## Thursday, October 14, 2010

### Clouds

Clouds contain microscopic droplets of water. The droplets are around 10 μm in diameter and there are hundreds of them in every cubic centimeter (see here). We discussed the absorption of long-wave radiation by water in an earlier post. We presented a graph of water's absorption length versus radiation wavelength. The graph shows that the absorption length for 5 μm to 60 μm radiation is less than 20 μm. A 100-μm layer of water will absorb 99% of long-wave radiation.

Let us consider the fate of a single long-wave photon as it enters a cloud. The photon may pass all the way through the first centimeter of the cloud without touching a single droplet, or it might strike a droplet. If we assume our droplets have cross-section 100 μm2, the chance of our photon striking any particular droplet while passing through the first cubic centimeter is one in a million. The chance of the photon striking any one of a hundred droplets in the same cubic centimeter is a hundred times greater, or one in ten thousand. In every 100 m of cloud, the expected number of droplets our photon will strike is one (that's one in ten thousand per centimeter times ten thousand centimeters).

When the photon strikes a droplet, it can reflect off the surface or it can enter the liquid. According to this graph from Querry et al., the reflectance of water to long-wave radiation is 2% when the photon approaches at 10° to the perpendicular, 10% at 60°, 20% at 70° and 80% at 87°. The chance of a long-wave photon reflecting from a random spot on the surface of a droplet is around 10%. We also note that most reflection takes place when the photon strikes the droplet at a small angle to the surface, so the photon will continue into the cloud after reflection.

If a photon enters a 10-μm droplet, it will pass through something like 10 μm of water and strike the opposite, inner surface. Here it might reflect back into the droplet. But let's keep things simple: if the photon strikes a droplet, it will most likely enter the droplet, pass through 10 μm of water and keep going.

With 100 droplets in each cubic centimeter, our photon must pass through one droplet on average for each hundred meters of cloud. In 1 km of cloud, our photon will pass through ten droplets. In order to pass through the cloud, it will have to pass through roughly 100 μm of water. But we know that 100 μm of water absorbs 99% of all long-wave radiation. So a 1-km cloud is opaque to long-wave radiation.

Clouds have a different effect upon short-wave radiation, such as sunlight. Short-wave radiation reflects off the air-water surface far more readily than long-wave radiation, and once inside, it is more likely to reflect off the internal surface as well (internal reflection within droplets is what causes rainbows). But on the other hand, the absorption length of visible light in water is tens of meters. The water in a thick cloud will not absorb short-wave radiation. Instead, short-wave radiation changes direction as it encounters water droplets. It can change direction by reflecting off the outer or inner surface of a droplet. It can change direction by refraction as it enters or leaves a droplet. The reason thick clouds are dark is because sunlight is being turned around and re-directed back into space by the cloud's water droplets, not because sunlight is being absorbed by the clouds.

A short-wave photon passing through 1 km of cloud will encounter only ten droplets. These ten droplets are enough to absorb long-wave radiation, but they are not enough to guarantee that the short-wave photon gets re-directed back out of the cloud. With the help of some pencil drawings, we estimate that it would require encounters with one hundred droplets to guarantee that the photon came back out of the cloud on the same side that it went in. A 1-km thick cloud might reflect 10% of the short-wave radiation incident upon it, while a 10-km thick cloud will reflect 90%.

A 1-km layer of cloud acts like a black-body absorber of long-wave radiation. By radiative symmetry, this same 1-km layer must also be a perfect black-body radiator of long-wave radiation. We estimate that this same 1-km of cloud will allow 90% of short-wave radiation to pass through, while reflecting 10%. A 10-km layer of cloud, meanwhile, will allow only 10% of short-wave radiation to pass through, while reflecting 90%.

Heat radiated by the Earth will be absorbed by clouds. These same clouds will radiate their own heat out into space. Sunlight will be reflected off their top surfaces. In our next post, we will consider the effect of a 1-km layer of cloud at altitude 10 km upon the Earth's total escaping power and upon the amount of heat arriving from the Sun.

UPDATE: Cirrus clouds appear to be examples of our 1-km cloud layer. According to NASA, cirrus clouds are opaque to long-wave radiation, but largely transparent to short-wave radiation.

UPDATE: We see here confirmation of our estimates: clouds reflect from 10% to over 90% of short-wave radiation (albedo is 0.1 to over 0.9).