Weather: Taking a mathematical view of the rain
Tuesday 13 January 1998
I have just been reading a 19th-century book on the weather, and it has cleared up a misconception that I have long held about the formation of rain. The fourth edition (1887) of Robert H Scott's Elementary Meteorology explains that rain can form in three ways: 1: the ascent of a current of damp air, chilled by expansion as it rises; 2: the contact of warm and damp air with the colder surface of the ground; 3: The mixture of masses of cold and hot air.
Of these, he says, the third is the least effective, and to prove the point, he quotes the words of Dr Hann, written in 1881:
"If a mass of saturated air at 77F be mixed with another at 32F, as the weights of vapour per cubic foot are 10.0 grains and 2.1 grains in the two masses respectively, the arithmetical mean temperature of the mixture (if the air be supposed dry) would be 54.5, and the arithmetical mean of the weight of vapour 6.1 grains. But at 54.5 a cubic foot of air can only contain 4.8 grains, so that from every cubic foot 1.3 grains would be condensed by a fall to that temperature. Now the condensation of 1 grain of aqueous vapour sets free 0.093 units of heat, which are available to warm up the air, so that the temperature cannot fall as low as we have supposed.
"In fact, calculation shows that the temperature of the mixture would be 58.7 (instead of 54.5), and the weight of vapour condensed would be only 0.6 grains per cubic foot. The amount of rain such a condensation would produce would be quite insignificant; a column of air 1,000 feet in height yielding only about 600 grains, or one-twelfth of a pound of water, per square foot of section. This would correspond to a rainfall of .048 inch - not quite half a tenth of an inch."
The calculation - based on condition more extreme than are likely to occur naturally - is convincing and shows that the simple model of warm and cool air mixing to form rain simply does not explain things. The modern theory of supercooled water droplets forming ice crystals in columns of warm rising air was only developed much more recently, but it is interesting to know that even 100 years ago they knew the old model was wrong.
They also understood the effect of the latent heat reduced when water vapour turns to rain. Water cannot turn into vapour at the same temperature without the application of some form of energy. This is why evaporation has a cooling effect. And when vapour condenses into water again, it releases the same amount of energy, which may warm up the air.
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