which is determined by the droplet size distribution, the net flux divergence, and the partition- ing of the long wave absorption between cloud droplets and water vapor [Davies, 1985]. The value of this ra- diatively forced equilibrium, together with the effects of convection and entrainment, determines the conden- sation growth for droplets smaller than • 5 pm near cloud top. Droplets larger than 10 pm can grow even under very low supersaturations through direct cooling to space. The details of this radiatively driven droplet growth may be particularly important in 200- to 400-m-thick layer clouds. Given droplet concentrations between 50 and 150 cm -s, condensation growth produces 10-14 pm mean droplet radii near cloud top, with maximum liquid water mixing ratios of Wl • 0.4- 0.8 g kg -1. Au- toconversion rates (the rate of formation of embryonic precipitation particles) given these distribution param- eters are several orders of magnitude below those typ- ical of deeper cumuli with larger liquid water contents [Austin et al., 1995], but both observations [Nicholls, 1984; Austin et al., 1995] and models [Nicholls, 1987; Baker, 1993; Austin et al., 1995] indicate that signif- 14,231 14,232 AUSTIN ET AL' CONSTRAINTS ON DROPLET GROWTH icant precipitation can be produced by these clouds. One factor that can accelerate precipitation develop- ment for shallow clouds is the condensation growth of cloud droplets in updrafts [Kovetz and Olund, 1969; Jonas and Mason, 1974]; radiative cooling at cloud top might be expected to have a similar impact on acceler- ating the onset of precipitation in layer clouds. Below we will use model-derived radiative profiles dEn dz ,i) • (/•drops,i q- /•gas,i)4 (Bi - 0.5(El + + E•-)) = (ndrops,i q-ngas,i) 4Ed,i __ (dEni q- (dEni - k, am' ) drop s am' ) gas (1) where /•drops,i and /•gas,i (m -•) are the absorption co- from typical marine stratocumulus clo ds to tudy the efficients i  band i;Bi, Ei +, and E? (W m -2) are the condensation gr wth ofradiatively-cooled clou  roplets, Planck function and the upward and downward i ra- the evolution f the equilibrium supersaturation, and diances; the subscript n denotes the net upward flux the initiation of precipitation. The cloud droplet ra- diative environment will be specified using the model of $ierns et al. [1993] (referred to below as SLB); it gives profiles of temperature, vapor mixing ratio, and the upward and downward long wave fluxes for a noc- turnal cloud in which a subsiding layer of overlying air determines the radiative balance at cloud top. In Section 2 we derive an expression for the equi- librium supersaturation, Seq, and calculate an upper bound on Seq for typical cloud radiative and microphysi- cal conditions. We use the SLB model to obtain vertical profiles of the total net emission (termed the "radia- tive exchange" by Roach [1976]) and the fraction of the total cooling due to cloud droplets and model droplet condensation growth given this radiative environment. In Section 3 we calculate droplet coalescence near cloud top and compare it to coalescence in updrafts typical of nocturnal stratocumulus clouds. Section 4 contains a discussion of these results. Ei + - E•-; and z is the height (see the Notation list for a full list of symbols). The thermal emission and the upward and downward irradiances combine to give the radiative exchange de- fined by Roach [1976] as the net power leaving a droplet of radius r per unit area per unit absorption efficiency: Ed,i- (Bi- 0.5(El + q- E?)) (2) Given n(r)dr (kg -•), the mixing ratio of droplets with radii between r and r + dr, and Qa(r,i), the absorption efficiency for droplets in spectral band i, we can write that portion of the flux divergence due to droplets in terms of Fd(r, i) (W), the net power radiating from a droplet of radius r in spectral interval i: ( ) /0 dEn,i -- Pa n(r)•rr2Qa(r, i)4Ed,idr dz drops -- Pa n(r)Fd,i (r)dr (3) 2. Condensation Growth and the Equilibrium Supersaturation where Pa is the density of dry air. 2.2. Droplet Growth Equations 2.1. Radiative Fluxes: The Two Stream Approx- imation We will calculate radiative cooling due to absorption and emission by cloud droplets, water vapor, and car- bon dioxide using the five-band model of Roach and 5'lingo [1979]. The wavelength ranges for the five bands are given by Table 1; the model includes parameter- izations for the transmissivity of water vapor, carbon dioxide, and cloud droplets and can be run at arbitrary vertical resolution. Given model values for Ei + and F,•- we can write the net flux divergence in band i as the sum of separate contributions from the droplets and gas [Bott et al., 1990]: We can calculate droplet growth given Ed,i, the size and composition of the cloud condensation nucleus, and the thermodynamic state variables. The droplet growth equation including radiation is [Roach, 1976] dr ( LvFd ) r• - G (r) S - CK + CR q- 4•rrK,RvT2 (4) where r is the radius, S is the supersaturation, t is the time, and the Kelvin and Raoult terms (CK, Ca) are given in the Notation list. We have dropped the band subscript i from Fd to denote summation over the five bands. For a closed parcel exposed to net flux divergence the conservation equations for energy and water are Table 1. Band Wavelength Ranges for Roach and $1ingo [1979] band wavenumber (cm -• ) A (/•m) 1 0-400 25 - c• 2 400-560 17.9 - 25 