Competition among CCN for Water Vapor

The sudden increase in the growth rate of a CCN once the critical supersaturation is reached - called CCN activation - is a strange phenomenon. The situation is much simpler if all the CCN of a given air parcel are chemically identical and all have the same dry mass mx. In this case, they all have identical Kohler curves, identical growth rates, and will reach the critical supersaturation radius at the same time. Expansion cooling increases the RH of a rising air parcel. If the vertical ascent pushes the ambient supersaturation S above the critical supersaturation Sc of the CCN, the situation is unsustainable. Remember S = (e - es)/es, where es is the is the Clausius Clapeyron saturation vapor pressure for that temperature. There are no values of S above the critical supersaturation for which there is an equilibrium radius (i.e. no corresponding value of r on the Kohler curve). You must be in the growth regime where C > E. The CCN experience rapid condensational growth. For r > rc, the droplet growth decreases the S required for the droplet to be in equilibrium. By growing, the droplet therefore finds itself in an environment which is even more supersaturated (distance from Kohler curve increases, or the difference between the actual S of the cloud, and the equilibrium Kohler curve S for droplets of that radius, increases). By this positive feedback process, whereby some small increase in r results in a further increase in distance from the Kohler curve, the droplet grows very rapidly. It can suddenly increase its radius by about 10-100, or volume by 1000 - 1,000,000. This is shown in Figure 6.16. This helps explain why new growing liquid water clouds have sharp boundaries: the increase in size in going from a CCN to a cloud droplet is associated with a large change in radiative properties, and especially in the scattering cross-sections of the water droplets (though as time goes on mixing and ice formation can diffuse the edges of clouds). The process continues until the activating cloud droplets absorb enough water that the ambient S of the cloud is dragged back to the Kohler curve.

RH suppression in very polluted environments

In general, one would expect the critical droplet radius to be the boundary between the region where the CCN passively respond to any changes in background RH of the atmosphere, and the situation where the cloud droplet size and the background RH interact (e.g. during activation). However, in extremely polluted environments, with extremely high concentrations of CCN, one might expect that there are enough CCN to suppress the background RH by absorbing water vapor from the atmosphere. Another way of saying this is that the amount of condensed water in the CCN can comparable with the amount of water in the vapor phase (expressed in (g water vapor)/(g of air) or some other measure.) Instead of getting clouds, one ends up with a "haze": no definite cloud boundaries but a white featureless sky. A hazy sky would not always imply that background RH (and cloud formation) are being suppressed, but this may be true some of the time. It could also prevent cumulus cloud formation by reducing shortwave solar absorption at the surface, so preventing convective instability.

Difficulty in Determining Critical Supersaturations Sc in Real Clouds

Atmospheric scientists use simple models to analyze the activation of CCN into cloud droplets. These models will often have some fixed updraft speed, and some assumed distribution of CCN dry masses and chemical compositions. These models can produce plots such as shown in Figure 6.16, and predict the fraction of CCN activated. Unfortunately, it is difficult to directly compare the activation of CCN as described in these models with observations. The critical supersaturations Sc achieved during CCN activation are likely of the order of Sc = 0.001, or 0.1 percent. In order to measure the critical supersaturations of real clouds, aircraft would have to fly through cloud base (within a few meters of the bottom), and measure both temperature water vapor pressure to an accuracy of better than one in a thousand, and at very high temporal resolution. This is not achievable with current instrumentation on aircraft (to the best of my knowledge). We can measure the number of cloud droplets in a cloud, and the number of CCN below a cloud, but these are only the end points of the activation process. The process of cloud activation itself, and in particular the supersaturations which are driving the process, are not directly observable.

