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In high-pressure areas, air falls, but why does high pressure leads to clear, sunny skies?

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The weblink above gives a good overview. To clarify though, high and low pressure are terms that are more general than the H and L that you see on a surface weather map. The H and L that you see on the weather map refer to specific evolved states of air columns, anticyclonic (labelled as H) and cyclonic (labelled as L). The anticyclonic formation leads to air sinking from high elevation (low pressure) to low elevation (high pressure). See for example: https://slideplayer.com/slide/8992497/27/images/18/Role+of+Airflow+Aloft+in+Cyclonic+Activity.jpg I don't own this slide, I am just referencing something I found on google that is correct.

This movement of an air packet from low pressure to high pressure causes it to be compressed by the surrounding air, and the compression is ultimately caused by the transfer of kinetic energy of surrounding air into the air packet, thus raising its temperature above the dew point (condensation temperature), and clouds evaporate. A similar phenomena contributes to the so-called rain shadow (pattern of little rain) on the leeward side of mountains.

Conversely, compressed ground air that is rising due to convection, orographic lift, or at a weather front, especially a cold or warm front, undergoes expansion as it pushes against the lower-pressure higher air. The random collisions transfer energy to the surrounding air, and the rising air packet cools. If the air rises fast enough and there is sufficient moisture, the temperature will drop below the dew point (condensation temperature) of water, and a cloud will form.

This is why mountains get so much more precipitation than other areas. Many people misconceive the origin of snowfall over mountains as being from their height, but there is air at the same height all over the surface of the earth. The key difference is that the air moving across the continent interacts with mountains the same way a marble encounters a ramp. The air is pushed up and expands, leading to cooling and cloud formation. All of this can be demonstrated with the cloud-in-a-bottle experiment.

These patterns are certainly at play in so-called mid-latitude systems. None of this is to exclude other weather factors at work.

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