Terra -- with weather systems, we typically use what is called the hydrostatic approximation. It comes out of Newton's second law and a force balance diagram, essentially, and isn't specific to just meteorology. It states that the change in pressure over a change in height is equal to the negative of the gravitational constant times the density. You can plug the ideal gas law into this equation (for density; p = rho*Rd*Tv) to get a relationship with regards to temperature. Further manipulation is possible until we get to something called the hypsometric equation. I won't go into the full details here, though.
This equation allows us to use thickness principles -- essentially, how far of a distance is it between two pressure levels -- in conjunction with atmospheric laws to understand the temperature structure of cyclones. Low pressure systems can have either warm or cold cores, where the temperature in the center of the system is either warmer or colder (respectively) than that outside the center. How these cyclones vary in intensity with height comes from this equation and shows how a warm core surface cyclone must weaken in intensity with height -- as we see with hurricanes. That's but one of many examples -- there are lists around in meteorology texts that deal with many more -- but the jist of everything is that with a cyclone, the strength of the warm or cold temperature anomalies will determine how far the pressure falls.
Note that in a simplified scenario, we can use the ideal gas law; but in reality, the more applicable equation is the moist gas law. Nevertheless, the implications are about the same.
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