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How do we derive equations which relate the temperature and pressure anywhere in the atmosphere? (assuming earth is still so no coriolis effect and flat if that makes the problem any easier). Like suppose we had an even temperature and pressure distribution but shocked a volume of air to a higher temperature. We can't just use $PV =nRT$ because it's an open system and $P$ isn't constant throughout the volume, and we aren't necessarily at equilibrium. Are there some sort of field equations we can use to model how the system evolves?

Ultimately I wanted to be able to explain basic weather phenomenon, such as why high pressure systems are colder in temperate climates? Maybe I'm overcomplicating it?

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Why high pressure systems are colder in temperate climates?

That's not necessarily the case. High pressure systems in the summer oftentimes result in brutally hot weather. A summertime warm core high pressure system has higher than nominal pressure well up into the atmosphere. The falling air at the center of a warm core high pressure system heats adiabatically, resulting in warmer than average temperatures, and the long period of time when the sun is shining versus the short nighttime exacerbates the situation.

It's wintertime when high pressure systems are often associated with cold weather. The core core high pressure systems that set up in high latitudes can make their way to temperate latitudes in the winter. The high pressure in these systems is rather shallow; pressure at higher altitudes can be below normal. This reduces adiabatic heating. The clear skies that usually accompany with high pressure cells mean that nighttime cooling can be quite strong, much stronger than the weak daytime cooling that occurs in winter.

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