3 560-800 12.5- 17.9 4 800-1150 8.7- 12.5 5 1150-2050 4.9 - 8.7 dT -Lv dwv RdT dp 1 den Cpln dt= • -[ (5) p dt Pa dz /o c• dr dwv_ dwl = _4•rp 1 n(r)r2 •dr (6) dt dt where w is the vertical velocity, Wv and Wl are the vapor and liquid mixing ratios, and Cpm - Cpd + (Wv + Wl)Cw is the heat capacity of the cloudy air. AUSTIN ET AL.: CONSTRAINTS ON DROPLET GROWTH 14,233 Together with the hydrostatic equation, (4)-(6) can be solved given an initial droplet number distribution n(r). The supersaturation S can be diagnosed at each time step through the definition S - e 1 - P - 1 (7) es(T) es(T)(1 + e/Wv) If the updraft velocity and/or the flux divergence are steady on timescales longer than a few seconds, it can be shown that the supersaturation calculated by the in- tegration of (4) - (6) relaxes to a quasi-equilibrium value Seq [Roach, 1976; Davies, 1985]. We use the approach of Davies [1985] to derive an expression for Seq in the Appendix; in a closed parcel it is determined predom- inately by the vertical velocity, the net flux divergence and the integral radius, I: Seq • al Ts W -}- CK -- Cr a3 /G/ (1 den) - 47rI• •7 Pa dz drops a3 (1 den) + 47rpla2•CpmI Pa dz total (8) where I - f rn(r)dr, al, a2 and a3 are slowly vary- ing functions of the pressure and temperature and rs - 1/(47rpla2GI) is the relaxation time (see, for example, Cooper [1989]). The overbars in (8) represent an av- erage weighted by the integral radius, while the angle brackets represent an average weighted by the droplet flux divergence (see the Appendix). We show in the Appendix that for typical stratocu- mulus droplet and aerosol distributions, Cr is 2 orders of magnitude smaller than Ck. With this approxima- tion, and assuming w - 0, (8) can be written in a more compact form: Seq • CK (9) • p-• •ZZ total pla2•Cpm T where F - (dEn/dZ)drops/(dEn/dz)total is the fraction of the total flux divergence due to the droplets [Davies, 1985]. We will follow Fukuta and Walter [1970] and use c• - 1, /• - 0.04 in (8)-(10), which produces as • 0.1 /•m, az m 4 pm. As a result, K' is not a strong function of droplet radius, while G increases with increasing droplet size. 2.3. Radiative Fluxes: Cloud Flux Profiles The radiative exchange Ed,i at cloud top and the frac- tional absorption F due to droplets depend on the mag- nitude of the downwelling flux from the inversion, the cloud temperature, and the droplet size distribution. We will estimate these using the SLB model, which at- tempts to establish the interaction between the cloud- capped boundary layer and the overlying air. In this section we will fix the cloud layer thickness at 300 m and let the subsiding inversion evolve to steady state above the cloud for dry (inversion Wv - I g kg -1) and moist (Wv- 5 g kg -1) conditions. Adiabatic clouds of this thickness have large cloud top flux divergences be- neath a dry inversion and produce a broad range of precipitation rates, apparently modulated by the char- acter of the small droplet population. For example, aircraft measurements during the First International Satellite Cloud Climatology Project (ISCCP) RegiOnal Experiment (FIRE) show no drizzle formation in a 27g- to 300-m-thick layer with a droplet number concentra- tion of 150 cm -3, while in adjacent clean air, peak rain rates of 5-8 mm day -1 were observed in a layer of the same thickness but with droplet concentrations below 50 cm -3 [Austin et al., 1995]. We initialize inversion air at a fixed temperature and vapor mixing ratio at a height of 3 km and let it descend with velocity Dz, where z is the height of the parcel and D, the large-scale divergence, is set at a fixed value D - 4 x 10 -6 S --1 (a value typical of the large-scale divergence off of the California coast in July). The in- versioh air warms adiabatically and emits and absorbs long wave radiation as it descends; the inversion tem- perature above cloud top is controlled through the wa- ter vapor mixing ratio, which determines the emissivity, and the subsidence, which determines the time available for the parcel to experience diabatic heating or cooling as it descends. Radiative fluxes are computed at each time ste•) using the Roach and Slingo [1979] model. The model requires everal hours to evolve un.•il the overly- ing, subsiding air is in a quasi-steady state. Following this,:gl•e radiative flux divergence changes very slowly; for our purposes, we consider this the steady state ra- diative flux profile. Figures la and lb show vertical profiles of water va- por mixing ratio, temperature, and liquid water mixing ratio for the 300-m-thick cloud layer and an inversion in which Wv - I g kg -1. The total number concentration is constant at NT -- 50 mg -1 , and the adiabatic in- crease of liquid water with height produces a maximum volume mean radius at cloud top of rvol -- 13.4 pm, where rvol is defined by rvol-- ( 3Wl ) 1/3 47rplNT (10) The fluxes for these thermodynamic profiles are calcu- lated on a grid with variable vertical spacing: Az = 2.5 m in the 10 m below the inversion, followed by two layers with 5-m spacing, 2 with 10-m spacing, and the remainder set to Az = 20 m. Figure lc shows the corresponding values of the up- ward and downward irradiances and the total radia- tive exchange, Ed, summed over the five bands. The warm, dry inversion produces a total downward flux of • 280 W m -2, while the upward flux at cloud