Usually it only rains when the temps are above 35 or 40, cooler than that we still get snow. No problem at all for the heat to remain in the chimney for about 10 or 12 feet.With a chimney as tall as yours above the roof, I'd be surprised if there was any heat left in the rising smoke.
EDIT: If you were referring to the cold winter months, it rarely gets below 10* here and that would be only during a winter storm system passing by. Usual low temps in the early morning are above 20* and daytime temps barring a storm are close to 30 or above. The heat will rise.
In the summer months the sun will heat that pipe well over 130* and purge the hot air out the top. With about 20% humidity the air is very dry. Moisture would wick out rather fast, the chimney will cool at night and draw in cool dry air and repeat the heat process again the next day.
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