Adiabatic Liquid Water Content

Liquid Water Content (LWC) is usually expressed as the mass of condensed water per unit volume of cloudy air. Note that this is really a cloud liquid water DENSITY (the word "content" in this context is ill advised I think). The recipe for determining the adiabatic LWC is as follows. Make some assumption for the T and pd of the air parcels that enter the cloud at cloud base. At cloud base, there is no condensate (wl = 0), and the air parcel is saturated, so the total water wt(cloud base) = w(cloud base) = epsilon*es(Tcb)/pd, where Tcb is the temperature at cloud base. Then assume that the total water mass mixing ratio (sum of the vapor mass mixing ratio and the liquid water mass mixing ratio wt = w + wl) is conserved in clouds. This would be true under the following "adiabatic water" conditions: (i) there is no irreversible removal of cloud water via precipitation formation and fallout, (ii) after entering the cloud at cloud base, the cloudy air parcel has not experienced mixing with air parcels with dissimilar wt (i.e. no mixing), and (iii) the cloudy air parcel has not experienced evaporation from falling precipitation. You then use LWCad = rhod*(wt(cloud base) - ws(T))*1000 to get an adiabatic LWC in g/m3, where T is the temperature of the cloudy air parcel, and rhod is the dry air density of the cloudy air parcel. Note that the water vapor mass mixing ratio w = ws(T) for cloudy saturated air parcels.

The adiabatic LWC is the maximum theoretical upper limit of LWC in the cloud. If a cloudy parcel has LWC ~ LWCad, then you can assume that the parcel has experienced undilute ascent from cloud base. In most cases, LWC < LWCad. This is due to a violation of one of the adiabatic water conditions mentioned above, or that the parcel was entrained along the sides of the cloud at a lower wt (or some combination of all of these reasons).

The total water wt is a tracer in clouds under adiabatic water conditions. LWCad itself is not conserved within clouds, even if there is no precipitation or mixing, because the dry air density rhod is not a tracer, and because ws(T) decreases as an air parcel gets colder. LWCad therefore increase with height inside a cloud, due to the decrease in ws(T). In general, you would also expect LWC to increase with height inside a cloud, at least until it starts to precipitate. The definition of Adiabatic LWC is awkward in the sense that it is defined in terms of a conserved quantity wt under particular conditions, but it itself is not a conserved quantity under those conditions.

Relevance of Adiabatic Thinking to Real Clouds

Most of the time, the observed LWC in clouds is quite a bit less than the adiabatic LWC. This is to be expected, since clouds turbulently mix, and sometimes precipitate. However, it is likely that vigorous thunderstorms possess a quasi undilute core. When cloudy parcels mix with subsaturated environmental air, cloud droplets evaporate, and the cloudy parcel is chilled. Mixing therefore reduces the temperature, buoyancy, and updraft speeds of clouds. In order to produce hail, updraft speeds must be on the order of 20 m/s. It is very likely that the only way to produce such large updraft speeds is if the amount of mixing that an updraft parcel experiences is relatively small. Obviously such updraft parcels are not adiabatic with respect to water, since thunderstorms precipitate. Nevertheless, one would expect the equivalent potential temperatures of updraft parcels within the core of a strong thunderstorm to be roughly conserved. But thunderstorms obviously represent only a tiny fraction of real clouds. Most convective clouds are killed by mixing before they produce any precipitation, let alone hail. Vigorous thunderstorms can only develop if they have an updraft core which experiences limited mixing, or if they grow in a background atmosphere which is very humid, so that the evaporative cooling experienced by cloudy parcels when they mix is reduced.

Assumptions Made in Cloud Droplet Condensational Growth

(1) The cloud droplet is dilute enough that there is no solute effect (fH2O ~ 1). This should be pretty good because there is a 1000 - 1,000,000 times volume increase in going from a CCN to a CD.

(2) The droplet is large enough that there is no curvature effect. Cloud droplet radii are typically 5 - 10 microns. The curvature effect e(r,T)/es(T) in this size range is small. This is shown in Figure 6.2. The curvature effect is negligible for r > 1 micron. It is relevant to the CCN size range but not to the CD size range. Note: you do not HAVE to make assumptions (1) and (2) to solve for dr/dt of a droplet. And during the activation process where size ranges may be smaller, it may be important not to. However, if you don't make assumptions (1) and (2), the saturated vapor pressure es(r,fH2O,T) of the cloud droplet becomes a function of r (note that fH2O is a function of r), and obtaining a solution for r(t) becomes much harder.