top is • 320 W m-2; droplets in this cloud experience values 14,234 AUSTIN ET AL' CONSTRAINTS ON DROPLET GROWTH w v (dashed) (gkg '1) 2 4 6 8 ,, o 2•t0 285 290 295 300 Temperature (solid) (K) w, (dashed) (gkg '1) 0 0.1 0.2 0.3 0.4 0.5 0,6 i i i ! i i b) • • // E d (dashed) (W m '2) 10 20 30 0 0 5 10 15 280 320 360 400 rvo • (solid)(gin) E + (solid) E' (dotted) (W rn '2) Figure 1. Vertical soundings for a 300-m-thick layer cloud beneath a quasi-steady state inversion (inversion wv - 1 g kg-1). (a) temperature T and vapor mixing ratio wv; (b)volume mean radius r•oi (with total number concentration NT -- 50 mg -1) and adiabatic liquid water mixing ratio wi; (c) total upward and downward irradiances E+ and E- and the total radiative exchange Ed of the radiative exchange varying from 2 to 35 W m -2 through the upper 80 m of the cloud layer. The net flux divergence dEn/dz (not shown) varies from 0.1 to 2.6 W m -3 over the same 80-m scale. Increasing the emissivity of the overlying air signifi- cantly reduces both the flux divergence and the radia- tive exchange. This is shown in Figure 2a, which gives the quasi-steady state radiative profiles for a subsiding inversion with w• - 5 g kg -x atop the mixed layer of Figure 1. The peak value of the radiative exchange is reduced by roughly 50% for this cloud, to 18 W m -2 at cloud top, while the maximum net flux divergence is now 1.6 W m -3 at cloud top (not shown). Figure 2b shows the impact of droplet removal on the radiative exchange. The liquid water content has been reduced by 50% at each cloud level by remov- ing droplets, consistent with, for instance, precipita- tion scavenging. The lower cloud emissivity reduces the downwelling irradiance and increases the radiative exchange near cloud top; Ed remains above 5 W m -2 through the upper 100 m of the cloud. The 50% re- duction in cloud water also halves the flux divergence through (3), with a new cloud top flux divergence of 1.3 Wrn -•. In Figure 2c, the fraction of the cooling due to droplet emission F is shown for the adiabatic layer of Figure lc and the subadiabatic layer of Figure 2b. It is estimated using (1): •--T) drops • •drops (11) (•z)total •drops q- •gas For the subadiabatic and adiabatic clouds the droplets account, respectively, for approximately 80% and 87% of the net flux divergence; these relative contributions change little through the upper 100 m of the cloud. 2.4. Constraints on Seq At fixed temperature and pressure, we expect Seq to be maximum in (10) for large flux divergences and small __ __ values of F, G, and I. Table 2 lists values of G and I for a range of drop size distributions specified by the modified gamma function of Berry and Reinhardt [1974a]' n(rj) = (12) (l+y)(l+v) (r?) v [ NT •vo••'  • _-•-- exp --(1 + 3 rvol rvol ] where rj is the radius of size class j, NT is the total number mixing ratio, rvol is the volume mean radius, and y is a parameter related to the dispersion of the number distribution. Table 2 also lists two other mea- sures of the distribution moments, rg and rb, which will be discussed in Section 3. Observations from the FIRE indicate that the droplet spectrum below 23 /•m radius can be fit to distribu- AUSTIN ET AL.' CONSTRAINTS ON DROPLET GROWTH 14,235 E d (dashed) Ed (dashed) (W m '2) (W m '2) 0 10 20 30 0 10 20 30 280 320 360 400 280 320 360 4•10 E + (solid) E'(dotted) E + (solid) E'(dotted) (W m '2) (W m '2) subadiab. .......... adiab 0 0.2 0.4 0.6 0.8 1 F Figure 2. (a) As in Figure lc but for a moist inversion (Wv - 5 g kg -1). (b) As in Figure lc but for a cloud with awl profile that is 50% of adiabatic. (c) Fraction F of the net flux divergence due to droplet cooling. tions of the form of (13) with values of v ranging from 0.2 _< v _< 2.2 [Austin et al., 1995]. The parameter F is related to the dispersion (fr/Y) where fr is the standard deviation of n(r) and Y is the mean radius by (13) 7.14(1 + Thus 0.2 _< • <_ 2.2 is equivalent o 0.2 <_ fr/• • 0.35, in approximate agreement with dispersion values found' in North Atlantic stratocumulus by $lingo et al. [1982]. The radiative profiles of Figures 1-2 were computed for distributions of the form of (13) with • - 0.3, NT -- 50 mg -1, and 25 droplet size categories with rj between 1 <_ rj <_ 25/•m. Substituting the other distributions of Table 2 (while keeping the dry inversion, 300-m cloud thickness, and adiabatic liquid water profile) changes the cloud top cooling rate by less than 20%. Although I, F, and G will vary with local changes in the droplet distribution, other coefficients in (8) re- main approximately constant over a broad range of temperatures, pressures, and drop size distributions. The thermodynamic coefficients al, a2, and a3 vary by less than 15% in the temperature and pressure range 273 K < T < 293 K, 1000 hPa < p < 800 hPa. The microphysical coefficients Cr, Ck, K • depend weakly on the drop size distribution, but can be considered con- stant for the range of distributions given by Table 2. Table 3 shows values for each of these parameters for a cloud top temperature and pressure of T=283.5 K, p=954.5 hPa. The equilibrium supersaturation is held within a com- paratively narrow range for the coefficient values of Ta- bles 2 and 3. The second term in (10) remains greater than zero for all values of G, so that the minimum Seq is set by Ck -- 10 -4. To compute an upper bound on Seq Table 2. Gamma Distributions for Fixed wi -- 0.5 g kg -1 Case rvo] Wl NT v Err/• rg (/zm) (g kg -1) (rag -1) rb (•um) I (mkg -1) (/tin 2 G 1 10 0.5 119.36 0.45 0.31 11.91 2 10 0.5 119.36 0.30 0.33 12.09 3 10 0.5 119.36 0.00 0.37 12.60 4 11 0.5 89.68 0.45 0.31 13.10 5 11 0.5 89.68 0.30 0.33 13.30 6 11 0.5 89.68 0.00 0.37 13.86 7 12 0.5 69.08 0.45 0.31 14.29 8 12 0.5 69.08 0.30 0.33 14.51 9 12 0.5 69.08 0.00 0.37 15.48 10 14 0.5 43.50 0.45 0.31 16.67 11 16 0.5 29.14 0.45 0.31 19.06 9.40 9.57 10.00 10.34 10.53 11.00 11.28 11.49 12.00 13.16 15.03 1103.8 1093.9 1065.9 912.2 904.1 880.9 766.6 759.7 740.2 563.2 431.1 76.4 76.3 76.1 77.5 77.4 77.2 78.4 78.3 78.1 79.9 81.0 14,236 AUSTIN ET AL.