The idea of hollowing out asteroids, spinning them on their longest axis, and filling them with air, water, and soil is common in science fiction (Rendezvous with Rama, or 2312 for good examples). Completely enclosed, the asteroid is an entirely self-sufficient microcosm. It should look something like this:
I’m curious as to what the weather would be like in these terrariums. There are many factors that make the atmosphere in such an enclosed habitat interesting, but two that I would like to focus on to make this question tractable follow.
First, since gravity is being created through centrifugal force it varies a great deal from the inner edge of the asteroid (1G) to the center (0G). This means that there would be a strong Coriolis effect. What effects would this have? Would any clouds constantly be moving in the direction opposite the rotation to an observer on the inner surface? The variable gravity also means atmospheric pressure would be variable from the axis of rotation outward. Assuming the asteroid was filled to a pressure of 1 atmosphere at the surface, how would one compute the pressure at the center, and what effect would the variable pressure have on weather?
Second, the total atmospheric volume is small (at least compared to the earth). What does this mean for the weather? Are weather patterns likely to be less random and more periodic since there are no external influences?
For the purposes of the question the asteroid is spun such that the inner edge is at 1G. As for the dimensions of the asteroid, length and radius, I’ll leave that up to the reader. I would actually be very interested to know what effect changing the dimensions would have on the weather. Light is provided by a big light bulb that moves back and forth along a guideline through the center of the asteroid creating a 24 hour day/night cycle. At "dawn" it turns on at one pole, then moves across the asteroid to the other pole and dims at dusk until it reaches the brightness of a full moon. It then traverses back to the starting pole and starts again. The light is bright enough to mimic full day on earth and emits in the same electromagnetic spectrum. Excess heat is removed from the system by radiators at the poles which maintain the average temperature of the interior at a comfortable 22 degrees C. There is also an active water cycle with lakes and rivers being supplied by precipitation.