Interestingly, there actually are regions of the atmosphere (such as most of the stratosphere) in which temperature actually does increase with height. In these regions, because the hot air is on top where it is supposed to be, there is very little mixing between the layers of air.
In the lower atmosphere (troposphere) the air is heated from below by the earth, causing the lower regions to be warmer. When this warm air rises, however, it is able to expand and therefore it cools down. This whole process means that there is a lot of mixing in the lower atmosphere.
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