Rain

Our simulated sky never clears. Clouds fill the atmosphere almost entirely. They are forever forming in air that rises from the sea, and forever sinking to the ground, but they never come falling out of the sky all at once in the big drops we know as rain.

The droplets in our clouds are tiny. Those in our slow-sinking clouds are only 10 μm in diameter and descend at 3 mm/s. Those in our fast-sinking clouds descend at 300 mm/s. The graph we present in Falling Droplets implies that these fast-sinking droplets are 100 μm in diameter. Rain falls to Earth at several meters per second, so the same graph tells us that rain drops are at least 500 μm in diameter. A drop 500 μm in diameter contains a hundred times as much water as a droplet of 100 μm and a hundred thousand times as much water as a droplet of 10 μm. Could it be that cloud droplets collide and coalesce in order to form rain drops? If so, how long does this take, and under what circumstances does it occur?

The Wikipedia page on rain describes convective precipitation and orographic precipitation. In both these forms of rain, a cloud moves up, and encounters rain drops descending from above. If the descending drops are 500 μm in diameter, and the air is moving up at 1 m/s, the drops will remain at the same altitude. The cloud moving up and past them carries microscopic droplets that can collide with the stationary drops, coalesce with them, and so enlarge them until they are heavy enough to fall out of the rising cloud and descend to the Earth as rain.

But further reading suggests that rain formed of coalescing droplets is rare. A far more potent source of rain drops are ice crystals. In Cloud Physics, we learn of the Bergeron Process, whereby ice crystals grow, sink, melt, and become rain drops. Large rain-drops are melted hail-stones. Small rain-drops are melted snow-flakes.

In our simulation, whenever the concentration of water vapor exceeds the saturation concentration, we assume the excess water condenses. It turns out, however, that the surface tension of liquid water makes it hard for water to condense into floating, microscopic droplets. If we provide a solid surface for the water to condense against, such as a blade of grass or a glass mirror, the water will condense when it reaches the saturation concentration, but in a body of air high above the ground, the only such surfaces would be dust particles, and these may be rare. Each one will serve as a catalyst for condensation until a droplet forms around it.

But the same is not true of ice crystals. In air saturated with water vapor and below the freezing point of water, an ice crystal can form on a grain of dust, and after that it will continue to grow. Water vapor deposits directly upon the surface of the crystal, thus changing state from gas to solid in one step, and the newly-created ice surface is an ideal foundation for further growth.

Now, suppose a cloud of microscopic water droplets rises until its temperature drops to −20°C. We might assume that the droplets will freeze. But pure water droplets resist freezing until they drop to −40°C. Ice crystals form in the midst of the cloud of super-cooled water droplets. As water vapor is deposited on the crystals, the concentration of water vapor in the air drops.

And here we encounter another curious physical phenomenon. The saturation concentration of water vapor with respect to an ice crystal turns out to be lower than the saturation concentration of water vapor with respect to super-cooled liquid water. Water vapor will deposit on the ice crystals until the concentration of water vapor drops to the saturation concentration of water vapor with respect to ice crystals. Because this concentration is below the saturation concentration with respect to super-cooled liquid water, the water droplets actually start to evaporate. The droplets evaporate, and their water is deposited onto larger and larger ice crystals.

Once the ice crystals are large enough, they start to fall, and they eventually fall into air that is warm enough to melt them. They turn into drops of water and fall to Earth as rain. It is this process that we will attempt to simulate in the next version of our Circulating Cells program.