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In this picture (from EasyPPL mock exam) and in a few other sources (e.g. https://www.britannica.com/science/katabatic-wind) it is stated that at night, the ground radiates heat and the air passing over it cools and descends.
I have also been taught that during the day, the ground re-radiates heat energy from the sun and that is how the air is warmed.
I'm trying to understand how come the air descending down a mountain, which is radiating heat... cools?
To be accurate, if the mountain's temp is above absolute zero (certainly true on earth), it's radiating heat even in the daytime, but in the daytime the incoming solar radiation would likely be greater than the outgoing radiation.
Air descends down a mountain slope because air is colder (and therefore denser) at higher altitudes (but only up to about 11km altitude). The air would naturally warm (not cool) as it descends (due to increasing air pressure with lower altitude), even without the mountain. If the mountain is warmer than the descending air, the air will also absorb some heat by conduction from the mountain as it travels down the mountainside. The mountain is also radiating heat, and some of the radiated heat will be absorbed by the katabatic flow, but much will be radiated into the higher atmosphere or even into space. This absorbed heat will certainly contribute to warming the katabatic flow, and when (or if) the flowing air is eventually warmed enough so that it's no longer denser than the ambient air, it will stop flowing downhill. Of course, the momentum of the descending air mass will continue to drive the katabatic flow for some additional distance down the mountain.
It is common on clear and windless nights to see frost form on roofs when the air temperature is 4°C to 5°C (about 40° F). That's because the official air temperature is the temperature of the air about 1.5 meters above the ground. The surface cools faster than does the air. The surface cooling can help cool the air under the right conditions (clear sky and little to no wind). Under the right conditions, this temperature inversion can continue for several tens of meters or even several hundreds of meters above the surface.
The cooled air will remain trapped in a valley that everywhere is lower in elevation than surrounding the hills or mountains. But if there's an escape path, that cooled air will be denser than will the surrounding air as the product of density and temperature is proportional to pressure. This eventually makes the surface-cooled descend. The result is a katabatic wind.
Katabatic winds can range from very cold to very warm. As the air descends it warms at the dry adiabatic rate. The temperature of the wind at some lower elevation depends on elevation drop and on the temperature at the higher elevation where the wind formed. The Santa Ana winds in California tend to be rather hot winds because the high desert on which those winds form is only a bit cooler than the coastal regions. The katabatic winds in Antarctica on the other hand form in regions where it is extremely cold. The adiabatic heating there is not sufficient to make those antarctic katabatic winds become warm.