(3) The cloud droplet density is not too high. Typical values of LWC in clouds are 1 g/m3. Typical values of air density in clouds are 1 kg/m3. Since the density of liquid water is about 1000 times the density of air, cloud droplets occupy about one millionth the volume of a cloud. This means that the typical distance between cloud droplets is about 100 times the cloud droplet radius. Growing cloud droplets reduce the water vapor density around them from the background value in the cloud. This is neccessary to create a water vapor density gradient which will drive a diffusional flux of water vapor molecules toward growing cloud droplets. The reduction in water vapor density in the neighborhood of a growing cloud droplet will typically extend for a distance about 10 times the radius of a cloud droplet. Since cloud droplets are about 10 times further apart than this, you can effectively think of condensational growth of cloud droplets as non-interacting. The perturbations in the water vapor density of one droplet do not directly affect the growth rates of other droplets. You can solve for dr/dt of an isolated droplet and assume it is reasonably accurate in a real cloud.

(4) The water vapor density field phov(x) around a growing droplet is isotropic, i.e. independent of direction. This is OK for spherical droplets (not OK for non-spherical ice crystals), and when the cloud droplet density is not too high.

(5) Cloud droplet growth is diffusion limited. The diffusional flux of water vapor toward growing cloud droplets will create small supersaturations adjacent to the droplet surface. This is neccessary for C > E, averaged over the surface of the droplet. However, this supersaturation is tiny. It is OK to assume that the water vapor density immediately adjacent to the droplet is equal to the saturation density of water at the droplet temperature, rhov(r) = rhov_s(T), or e(r) = e_s(T).

(6) The water vapor density rhov(x) is independent of time. This would strictly speaking be not true for a growing droplet. For example we have made the assumption that rhov(r) = rhov_s(T) at the droplet surface. However, if a droplet is growing, this boundary condition constraint moves outward also as r changes, so rhov(x) also changes with time. However, dr/dt is relatively slow compared with the timescale over which diffusion will affect phov(x), so it is OK to assume that phov(x) is stationary. A related assumption is that the S of the cloud is constant, so that rhov(x = infinity) is fixed.

(7) The temperature of the cloud droplet is the same as the temperature of the background atmosphere. This is not true, since a growing cloud droplet is subject to condensational heating. This would increase the cloud droplet temperature, increase evaporation from the cloud droplet, and decrease its rate of net growth (i.e. C - E becomes smaller). In principle, it could be possible that the rate of condensational growth of a cloud droplet could be limited by the rate of diffusion of water molecules toward the droplet, or by the rate at which the cloud droplet gets rid of its excess heat. It turns out that the diffusional flux of water molecules is more important, but taking into account the heat flux will reduce dr/dt of a growing cloud droplet. But this correction is usually small, and we have not considered it.

Rain Formation

Clouds come into existence mainly because upward motion increases the relative humidity of rising air parcels. What then occurs is activation, the rapid transition of a CCN to a cloud droplet (see above discussion on competition of CCN for water vapor). There are always some CCN in an air parcel, though the number is highly variable. Typically, there are more CCN in continental air than in marine air, so cloud droplets over continental air are typically smaller - there is more competition among cloud droplets for the available supersaturated water vapor. Cloud droplets are about one millionth the volume required to make a precipitation sized droplet. How do you concentrate the available cloud water into a smaller number of larger droplets, that are then big enough to have a significant fall speed? This is the essence of the rain formation problem, and nature has two solutions.

(1) One way is through warm rain collision/coalescence (CC). This is just collisions between water droplets. The usual problem here is that the timescale to make a precipitation sized droplet is longer than the lifetime of a cloud (the lifetime over which the upward velocities needed to sustain the cloud are maintained). Exceptions occur when the cloud droplet density is very high. The time required to make rain via CC is a nonlinear function of the cloud droplet density. This is more likely over the warm tropical oceans, where the air parcels entering clouds have higher water vapor mass mixing ratios r, and for a given cooling, would produce more cloud water.