- CONSTRAINTS ON DROPLET GROWTH Table 3. Coefficient Values for (8) Coefficient Value T, K 283.5 p, hPa 954.5 al, m-1 5.2 x 10 --4 a2 283.1 as, j-1 K kg 0.067 pl, kg m -3 1000 Ck 1.0 X 10 -4 C--• 9.7 x 10 -7 Cpm, J kg- 1 K- 1 1020 K •, J (m s K)- 1 0.024 pa, kgm -3 1.17 r.,, s 4.6 using the conditions suggested by Figures 1 and 2, we take F - 0.87, (dEn/dz)total=10 K hr -1 (3.3 W m -3) and the drop size distribution of Case 11. This yields Seq - 3.7 x 10 -4. The equilibrium supersaturation de- creases as NT increases' inserting (I, G) for Case 1, Table 2 halves the radiative contribution to Seq and re- duces it to S•q = 2.2 x 10 -4. These low S•q values preclude any significant radiatively-induced aerosol ac- tivation at the tops of the cloud layers presented here; roughly 97% of the CCN distribution described in the Appendix require supersaturations larger than 0.037% for activation. 2.5. Condensation Growth Calculations The equilibrium values of the supersaturation for the examples given above are small enough so that in the absence of upward motion, droplets with radii less than m 5/•m evaporate throughout much of the upper part of the clouds shown in Figures 1-2. Figure 3 shows the growth histories of 25 droplet classes (solid lines) com- puted by an explicit integration of (4)-(6) for conditions taken from Figure 1. For this case we locate the drop size distribution 40 m below cloud top, in a parcel of adiabatic cloud base air that remains at that height as it cools. The value of Ed is 7.3 W m -2 at this height, while the initial drop size distribution has an adiabatic liquid water content of 0.15 gkg -1 and rvo1-13.3 /•m. The droplets and vapor produce a combined net flux divergence of 0.93 W m -3, with an initial cooling rate (including the effects of phase change) of-1.6 K hr -1. Over the course of•21 minutes in this environment the cooling decreases the parcel temperature by 0.25 K and lowers the parcel saturation level (the pressure at which the parcel would be exactly saturated) by approx- imately 10 hPa. The solid lines in Figure 3 show the growth histories of individual droplet categories during this time. The critical radius and supersaturation for this aerosol mass are 1.48 /•m and 5 x 10 -4 respec- tively, so that the smallest droplets assume their un- activated equilibrium radii, while droplets smaller than 5/•m evaporate until they deactivate. Droplets larger than 10/•m grow at between 0.10 and 0.14/•m min-1, 10 6 5 S (x 105) (dashed) 12 14 16 18 20 ,. I . I I I v 2'o 2'5 3'0 radii (solid) (gin) Figure 3. Evolution of an adiabatic drop size distribu- tion 40 m below cloud top. Solid lines show the droplet radii for size classes 1-25 as a function of time and par- cel saturation pressure. The dashed and dotted lines show the equilibrium supersaturation calculated by (8) (dashed line) and (10) (dotted line). with a growth rate that varies linearly with Ed for fixed droplet size. The supersaturation decreases during the cooling, as larger droplets increase the distribution mean radius (and therefore increase I) by 1.1 /•m from 12.2/•m to 13.3/•m. The flux divergence decreases by • 4% over this time period (not shown), as reduced thermal emis- sion is offset by an increase in the droplet absorption coefficient ndrops. The dashed line on Figure 3 shows Seq given by (8); it differs from the actual supersatu- ration by less than 0.1% over the course of the inte- gration. Also shown is the approximate value given by (10), which is within 2% of the more accurate value. Figure 4 shows the contribution of individual terms in the droplet growth equation (4) to the growth rate at the beginning of the integration. Both the super- saturation term ("S") and the radiative cooling term (CE = LvFd/(4•rrK•RvT2)) contribute to the growth of 5-/•m cloud droplets, while the aerosol mass (term R) has little impact on the threshold size at which drops begin to evaporate. Droplets smaller than 3/•m will not evaporate provided S•q > 0.03%, which would be pro- duced by a radiative exchange greater than • 25 W m -2 for this drop size distribution. From Table 3 and (8), S•q > 0.03% would also be produced by updrafts at cloud top larger than m 0.1 ms -1. 3. Coalescence With Radiative Cooling As the value of the radiative exchange increases be- yond 10 W m -2, the radiatively driven growth rates of AUSTIN ET AL.