Friday, September 16, 2011

Simulated Clouds, Part II

In Part 1, we gauged the thickness of a cloud by how deep a layer of water it would make if we combined all its water droplets into a pool of the same area as the cloud. A thin cloud might contain 1 mm of water, while a thick storm cloud might contain 100 mm.

We also concluded that even the thinnest of clouds is opaque to long-wave radiation, and therefore a good radiator of its own heat. Meanwhile, clouds do not absorb short-wave radiation from the sun at all because water is transparent to sunlight. Instead, they reflect sunlight back into space. For the purpose of our Circulating Cells simulation, we decided that each 330 μm thickness of water will reflect 10% of sunlight. Perhaps that's too much reflection, perhaps it's too little. We can adjust the 10% reflection depth later if we need to.

Suppose we have a 1-mm cloud layer up near the tropopause, and a 10-mm cloud layer nearer the ground. The combined thickness of both clouds is 11 mm, from which we deduce that only 3% of sunlight will penetrate to the planet surface. This is a calculation we can perform easily in our simulation. We add the thickness of the clouds above each surface block, and apply our formula for reflection to obtain the fraction of sunlight arriving at the surface.

More complicated than the incoming sunlight is the absorption and radiation of heat by separate cloud layers. The surface radiates heat as if it were a black body, but our simulated atmospheric gas has a transparency fraction, which tells us the fraction of long-wave radiation passing through the gas. The rest of the radiation is absorbed. Suppose our transparency fraction is 60%, then 60% of the heat radiated by the surface will reach the bottom layer of cloud, where all of it is absorbed. The cloud itself radiates heat in proportion to the fourth power of its temperature, as if it were a black body, and of this heat 40% is absorbed immediately by the gas above, below, and even at the center of the cloud. The remaining 60% passes down to the surface and up to the upper layer of cloud. The upper layer of cloud absorbs all the radiation from below, and itself radiates heat in proportion to the fourth power of its temperature, as if it were a black body. Of the heat radiated by the upper cloud, 60% will pass back down to the bottom layer of cloud and out into space.

Thus we see that we have long-wave radiation flowing in both directions because of the clouds. If we had just one, thick, cloud layer, our calculation would be simpler. But we have fifteen rows of cells in our simulation, so we could have seven layers of cloud, each separated by a row of gas cells. Our way of handling this problem will be as follows.

For each column of cells, we start at the top and make our way down to the surface. When we encounter a cloud, we calculate how much heat it radiates downwards from its bottom surface. We proceed until we reach another cloud, and here we allow the downward heat to be absorbed at the top surface of the cloud. We continue to the bottom surface of the cloud, and keep going with the same procedure until we get to the surface. By this time we have added up the total cloud thickness and we can determine how much sunlight has reached the surface as well.

Now we start from the surface and go upwards. The surface radiates heat, and this is absorbed by the bottom surface of the lowest cloud. The top surface of this cloud radiates heat upwards. If there is another cloud above, its bottom surface will absorb the upward-going heat, but if there is no other cloud, the heat passes into space.

During this entire process, we keep track of the amount of heat that is added or subtracted from the surface and from each gas cell. Once we are done, we adjust their temperatures to account for the heat lost or gained.

Thus we see that our clouds will introduce new sources of radiation into space that are at a lower altitude than the tropopause that is currently doing all the radiating into space of our simulated atmosphere. On the other hand, the clouds obscure the hottest radiating surface of all, which is the ground.

Our calculation of up-welling and down-welling radiation might slow down our simulation a great deal. But we're not in any hurry, so we won't worry about the computation time.

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