P. Squires Evolution of Cloud Droplet Laboratory of Atmospheric Physics Spectra: The Ellect ol Desert Research Institute University of Nevada System Nuclei Spectra Reno, Nev. 89507 Abstract complexities arise in the case of stratocumulus and even The particles on which cloud drops nucleate are thought to more so in the case of fog. be of mixed and somewhat variable constitution. Conse- Thus the specific relationship between nuclei and quently, in the discussion of cloud formation, a number of cloud droplet spectra appears in its simplest form in the simplifications and approximations are introduced that can case of cumulus, where a simple kinematic and thermo- in fact be justified a posteriori. Provided that the spectrum of critical supersaturations of the particles is known (rather dynamic model may realistically be taken as represent- than, for example, their size distribution), the resulting ing the major atmospheric event. theory can be used to predict the concentration of droplets in clouds with errors of order 50%. Thus, the theory can 2. Theoretical approaches explain, for example, the systematic difference between maritime and continental clouds. Some of the errors may The theoretical discussion of cloud formation usually be attributed to sampling problems and some to instrumental begins after the condensation nuclei have become haze problems; a more accurate check on the theory would require droplets. There exists a veritable hierarchy of approxi- the use of improved instrumentation to characterize the mations that result in simplified equations. Thus, the aerosol, and the conduct of a cloud-forming experiment under rather well-controlled conditions. transfer of vapor to a growing droplet and of heat from it is treated as a steady state process. Ventilation 1. Introduction effects due to sedimentation and the interactions of the Clouds form by a kind of macroscopic heterogeneous diffusion fields of one droplet with those of its neighbors nucleation of unstably growing water droplets on a are ignored. Continuum physics fails to describe mass selected group of particles, the so-called cloud nuclei or and heat transfer in the immediate vicinity of a drop- cloud condensation nuclei, as first discussed quantita- let with a radius comparable with the mean free path tively by Howell (1949). This nucleation type event is of air and vapor molecules. Following Fuchs (1934), this therefore one of the meeting places between two fields is dealt with approximately by introducing the notion of inquiry: aerosol physics, including the formation, of a "boundary sphere" at a distance from the drop growth, and elimination of particles; and cloud physics, surface of the order of a mean free path. Within this including the hydrologic cycle and the profound thermo- sphere, transfers are supposed to occur by molecular- dynamic influence of clouds on the lower troposphere. kinetic processes. These considerations have the effect of making the rate of growth of a small droplet de- The relationship between these particles and cloud pendent on the thermal accommodation coefficient droplet spectra has been studied most intensively in (which is usually assumed to be unity) and on the ac- connection with convective clouds. Cumulus clouds have commodation coefficient for water molecules, which will a basic kinematic simplicity in that cloud formation may be referred to here as the "sticking" coefficient. Experi- reasonably be supposed to occur as a result of the mental values for this coefficient center at about 0.04, steady, adiabatic expansion of moist air. Radiative ef- though they cover a wide range. In addition, a number fects may be disregarded because characteristic updraft of approximations are usually used to describe how the velocities correspond to cooling rates of order 10?oC per equilibrium vapor pressure of a solution droplet de- day. No doubt other more complex nucleation events pends on its radius and on the nucleus. For example, occur within a cumulus cloud as a result of the in-mixing it is typically assumed that the nucleus is entirely solu- of the air into which it grows; but since the real moisture ble, that the van't Hoff factor does not change as the supply for cloud formation comes from the well-mixed, droplet grows and becomes more dilute, and that the relatively homogeneous subcloud layer, the microphysical surface tension and density of the solution are equal stage is essentially set at the time of the original nuclea- to those of water. These simplifications are justified only tion of unstably growing droplets. if the solution is dilute. Indeed, as discussed below, the This simplifying element is absent in many other "dilute solution" assumption would appear to represent cloud systems. For example, in the case of a middle-level an acceptable approximation. cloud forming in an area of general rising motion, char- acteristic cooling rates due to expansion are not so large The analysis traces out how the relative humidity that radiative effects are obviously negligible. Similar rises at first more or less linearly as a result of the