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?