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I have heard that "Wind is caused by differences of atmospheric pressure" and "Wind is caused by differences of atmospheric temperature" both but I don't know for case that doesn't follow one of this. I will simulate that case.

Case 1 : same temperature but different pressure

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Each room has air inside and has same volume.

Case 2 : same pressure but different temperature

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Each room has air inside and has same volume.

If I removed thin adiabatic wall instantly, which case does wind occurr or both ?

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Movement of air (the wind) is caused by difference of pressure. Air moves from the areas with high pressure to areas with low pressure.

Well, as a result the pressure decreases in the area the air moves from and increases in the area the air moves to.

As a result the pressure would become equal and the wind would stop. But how come Earth's atmosphere has not reached equilibrium during it's billion years history and from time to time the wind does blow?

Even if at some moment the atmosphere is in equilibrium, because the Sun heats the air not evenly, the temperature (and the pressure) of some volumes of air would change and air masses would start moving.

Now to your examples.

If you remove the wall in the first case (pressures are different), the gas will move very fast from left side to right side. There will be a kind of gust of wind - very strong, but short.

In the second example, nothing would happen at the moment you remove the wall. Left half of gas would gradually heat up, right side would cool down. By the time equilibrium is reached concentration of gas in both halves will become equal ($1.5 n_0$), so there would be some net movement of gas from left to right, but that's not a wind.

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    $\begingroup$ Box one has convection while box two has diffusion, to put more precise terms to "wind" and "not a wind." $\endgroup$ – tpg2114 Apr 25 '18 at 14:08
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    $\begingroup$ So wind is caused by differences of atmospheric pressure, but why most people say that "Wind is caused by differences of atmospheric temperature" ? $\endgroup$ – Ro Theory Apr 25 '18 at 17:47
  • $\begingroup$ @RoTheory Isotermal atmosphere would very quickly come to equilibrium and there would be no macro movements, no wind. It is the fact that Sun heats air unevenly that creates pressure differences. Temperature differences is a rootcause of pressure differences, but if your consider some volume of air it moves not towards the area of higher or lower temperature, but towards the area with lower pressure. $\endgroup$ – lesnik Apr 25 '18 at 19:27
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Movement of air also happens by a difference in temperature. After all $pV=nRT$ and temperature can create a pressure differential.

Take katabatic winds:

Warm, dry katabatic winds occur on the lee side of a mountain range situated in the path of a depression. Examples for these descending, adiabatically warmed katabatic winds are the Foehn winds.

Cold and usually dry katabatic winds, like the Bora, result from the downslope gravity flow of cold, dense air.

The role of temperature in the creation of wind is clear. Take also the stack effect:

Stack effect or chimney effect is the movement of air into and out of buildings, chimneys, flue-gas stacks, or other containers, resulting from air buoyancy. Buoyancy occurs due to a difference in indoor-to-outdoor air density resulting from temperature and moisture differences.

You are keeping the pressure constant in the second case, so you will not get a wind. There is always convection from a hot part of a room ( the fireplace) to the cold, even though at same pressure to start with, heat input changes pressure. It is all in the gas law.

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in the second case the air on the right side has half density relative to the air on the left and if there is gravity field it would dictate the direction of air movement that we call wind

even in zero gravity there will be heat transfer from the right side to the left and corresponding expansion of air from left to right

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