steady expansion of the air and how its rise is slowed as suc- i Paper presented at the International Cloud Physics Con- cessive groups of nuclei become unstable and begin to ference, 26 July 1976, Boulder, Colo. absorb significant quantities of water. The maximum Bulletin American Meteorological Society 29 Unauthenticated | Downloaded 10/03/21 10:33 PM UTC 30 Vol. 58, No. 1, January 1977 value is reached when the absorption of water and re- of the growing drops is most critical for the cloud-form- lease of heat match the rate of expansion, and thereafter ing process at the time when they approach their critical the supersaturation declines. To a first approximation, size. In the case of a nucleus consisting of a pure soluble the concentration of cloud droplets that form is equal electrolyte, the effective molar ratio of solute to solvent to the concentration of nuclei with critical supersatura- has by then fallen to about Sc/2 (Sc in absolute units), tions less than the maximum value reached in the so that if the critical supersaturation is of order 10"2, ambient air. the critical droplet solution is indeed somewhat dilute, It is generally believed that atmospheric condensation though this does not always mean that the van't Hoff nuclei are not the pure soluble electrolytes that are factor is very close to its value at infinite dilution (e.g., usually discussed in connection with cloud formation in the case of H2SO* it is still about 2). Thus, since cloud but are mixed particles consisting in part of insoluble formation usually involves only particles for which Sc components. Some evidence on this question has been is of order 10~2 or smaller, in the case of critical droplets obtained by Twomey and Severynse (1964) and Twomey there is some a posteriori justification for the approxi- (1965) who estimated the sizes of natural cloud nuclei mations that depend on the "dilute solution" assump- by means of diffusion battery measurements. The diffu- tion. Also, the presence of an insoluble component will sion coefficient of these particles was smaller than would by that time have only a reduced influence. be expected if they were wholly soluble; this finding There is some plausibility, therefore, to the commonly led to the conclusion that they contained an insoluble adopted notion that the most important single property component. of a complex nucleus is its critical supersaturation. However, Katz and Kocmond (1973) measured the Thus, if the spectrum of critical supersaturations is size distribution of an NaCl aerosol prepared in the known, the theoretical discussion of cloud formation laboratory using an electrical aerosol analyzer and found may reasonably be carried out as if the nuclei were that even with a pure soluble aerosol, the particles with simple soluble particles of the appropriate size. a given critical supersaturation, Sc, were 2-3 times larger than theory would indicate—indeed, somewhat larger 3. Experimental checks than indicated by the work of Twomey and Severynse The first attempt to check whether theory correctly for natural particles. This work tends to call in question predicted the way a cloud forms was made by Twomey the basic theory of Kohler that relates the size of a and Squires (1959) in the simplest way possible, by mea- soluble particle to its critical supersaturation, and there suring the spectrum of the critical supersaturation Sc is a need for further investigations along these lines. of cloud nuclei in the air below a cumulus and the In contrast to the results of Katz and Kocmond, H. C. concentration of droplets in the cloud. This type of Gerber (personal communication, 1976) has recently ob- experiment has been repeated several times, and, in tained some results that confirm Kohler's theory. general, there has been agreement between observed and The many approximations used in discussing cloud computed droplet concentrations to within about 30- formation, some of which are fairly crude, are justified 50%. This result is about as good as could be expected by two considerations. First, the question of the nature in such an experiment considering the approximations of the nuclei. Obviously, a complete physical and chemi- used in the theory, the difficulty of the measurements, cal description of the nucleus population is beyond our and the serious sampling problems that arise. Strictly reach. Methods exist for exploring the size distribution speaking, the aerosol measurements should be made just over the relevant range of sizes (down to about r = below cloud base, and the cloud droplet measurements, 0.01 /um), but even the average chemical constitution is together with vertical velocity measurements, should be poorly understood, and it is totally unknown whether made very soon afterward just above this point. This is droplet growth rates may be affected by the presence in impracticable with a single aircraft, so that inevitably the nuclei (or some of them) of very small proportions the results tend to be confused by fluctuations in at- of surface active materials, as suggested, for example, by mospheric properties.