(2) The usual way is via the ice phase, or Bergeron process. Supercooled droplets can nucleate into an ice crystal if they contain some Ice Condensation Nucleus (ICN) { e.g. piece of dust, pollen, AgI, etc.) The number of ICN in a cloud increases dramatically as the temperature gets colder. i.e. dust might work below -10 C, etc. Every material has a different threshold temperature at which it would trigger ice formation in supercooled water. Good ICN work at higher temperatures closer to 0 C, but are rare. Once a single ice crystal forms in supercooled cloud, it grows rapidly, since it's supersaturated (water vapor pressure e would roughly equal es as long as water droplets are present, and es/esi > 1).

Ice can grow via both depositional (condensational) growth, or also by "accretion": collisions with supercooled droplets, which freeze by contact. Accretion will dominate if the density of water droplets is sufficiently high (typically more common with faster upward velocities, as in thunderstorms). Accretion usually leads to spongy ice (e.g. graupel), rather than nice symmetric single crystals, which require nice slow depositional growth. Snowflakes are usually aggregations of single crystals. Ice crystals stick together better at temperatures closer to zero. Larger snowflakes are favored at warmer temperatures closer to 0 C where the sticking probability is higher, and where the density of snowflakes is higher. It is also likely that larger snowflakes are more common at weaker winds, since strong turbulence, and local wind shears, could tear large snowflakes (which are delicately held together by ice bridges) apart.

Why are cloud droplets larger than 20 um neccessary to initiate collision/coalescence?

There are two reasons. One reason is that it is difficult for cloud droplets to collide if they are all exactly following streamlines (by this I mean the droplet is behaving like it does not have any inertia and has the same velocity as an air parcel would have at the same position). Keep in mind that cloud droplets are usually separated by a distance comparable with 100 times their radii. Droplets can be brought into contact if there are regions where streamlines converge together. This is called confluence. The existence of such regions is thought to be enhanced by turbulence. But the main mechanism by which cloud droplets collide is thought to be differential fall speeds. Fall speeds increase dramatically at small r (going as r squared). The greater the difference in fall speed, the greater the likelihood two droplets will collide. Having some larger particles therefore dramatically increases the likelihood of two droplets being brought into contact. The second reason is that the collision efficiency between two droplets increases rapidly if at least one of them is larger than 20 microns. These two reasons are a bit intertwined since the collision efficiency of two droplets, in addition to being a function of the two radii, will also be a function of the relative velocity of the two droplets. It is assumed that each droplet is at its terminal velocity (since this would be true under laboratory conditions, and also usually true in real clouds).

Keep in mind that condensational growth can only create droplets larger than 20 microns in a reasonable time (20 minutes or so) if the updraft speed is fast enough to maintain a supersaturation of at least 0.5 %. This requires about 5 m/sec updraft, and is only attained in vigorous deep convection. What else then would be the usual mechanism to create 20 micron droplets so a cloud can precipitate? The counterintuitive explanation is mixing, which can reduce the number of cloud droplets, and reduce competition - subsequently allowing condensational growth to make larger droplets.

Why are cloud droplets separated by at least 100 times their radii? The mass mixing ratio w of water vapor typically does not exceed 0.03 or so. These large value values only occur near the surface, so a better upper limit for clouds would be 0.01. The cloud liquid water content is usually a small fraction of the water vapor, so a good upper limit for it could be 0.001 (kg liquid water/kg dry air). Liquid water is about 1000 times more dens than air. An upper limit for the fraction volume of cloud water would therefore be 1 in a million (10 to the 6). Cloud droplets would therefore be separated by at least distance of 100 times their radii.

Why is hard to predict whether a given cloud will precipitate?