- CONSTRAINTS ON DROPLET GROWTH 14,237 o_ o- -1 -0.5 0 0.5 I 1.5 2 / 15 volume cjrowth rate (gin3 s-1) Figure 4. Volume growth rate 47rr2dr/dt as a func- tion of radius for a radiative exchange Ed-- 7.5 W m -2 and an equilibrium supersaturation f 1.5 x 10 -4. The symbols designate the terms from (4): S = S, K = CK, R = Ca, E=cooling term, T = S + K + R + E the larger droplets exceed those produced by typical up- drafts in stratocumulus clouds. Figure 5 shows the indi- vidual terms of (4) (given a total radiative exchange of Ed = 17.5 W m -2) compared with the supersaturation term assuming S=10 -3 (generated, for instance, by a 0.35 m s -1 updraft acting on the initial drop size distri- bution of Figure 3). Radiation has a substantial impact on the growth of the larger droplets; it is the droplets in the 20-25 /zm size range that determine the initial coalescence growth rate of the condensation-produced droplet size distribution. Coalescence is particularly volume growth rate (gm 3 s '1) Figure 5. As in Figure 4 but for Ea- 17.5 Wm -2. Line "S" shows the growth rate due to a supersaturation of S- 1. x 10 -3 for comparison. sensitive to size increases in these droplet categories, because the collision efficiency increases from 0.02 to 0.2 as droplets grow through this 5-/zm radius range [Jonas, 1972]. Even modest condensation growth can have a rate-determining impact on the early stages of coalescence if it accelerates this droplet growth. In this section we will calculate coalescence growth for a sedimenting droplet population in a region of constant radiative exchange and in an updraft with a constant vertical velocity. Berry and Reinhardt [1974b] showed that a representative measure of the progress of coa- lescence in an evolving droplet population is given by the mass mean, or "predominant" radius, rg. This is defined by xg = / x2n(x)dx//xn(x)dx ( 3 ) (1/3) rg -- 47rpl (Xg) (1/3) (14) where x is the droplet mass. Values of rg for precipi- tating stratocumulus observed during FIRE range from 45 < rg < 75 /zm for midcloud rain rates of 1 - 10 mm day- 1. We will use Tg, the time required for rg to grow to 50/zm, as a measure of the rate of spectral evolution of an initial drop size distribution. Berry and Reinhardt [1974b] found that for a coalescing droplet population (without cooling or supersaturation growth), 1/Tg was linearly related to the liquid water content of the initial distribution and to rb, a measure of the initial norrnal- ized mass variance: ((x2)- (x)2) --1/(1 +.) (15) nvar x = -- r} nvar -- rvøl -- rvøl 1 q- F where the angle brackets define an average over the mass distribution. We calculate the evolution of rg for the initial dis- tributions of Table 2, integrating the stochastic col- lection equation using the algorithm and the kernel of Berry and Reinhardt [1974a] (with collection efficiencies taken from Hocking and Jonas [1970] and $hafrir and Neiburger [1963]). Coalescence growth is calculated us- ing 60 logarithmically spaced radius bins between 2 and 1800/•m radius, and for each 2-s time step we also com- pute condensation growth using the advection equation: Ot cond Or • We solve (17) using the semi-Lagrangian advection scheme of Bott et al. [1990] and the droplet growth equation (4) with S = CK = Ca = 0 (for the cool- ing cases), and S = Seq, CK : Ca = CE : 0 (for the 14,238 AUSTIN ET AL- CONSTRAINTS ON DROPLET GROWTH updraft cases). The pressure and temperature are fixed to their values in Table 3. For this calculation we make the approximation that 5 Fd -- 7rr 2• Qa(r, i)4Ed,i • 7rr2•aa(r)4Ed (18) i=l where the average absorption efficiency, Qa is taken from Roach [1976]' Qa - 1.18(1 - exp(-0.28r)) (19) with r given in microns. Although this approximation overestimates Fd by 5-30% for droplets with radii be- tween 5 and 15/•m, the errors decrease with increasing drop size, and are below 4% for drops larger than 20 Figure 6 shows the time evolution of rg for four dif- ferent values of the radiative exchange Ed (3.75, 7.5, 15, 25) W m -2 and three different initial drop size dis- tributions (Cases 1, 4, and 7 in Table 2). Also shown for Case I is the evolution with neither cooling or ver- tical ascent ("n") and with a constant vertical velocity of w - 0.5 ms -1 (initial Seq -- a•rsW - 8.3 x 10-4). Figure 6 shows that coalescence without cooling or ascent (line "n") proceeds very slowly for this kernel, with the predominant radius rg growing at roughly 6 x 10 -3 /•mmin -•. Placing the parcel in a steady 0.5 m s -• updraft (line "w") increases this growth rate by I I I I 12 14 16 18 % Figure 6. Stochastic coalescence with radiative cool- ing: predominant rg versus time for three different drop size distributions (Cases 1, 4, and 7) and four values of the radiative exchange Ed = 3.75, 7.5, 15, 25 W m -2. Solid lines are rvoi = 10/•m. Dotted lines are rvol = 11/•m. Dashed lines are rvoi = 12/•m. Also shown are growth curves with neither adiabatic ascent or radiative cooling ("n") and in a constant updraft of w = 0.5 m s -• ("w") without radiation. a factor of 21. The initial growth rate of the predomi- nant radius is independent of rg and proportional to the radiative exchange, varying for each distribution from 0.05 •um min -1 for a radiative exchange of 3.75 W m -2 to 0.3 •ummin -• for Ea=25 W m -2. For the smaller rvo• distributions at values of Ea less than 15 W m -2, there is a transition for 14 < rg < 16 •um, in which the growth rate increases by up to an order of magnitude. This change is due to rapidly increasing collection by droplets with radii greater than 30 •um (and collection efficiencies approaching unity). In Figure 7 we plot rg versus 1/Tg for four radiative exchanges and three