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The air at 10.000m altitude is mostly a lot colder than at the ground. Even if you are in the desert in temperatures of 40 degrees Celsius, the air above you will be significantly colder (below 0 degrees Celsius).

Considering that warm air is lighter than cold air, wouldn't one expect that the air on the surface of the earth is colder than in the atmosphere? Why doesn't the warm air just ascend while the cold air descends such that in the end the ground is cold while the high atmosphere is warm?

I suppose it has something to do with the pressure difference, but I can't figure out how exactly at the moment.

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The temperature indeed decreases with higher altitude, but the potential temperature does not.

The potential temperature is, as you guessed, dependent on pressure. It is the temperature of a parcel of air if it were adiabatically brought to surface (or rather, a reference) pressure. Have a look at the wikipedia page: https://en.wikipedia.org/wiki/Potential_temperature

A little thought experiment might help: Imagine an air parcel that is colder than the air below it. It might feel the urge to sink, so it tries to. It then notices, as it sinks a little, it is becomes warmer than the air below by adiabatic compression, so it will float back up to its original height.

Important to note also is that temperature does not determine whether an air parcel sinks or floats; its density does.

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I think it's simply because the ground is warm and the higher atmosphere is further away from the ground.

Most of the sunlight are reflected or absorbed by the ground and then re-emitted as infrared. So the lower atmosphere absorbs more infrared from the ground and is warmer.

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