Parcel theory -- how can there be buoyancy with miscible gases?

[raxp]

TL;DR Summary

Parcel theory holds that the reason hot air rises is that its density is lower than the surrounding cold air, leading to Archimedean buoyancy. While this explanation is perfectly reasonable for a hot air balloon where there is a mechanical interface between the hot and cold gas, for warm and cold regions of an ideal gas, there can be no mechanical interface between the two.

Parcel theory holds that as air is heated, it expands. Its density hence decreases and the hot air "floats" upwards, pushed by the colder, more dense air surrounding it.

It is an experimental fact that hot air rises, but the explanation from buoyancy seems suspect. In a gas, all motions are uncorrelated, and the collision cross-section for each molecule is minuscule. How can there then be a mechanical force exerted between two regions of the same gas, differing only in their temperature? Each single molecule does not "know" to which parcel it belongs and may pass freely between them, unlike the case where there is some mechanical interface (the fabric of a hot air ballon, say) between the two regions.

 

[Baluncore]

How can there then be a mechanical force exerted between two regions of the same gas, differing only in their temperature?

When the "Mean Free Path" is very short, and the size of the parcel is large, the gasses do not mix rapidly. There is plenty of time for the parcel to rise or fall before it becomes mixed.

At sea level, the parcel dimension is 100 m or more, while the MFP is less than 100 nm. That is a difference of 9 orders of magnitude.
See the equations and table at the bottom of this section.
https://en.wikipedia.org/wiki/Mean_free_path#Kinetic_theory_of_gases

 

[Drakkith]

At sea level, the parcel dimension is 100 m or more, while the MFP is less than 100 nm. That is a difference of 9 orders of magnitude.

Ooh. I had no idea the mean free path for air at sea level was so short!

 

[sophiecentaur]

Ooh. I had no idea the mean free path for air at sea level was so short!

Not intuitive; I agree. But other numbers count too. There are zillions of molecules involved in diffusion between / within parcels of air. Most of our experiences of what a Science Teacher would call Diffusion (smells that drift around the room) do not involve stationary air.

 

[A.T.]

How can there then be a mechanical force exerted between two regions of the same gas, differing only in their temperature?

Two neighboring regions of the same gas at non-zero pressure always exert equal but opposite forces on each-other, even when at the same temperature.

Each single molecule does not "know" to which parcel it belongs and may pass freely between them,

Yes, but that mixing is much slower than the propagation of forces via repeated collisions.

 

[sophiecentaur]

Two neighboring regions of the same gas at non-zero pressure always exert equal but opposite forces on each-other, even when at the same temperature.

Caveat here: But an N3 pair doesn't imply equilibrium