values of the updraft velocity, with the initial distributions of Table 2. The rate of spectral evolution varies linearly with the value of the radia- tive exchange for these initial distributions. The w = 0 curve gives a baseline against which the impact of cool- ing and ascent can be compared: a radiative exchange of 15 W m -2 or an updraft of 0.5 ms -• produces rates of precipitation development (given rvol ---- 10/•m) equiva- lent to those found in a quiescent cloud with a distribu- tion rvoi • 15/•m. The time required for the distribu- tion to evolve to rg = 50 /•m decreases from 81 min for Case I (no cooling) to 20.7 min with Ed -- 25 W m -2. Comparison of the slopes of the curve families in Fig- ure 7 shows different sensitivities to the initial distribu- tions for coalescence aided by supersaturation or radia- tive cooling. radiatively driven droplet growth is inde- pendent of droplet radius once Qa (r) reaches its asymp- totic limit (at r > 15 /•m). In contrast, saturation- driven growth for 20 /•m droplets in an updraft de- creases as the droplets grow, because of the 1/r depen- rg (pm) 12 14 16 18 I I I I I r b (gm) -0 Figure 7. Time Tg required to reach rg -- 50 /•m for the initial distributions of Table 2. Lines are least square fits through the cases, labeled by the appropri- ate value of the total radiative exchange Ed or updraft velocity w. AUSTIN ET AL' CONSTRAINTS ON DROPLET GROWTH 14,239 dence in (4) and because of the decrease in Seq as the integral radius increases in (8). 4. Discussion We have modeled radiatively driven droplet growth near cloud top for a 300-m-thick stratocumulus cloud beneath moist and dry inversions. Our results indicate an upper bound on the radiatively-induced supersatu- ration of less than 0.04%. Equation (10) suggests that this maximum might increase in thicker clouds with a larger total flux divergence, in clouds with lower values of I, or in layers in which vapor absorption constituted a larger fraction of the total layer absorption. Countering such an increase is the coupling of the integral radius and the flux divergence through (3); in the absence of precipitation, an increase in dEn/dz due to increased cloud liquid water content (and the resulting increase in •;drops) will be offset in (10) by the corresponding increase in I. If precipitation is considered, observations suggest a second limit, as clean marine clouds with low total droplet concentrations produce drizzle, which removes liquid water and reduces the emissivity (and flux diver- gence) of the cloud. Measurements from FIRE suggest precipitation scavenging sufficient to halve the cloud liquid water path in less than 25 min in 250- to 300- m-thick clouds with I m 650 m kg -• (NT -- 50 mg -•, rvo• -- 13 •um) [Austin et al., 1995]. This leaves a reduc- tion in F as the third possibility for increased values of Seq, but the results of Section 2.3 indicate that lower values of F occur in subadiabatic layers, with reduced liquid water paths and correspondingly lower values of dEn/dz. One source of uncertainty in these calculations is the value of the accommodation and condensation coeffi- cients. Growth measurements of water droplets held in electrodynamic balance [Sageev et al., 1986] seem to in- dicate a • I as used above. There is less support for the usual choice of/3 - 0.04; recent measurements of /3 [Hagen et al., 1989] show values that vary logarith- mically with droplet size for droplets grown in a fast expansion chamber, with /• decreasing from I to 0.01 as the droplet radius increases from I to 15 •um. Hagen et al. suggest that this variation is related to increas- ing concentrations of surface contaminants on the older (and larger) drops. If we use their average/3 for aged droplets (/3 - 0.01 + 7%) and recompute G we find a 30% reduction in G for the conditions of Table 3. This would increase the radiative contribution to Seq by • 30-40%. The stochastic coalescence calculations presented in Section 3 show a linear relationship between the radia- tive exchange, Ea, and the growth rate of the predomi- nant radius for fixed liquid water content. The growth rates shown in Figure 6 suggest that 10 min exposure to a radiative exchange of Ea - 7.5 W m • would increase the predominant radius of the distributions of Table 2 by • 1 •um. This is equivalent to a similar period spent in an updraft of 0.2 - 0.25 m s -•. For the initial dis- tribution with rvo• - 12 •um in Figure 6, this I •um increase represents a significant portion of the growth needed to move to the more rapid stage of coalescence at rg > 16 •um. An equivalent effect can be produced by higher values of the radiative exchange and propor- tionally shorter residence times in the upper part of the layer. A lower bound on the cloud top residence time can be found by assuming that the mixed layer circulation is organized, consistent with balloon and aircraft mea- surements of nocturnal clouds [Caughey and Kitchen, 1984; Nicholls, 1989]. These observations indicate that air parcels rise to cloud top, cool by 0.1-0.2 K, and then descend in downdrafts spaced 100-150 m apart. Given a convective velocity scale w. • 0.5 ms -• (inferred from either the observations or our mixed layer model), this implies a residence time at cloud top of 3-5 min, a flux divergence of 1.6 W m -3 (or Ed • 15 W m-2), and an rg increase of • I •um in a parcel