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Actually the density of the air *does* increase when droplets form (to the
extent that temperature changes can be neglected). (If the process is not
isothermal because of the liberation of latent heat upon condensation, it is
possible for the air density to decrease because of the temperature increase
results in the expansion of the air.)
Not only that, the droplets in clouds *do fall* wrt the ambient air like
humidifier mist. This doesn't mean that they necessarily must fall wrt
the ground however, although in many cases this happens as well.
A thought-experiment. Take some air with non-zero absolute humidity. Now
move all the water molecules together in groups to form a population of
droplets. The droplets are heavy, but the air left behind is lighter, so
the average density doesn't change and the cloud will not fall. But I
suspect that this reasoning is incomplete. Is humid air lighter or
heavier than dry air? And if the humidity is suddenly removed from air,
does the partial pressure fall, requiring that the parcel of air shrink
and become denser?
The pressure on relatively stationary air is due to the weight of the column
of air above the point of observation. This column's weight does not change
as the droplets form.
Clouds stay up for two main reasons. 1. updrafts, and 2. evaporation.
When clouds form along a frontal boundary they form as lower density moist
air is lifted up over a mass of higher density air (whose density is higher
because it is usually colder (although it may also be because the other mass
is drier, or both colder and drier). The moist air runs up the frontal
boundary which is shaped like a shallow inclined plane. The rising moist air
expands adibatically as it rises into a lower pressure environment. This
expansion cools the air. In addition, the air is somewhat cooled because it
is invading a cooler region (the troposphere's temperature decreases with
increasing height). As the air cools its relative humidity increases until
it is satutated. Cloud droplets then form. If the net vertical component of
the wind velocity is fast enough, the cloud droplets do not fall with respect
to the ground even though they are falling wrt the rising air. When there is
no frontal uplift mechanism clouds can still be supported by updrafts. For
instance, cumulus clouds form by a convective instability. Air spontaneously
convectively rises and condenses out droplets, which liberates latent heat
which heats the air enough relative to its surroundings to make it expand
enough to cause the bouyancy lift to propel it to higher heights where upon
it cools further, causing further condensation, causing further latent heat
release which causes still further uplift. Cumulus clouds are therefore much
more vertically stacked than other horizontally layered clouds like stratus
or cirrus clouds.
After a cloud forms it is possible that it may not fall down to the ground
even if there is no uplifting (or insufficient uplifting) mechanism to
support it. In this case the droplets do fall down towards the ground. As
the droplets reach the base of the cloud they enter warmer air (because the
lower lying air is warmer than the higher air). When the air is warm enough
for the air to no longer be saturated the drops begin to shrink via
evaporation. This dramatically slows their terminal velocity so that the
droplets tend to slow down and shrink at the cloud base until they have
completely evaporated. Thus the cloud base is automatically situated at the
altitude where the droplets evaporate (and the air is no longer saturated).
If there is a sufficient supply of condensation nuclei in the upper reaches
of the cloud new droplets can continue to form higher up in the cloud to
replace the ones that evaporate at the cloud base. This can not go on
indefinitely since eventually the falling droplets higher in the cloud will
deplete so much moisture from it that no new droplets will form. Also it is
possible that the cloud may run short of the condensation nuclei needed to
initiate the formation of the droplets. When this happens the cloud
eventually dissipates.