Why does wind inside a stadium blow opposite the wind above the stadium?

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In a (American football) bowl stadium, the flags on top of the goalposts typically fly in the opposite direction of the flags above the stadium, particularly when the wind is blowing heavily. Why is that?

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If the air at the bottom of the stadium is moving east, then the air at the top must move west, since air can’t just vanish. That same air has to stop and turn around and go back to where it started. The result is the air at the top going opposite to the air at the bottom.

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