completing this circu- lation. That residence time could be significantly extended by the entrainment of inversion air, since the cloud- top entrainment instability criterion [Randall, 1980] is not met for either the moist or dry inversions of Fig- ure I and Figure 2. Mixtures of inversion and cloud air will be more buoyant than surrounding cloud and will require more cooling to produce the 0.1-0.2 K temper- ature deficit of the descending plumes. As Figure 2b indicates, removal of cloud water has little impact on the value of the radiative exchange, and droplets that survive mixing with inversion air will grow as rapidly as their neighbors in adiabatic cloud, while the reduced emissivity will lower the parcel cooling rate. These mixed parcels, with reduced number concentrations, will also experience comparatively large equilibrium su- persaturations in cloud-top updrafts. A parcel with an integral radius of I - 200 m kg -•, experiencing a ver- tical velocity of 0.1 - 0.2 m s -• will produce Seq from 0.1% - 0.2% from (8); from Figure 5 this would more than double the growth rate for a 20-•um drop in a ra- diative exchange of 17.5 W m -2. A more complete treatment of the radiative contribu- tion to spectral broadening should consider correlations between perturbations in the radiative exchange, the vertical velocity, and the integral radius. Cooper [1989] has shown that given high (and uncorrelated) variability in I and w, mixing between parcels with different histo- ries could have a substantial broadening effect on mean droplet spectra. Fluctuations in Ed could act as another source of variability in I, as droplets residing in the up- per 10-20 m of the cloud are exposed to significantly different values of the radiative exchange. These radia- tive exchange fluctuations would be particularly large for entrained parcels with low I and a few large, sur- viving droplets, because the removal of overlying cloud would reduce the downward flux and increase Ed. We 14,240 AUSTIN ET AL.' CONSTRAINTS ON DROPLET GROWTH have also neglected the impact of supersaturation fluc- tuations at the scale of individual cloud droplets. Our Seq is the traditional far-field supersaturation, which may differ significantly from the value near the droplet surface. Srivastava [1989] has shown that this kind of supersaturation variability also has the potential to sig- nificantly broaden the drop size distribution. We plan to calculate the cumulative impact of tur- bulence, entrainment, and radiative cooling on large droplets cycling through cloud using large eddy sim- ulations. Recent work with a simple one-dimensional turbulence model, however, does suggest that conden- sation growth may play a significant role in the initia- tion of stratocumulus precipitation [Austin et at., 1995]. The results presented here indicate that radiative cool- ing has a similar potential, given a dry inversion and a cloud with peak rvol > 10/•m. Appendix- The Equilibrium Supersaturation A prognostic equation for the supersaturation can be found by differentiating its definition [Pruppacher and Ktett, 1978]' dS [ gw Lv dT t RdT RwT 2 t pRw dwv -} (A1) esRd dt where we have made the approximation (l+S)=l. To obtain (8), we begin by inserting (4) into (6) and integrate over the drop size distribution: dwv -- 4•pl [ISm- IGCk -}- IGCrJ dt Lv /G / / I den] (h2) - Pl RvT 2 •7 • dz 7 drops where we have used (3) to define the flux divergence due to droplets. The overbars in (A2) denote an average weighted by the integral radius, while the angle brackets denote an average weighted by the droplet cooling: xn (r) rdr xn (r)Fd (r)dr , <x)- (A3) X-- oo oo /0 n(r)rdr /0 n(r)Fd(r)dr For the distributions of Table 2 the use of separate integral radius-weighted averages for the coefficients in (A2) introduces errors in each term of less than 2%: GCk • GCk GCr • G Cr •7 • Kt (A4) dt When we substitute (5) into (A1) we obtain for the supersaturation dS • alw -- a24•rplI (•S - GCk + GCr) (G>(1 den) + a2pla3 •7 Pa dz drops + a3 (1 den) Cpm Pa dz total (A5) Solving (A5)'using a multiplying factor yields (8). We require an aerosol distribution to evaluate Cr and will make the simplifying assumption that each droplet has formed on an identical aerosol particle, consisting of ammonium bisulphate with a dry aerosol diameter of 0.2 •m. Observations indicate that sulphate is the prin- cipal constituent of the remote marine aerosol; observed mass distributions can be fit to a lognormal distribution with a geometric mean diameter of 0.2/•m and a geo- metric standard deviation of 1.7 [Ctarke et at., 1987; Twohy et at., 1989]. Particles with dry diameters less than 0.2 •m constitute half of the total aerosol mass and 95% of the total aerosol number available as cloud condensation nuclei for this choice of distribution pa- rameters. As the Cr values of Table 3 and Figure 4 indicate, our results are not sensitive to the choice of aerosol size. The result given by (8) differs slightly from that pre- sented in equation (20) of Davies [1985]. To permit a term by term comparison with his expression, we define a new average for the ratio {Kt/D t} that satisfies ø•n(r)Fd dr (A6) dr [TR,• K' L2• ] + With this definition and using the expression for F given in (10), (8) becornes Seq • Ck--Cr (A7) ([ a4 - (4•rI•) •7 Pa dz total X [Fpacpm - (1 - F) L2vps { KtR•T Writing Davies (20) in our