How Does Hot Air Rise and Affect Weather Systems?

[michelcolman]

You sure about that?

Good question. I was going to write "yes" but then thought about the fact that pressure changes travel at the speed of sound (which is basically the average speed of the molecules give or take a bit). So if lots of molecules at the bottom get some extra momentum, that could be felt at the top rather quickly.

Unlike some of the earlier posts by various people (who basically said I was wrong to think about individual molecules), your rhetorical question was actually helpful.

 

[michelcolman]

OK, it's getting late now, here in Europe, I'm going to bed, thanks for the helpful posts at the end, feel free to post more so I can read them tomorrow.

Things are making a bit more sense now. The macroscopic behavior can apparently be explained on a molecular level after all (which is all I was looking for).

 

[gmax137]

I too have wondered about this, and just like the OP, I know that hot air rises, the question is NOT does it happen. Sorry for most of the replies above that seem to say over & over again YES IT DOES. Geez you guys, the OP didn't say he doesn't believe in cool breezes or hurricanes...

To the OP: take a read thru the thread linked in Post #6. Some interesting thoughts there.

wow this thread really jumped while I was reading the other one...

 

[russ_watters]

Why would a molecule in the middle of the pocket go up faster than the boundary can move?

You are the the one suggesting that they would! The fact of the matter is that the average velocity of the atoms is equal to the group velocity. That's what an "average" is! You are the one suggesting individual atoms will move faster than the boundary, thus destroying the boundary.

After all, any momentum transmitted to that center molecule would ALSO have to travel, colision by collision, trough the pocket, right?

Yep. So what does that mean? It means that on average, those collisions will be stronger in an upward direction! Why? Density! There are physically more molecules per unit area above and they are moving slower, so the molecules below impart momentum upwards.

I can't see why all the molecules would all together start moving up more quickly than the boundary can move.

Obviously, that's a contradiction. The molecules, together, can't move faster than the boundary. They must move exactly equal to the motion of the boundary (with the caveat that as the bubble rises, it expands). Again, that's what a "boundary" is and that's what an "average" velocity is!

The "pocket" is the statistical volume containing the hot molecules. Am I correct in stating (guessing) that the "pocket" will be containing different molecules once it is higher up, so basically the motion of the pocket is the motion of a statistical distribution of speeds rather than a physical object?

No, that's what you keep asserting in your flawed understanding that you are clinging to. The whole point of a "boundary" is that the molecules inside stay inside and the molecules outside stay outside. As with other words with clear meanings that you aren't grasping or are just ignoring, that's what a "boundary" is!

You can also "see" lenticular clouds over mountain tops while in reality air is moving at high speeds THROUGH those clouds and vapor happens to be condensing and evaporating again at the same location, giving the appearance of a stationary cloud. Appearances can be deceptive.

Uh, yah, but you seem to accept that explanation! So why the difference? Is it because when you take a close enough look, it is obvious that thsoe clouds above mountains are not stationary molecules? Again, this is you accepting one obvious reality and not accepting another. I'm not sure why you are being so difficult about this.

Once again, I'm not saying it doesn't...

Well I don't know if that's progress but you most certainly did say exactly that in your first post: "There's no such thing as a "pocket" of hot air that somehow pushes other air away while going up."

 

[jachyra]

maybe the question can be rephrased to "what causes buoyancy at a molecular level"?

 

[jachyra]

actually my previous comment sounds a bit silly. let me explain myself.

what I meant is maybe he's looking for an explanation of buoyancy itself using a kinetic model...

 

[russ_watters]

maybe the question can be rephrased to "what causes buoyancy at a molecular level"?

That's reasonable, but I don't know if it would solve our problem. The OP's issue seems to be with the boundary, not with a point particle explanation of buoyancy. Boyancy on a point to point basis is a simple matter of integrating the pressures at all points around an object. You can do that with a bubble of air, but it doesn't directly address the issue of the boundary.

 

[Cyrus]

Q: Is the term buoyancy physically meaningful at the molecular level.

Answer this question and you can finish this thread off once and for all (The answer is no).

Fluid dynamics discussed here is for a continuum. Pop open any fluids book and it will tell you this in the first chapter.

 

[mikelepore]

what I meant is maybe he's looking for an explanation of buoyancy itself using a kinetic model...

Like the original poster, I found it easy to visualize buoyancy when when we have a submerged solid container. If I assume a simple case with flat and horizontal top and bottom areas, force (rho g A h1) pushes down on the top area, force (rho g A h2) pushes up on the bottom area, net force is upward. It is easy to visualize those particles rebounding off a solid container wall and thereby producing the force. Along those lines, every textbook, topic thermodynamics, subtopic 1st law, has several gas-in-cylinder examples where they say the gas molecules collide with a piston and thereby produce a force that causes displacement and therefore does work on the environment, if the piston isn't locked and is free to move. Consistently the basis was that gas molecules collide with a solid container wall. Now, in the chapter on fluids, it has bothered me that we start out with a boat or submarine or something, but then we suddenly take away the container, just leave whatever shapeless substance may have been inside the container, and buoyancy still works the same way. I have looked for the textbooks to explain why but I never found an explanation.

 

[Borek]

it has bothered me that we start out with a boat or submarine or something, but then we suddenly take away the container, just leave whatever shapeless substance may have been inside the container, and buoyancy still works the same way.

It was told many times - main reason for that is the speed at which the substance inside mixes with the outside is low enough. If you take very small volume (say, microliters) it will get mixed with the surrounding air/water so fast you will be not able to see any buoyancy effects. If we are talking about volumes measured in cubic meters, they live long enough and molecular scale effects don't play any significant role.