One reason is that the timescales for condensational growth and collision/coalescence (the two main processes that affect droplet growth) are comparable with cloud lifetimes. This means that the cloud droplet size distribution of an air parcel in the cloud is rarely in steady-state (or equilibrium) with the RH, temperature, and pressure of the air parcel. So given the physical variables, you can never say exactly what the cloud droplet size distribution should look like, and therefore, whether it will be producing precipitation sized droplets. The cloud droplet size distribution depends on the history and chemical composition of the air parcel, especially the values of supersaturation that were realized when CCN activation occurred. It is a complex, time dependent problem. Theory is a guide, but rarely gives easily determined realistic answers. In addition, the dynamical structure of a cloud depends on cloud microphysical processes (e.g. latent heat release and buoyancy), so that cloud development is complex two way process between microphysics and dynamics (latent heat release and buoyancy affecting vertical velocities, which affect supersaturation, which affect droplet growth, which affect buoyancy and latent heat release).

An atmosphere with really clean air: too clean to make clouds?

A typical raindrop radius is 1000 microns, or 1 mm (Fig 6.18). Such a raindrop would have a mass of 0.0042 grams. A typical LWC is 1 g/m3. Therefore 1 m3 of cloudy air can make about 240 r = 1 mm raindrops. Figure 6.5 shows that a typical CCN concentration is about 100 CCN/cm3, or about 1.0E+08 CCN/m3. If each CCN activates into a cloud droplet, then you would need about 400,000 cloud droplets to combine to make one raindrop. The process of rain formation in warm rain clouds would be much easier if the number of CCN were far fewer, so less collision/coalescence events were required to make a raindrop. If air were really clean, with on the order of 240 CCN/m3, then clouds may not exist. Each CCN would nucleate a single droplet that, with the reduced competition for supersaturated water vapor, would eventually grow large enough by condensational growth to fall out of the cloud. Of course, at such small cloud droplet concentrations, the cloud would be invisible. So, effectively, in the clean air limit, there would be no clouds. CCN activation would directly produce rain, and rain would appear to be falling on sunny days. Wouldn't that be nice? Of course, this could occur only on a planet which for some reason had an atmosphere that was exceptionally free of particles. This is probably impossible; any water or land surface would produce CCN in much higher concentrations than 240 CCN/m3. But it is at least theoretically possible to produce rain without clouds.

Homogeneous Nucleation: does it occur in the atmosphere?

Homogeneous Nucleation refers to the spontaneous freezing of a supercooled droplet, without the involvement of an ICN (some surface that is giving the supercooled water "instructions" on how to start making ice). This freezing becomes more likely as the free energy barrier to ice formation gets smaller. It is believed that the size of this barrier decreases with temperature so that by around -40 C, homogeneous freezing will happen quite readily even for tiny droplets of pure water. The spontaneous formation of a cluster of ice in supercooled water is a random stochastic process. It is more likely to occur if the droplet has a larger number of molecules. The temperature for homogeneous freezing therefore increases with the size of the supercooled droplet. It is also possible that the presence of dissolved salts or acids can lower the free energy barrier, and allow homogeneous freezing to occur at a higher temperature than it would for pure water. This salt or acid would come from the original CCN. However, one would not say that the salt in this case is acting like an ICN. One thinks of ICN as solid particles. Homogeneous freezing is only possible for very clean small droplets that were not earlier converted to ice at a warmer temperature by some ICN. Does homogeneous freezing occur in the atmosphere? The existence of homogeneous freezing is supported by reports of supercooled droplets down to -40 C at the tops of thunderstorms (e.g. from reports of aircraft icing). This suggests that there are indeed some droplets clean enough that there are no ICN present even down to temperatures this cold. (Remember that the number of ICN in a supercooled droplet increases as the temperature gets colder, and more types of surfaces are able to initiate ice formation.) However, it is very difficult to prove that homogeneous freezing has occurred. How would you definitively establish that a supercooled droplet did not contain some small piece of dirt that initiated freezing? Maybe this is possible in the lab, but is more difficult in the atmosphere.