notation Seq,Davies • Ck -- Cr - ( X [Fpacpm-a5(1- F)psLv AUSTIN ET AL' CONSTRAINTS ON DROPLET GROWTH 14,241 There are two offsetting approximations in (A8). The extra 1/T term that appears in both the numerator and denominator of (A8) (and is also found in the droplet growth equation used by Roach [1976]) arises from the assumption that the vapor flux toward the droplet can be written in terms of the vapor density gradient V ps. This is strictly true only under isothermal conditions. If the gradient V(nv/n) (where nv and n are the va- por number density and the total number density of moist air) is used instead, the droplet growth equation takes the form of (4), which is the same as that used by Cooper [1989] or Srivastava [1989] (W. A. Cooper, per- sonal communication, 1993). The 5% decrease in the bracketed term in (A8 / caused by the addition of the 1/T term is very nearly offset by the approximation that K•/D • • •K•/D•). As a result, (A7) and (A8) agree to within 1-2% for the droplet distributions of Table 2. Notation a• a• al a2 a3 a4 a5 a6 Bi CE CK Ca Cv Cpd Cw Cpm D D • e es Ed,i En,i accommodation length, K(2•rRMdT)•/2/(ap(Cv + R/2)). condensation length, (2•r/(RvT))i/2D//•. T cpRvT Ra ' (pa/ps) + (L•/(RvcpmT•)). Lv/(evT2pa). Lv/(CpmPaRv T2 + L•ps). (Lv/(RvT•)) - (l/T). a•/(Cp•pa + Lvpsa•). Planck function in band i. cooling term, LvFd(r)/(4•rK•RvT2). Kelvin term, 2fs/(p•rRvT). Raoult erm, vmM/(W ((4•/3)p•r 3- m)). molar heat capacity for water vapor. specific heat at constant pressure for dry air. specific heat for liquid water. specific heat at constant pressure for cloud parcel. diffusivity of water vapor in air. f•D [Fukuta and Walter, 1970]. vapor pressure. saturation vapor pressure. radiative exchange in band i. net upward irradiance in band i. upward irradiance in band i. downward irradiance in band i. r/(r + a•). r/(r + a•). surface tension. F Fd,i G(r) K K • I Lv Q•(i, r) m M NT nv n p r rb rg rvol R Rv S Seq Sv T T• w w1 Wv fraction of the flux divergence due to droplets. net power emitted by droplet in band i. TRy L• p•-i D_r•e, _[ - T 2 K•R• ß thermal conductivity of air. f•K [Fukuta and Walter, 1970]. integral radius, I- f n(r)rdr. latent heat of vaporization. absorption eificiency in band i for droplet of radius r. aerosol mass. molecular weight of aerosol. molecular weight of dry air. total number mixing ratio. vapor number density. total number density of moist air. total pressure. droplet radius. normalized mass variance radius. predominant radius. volume mean radius. universal gas constant. gas constant for water vapor. supersaturation e/es- 1. equilibrium supersaturation. dry virtual static energy. temperature. time required for coalescence to produce rg -- 50 •m. Van't Hoff factor vertical velocity. liquid water mixing ratio. water vapor mixing ratio. molecular weight of water. Acknowledgments. We would like to thank W. A. Cooper for pointing out the consequences of isothermal va- por diffusion discussed in the Appendix, and two anonymous reviewers for helpful suggestions. This article was type- set in DTF_• using the AGUTEX JGR style and Patrick Daly's AGU ++ article class. The research was supported by grants from the National Research Council of Canada, the Atmospheric Environment Service, and NOAA Grant NA37RJ0203. References Austin, P., Y. Wang, R. Pincus, and V. Kujala, Precipita- tion in stratocumulus clouds: observational and modeling results, J. Atmos. $ci., 52, 2329-2352, 1995. 14,242 AUSTIN ET AL.: CONSTRAINTS ON DROPLET GROWTH Baker, M. B., Variability in concentrations of cloud conden- sation nuclei in the marine cloudtopped boundary layer, Tellus, Set. B, J5, 458-472, 1993. Barkstrom, B. R., Some effects of 8-12 /•m radiant energy transfer on the mass and heat budgets of cloud droplets, J. Atmos. $ci., 35, 665-673, 1978. Berry, E. X., and R. L. Reinhardt, An analysis of cloud drop growth by collection, I, Double distributions, J. Atmos. $ci., 31, 1814-1824, 1974a. Berry, E. X., and R. L. 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Seinfield, and S. Arnold, Condensation rate of water on aqueous droplets in the transition regime, J. Colloid Interface $ci., 113, 421-429, 1986. Shafrir, U., and M. Neiburger, Collision efficiencies of two spheres falling in a viscous medium, J. Geophys. Res., 68, 4141-4147, 1963. Siems, S. T., D. H. Lenschow, and C. S. Bretherton, A nu- merical study of the interaction between stratocumulus and the air overlying it, J. Atmos. $ci., 50, 3663-3676, 1993. Slingo, A., S. Nicholls, and J. Schmetz, Aircraft observations of marine straocumulus during JASIN, Q. J. R. Meteorol. $oc., 108, 833-838, 1982. Srivastava, R. C., Growth of cloud drops by condensation, J. Atmos. $ci., j6, 869-887, 1989. Twohy, C. H., P. H. Austin, and R. J. Charlson, Chemi- cal consequences of the initial diffusional growth of cloud droplets I: Clean marine case., Tellus, Set. B, J1, 51-60, 1989. P. H. Austin and Y. Wang, Atmospheric Sciences Pro- gramme, •217 Geography, 1984 West Mall, University of British Columbia, Vancouver, B.C. V6T 1Z2 CANADA (email: phil@geog.ubc.ca; yinong@geog.ubc.ca) S. Siems, Department of Mathematics, Monash University, Clayton, Victoria, 3168 Australia (email: siems@cyclone.maths.monash.edu.au) (Received July 12, 1994; revised April 5, 1995; accepted April 5, 1995.)


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