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.