 

[Dadface]

In a given time "hot" molecules will undergo more collisions than "cold" molecules and the hot molecules will travel a greater distance as shown by the random walk analysis.The result is that heat will tend to spread from the heat source.The downward spread of hot gas molecules is limited by the prescence of the Earth's surface the molecules transferring some of their heat to the solid/ liquid parts of the earth.The upward spread is less restricted the result being that heat rises in the atmosphere.Of course other factors such as gravity and the nature and position of the heat source have their effects.

 

[michelcolman]

I too have wondered about this, and just like the OP, I know that hot air rises, the question is NOT does it happen. Sorry for most of the replies above that seem to say over & over again YES IT DOES. Geez you guys, the OP didn't say he doesn't believe in cool breezes or hurricanes...

Thank you.

 

[Borek]

In a given time "hot" molecules will undergo more collisions than "cold" molecules and the hot molecules will travel a greater distance as shown by the random walk analysis.The result is that heat will tend to spread from the heat source.The downward spread of hot gas molecules is limited by the prescence of the Earth's surface the molecules transferring some of their heat to the solid/ liquid parts of the earth.The upward spread is less restricted the result being that heat rises in the atmosphere.Of course other factors such as gravity and the nature and position of the heat source have their effects.

What if the hot air is not exactly on the surface, but somewhere higher, and is not in contact with the Earth surface? The same reasoning points to the conclusion that hot air spreads both up and down.

 

[Dadface]

What if the hot air is not exactly on the surface, but somewhere higher, and is not in contact with the Earth surface? The same reasoning points to the conclusion that hot air spreads both up and down.

Yes, the hot air spreads in all directions but any obstacles will tend to give the spread a net directionallity.

 

[michelcolman]

I will give a more detailed post later about what I understood about the process (which you might even agree with), but first I just have to reply to some of the individual arguments given against my previous posts...

You are the one suggesting individual atoms will move faster than the boundary.

The average speed of the individual molecules inside a gas is actually around 1.5 times the speed of sound, I looked it up. Of course they don't move very far at that speed, but that's just what I meant when I said that.

There are physically more molecules per unit area above and they are moving slower, so the molecules below impart momentum upwards.

Just like they impart momentum sideways on the side boundary, yet there you say it is happening incredibly slowly. And the bottom boundary should be going down too, then. In other words, you just described why hot air expands, not why it goes up. But at least we're finally talking about molecules.

Obviously, that's a contradiction. The molecules, together, can't move faster than the boundary. They must move exactly equal to the motion of the boundary (with the caveat that as the bubble rises, it expands).

People keep saying that the speed of the boundary is so slow that it can be treated like the surface of a ballon, much less than the speed of the pocket as a whole. That's what I meant with "faster than the boundary". I meant the speed of the boundary relative to the pocket.

The whole point of a "boundary" is that the molecules inside stay inside and the molecules outside stay outside. As with other words with clear meanings that you aren't grasping or are just ignoring, that's what a "boundary" is!

Suppose you make a model about the location of old people and young people over several centuries. For example, there are large "pockets" of old people in Florida. Would you then say that young people by definition stay inside the boundary around the young pockets? Since the boundary of the pocket is a clear word with a clear meaning? Or might you consider the fact that young people get old while new young people are born, so there's a constant migraton to florida even though the pockets stay at the same place?

Uh, yah, but you seem to accept that explanation! So why the difference? Is it because when you take a close enough look, it is obvious that thsoe clouds above mountains are not stationary molecules?

My point exactly. People would argue that the cloud is obviously stationary, because all you have to do is look at it, "why don't you just accept what everyone can see?" while in reality an entirely different story is going on. In my comparison, you would be the one arguing the cloud is stationary and all molecules stay inside the boundary, since the word "boundary" has such a clear meaning and you can't understand why I have so much trouble understanding such a basic concept.

Well I don't know if that's progress but you most certainly did say exactly that in your first post: "There's no such thing as a "pocket" of hot air that somehow pushes other air away while going up."

I also said that "the effect is obviously real" and used the word "pocket" for "some rigidly confined collection of molecules that can't escape". People started explaining the "pocket" as an area defined by the kinetic energy of the molecules inside and explained how it statistically tends to behave like a single object, which makes more sense.

 

[Borek]

Yes, the hot air spreads in all directions but any obstacles will tend to give the spread a net directionallity.

That doesn't explain why hot air tends to go up, then.

 

[russ_watters]

In a given time "hot" molecules will undergo more collisions than "cold" molecules...

Collisions with what? Obviously, the number of times hot molecules collide with cold molecules must be exactly equal to the number of times cold molecules collide with hot molecules!

 

[michelcolman]

I think I finally understand what's happening, thanks to a few pushes in the right direction from people who managed to understand the question I was really asking instead of telling me off for trying to think for myself.

I even think pretty much everyone is going to agree with this post, how about that

First of all, if the density is the same, hot air tends to expand since the hot molecules with higher kinetic energy can easily push the surrounding cold ones away (while speeding them up and slowing down themselves, aka adiabatic cooling). This is pretty easy to see.

So now let's look at a different situation and let's say a cube of hot air is surrounded by cold air at the same pressure (lower density inside the cube). Suppose the hot air has double the temperature (very hot air indeed) and half the density.

Without gravity, the edges of the cube only tend to change very slowly. An undecided molecule on the boundary will get twice as many pushes from the cold side, but the pushes from the hot side will be twice as energetic. There is no net tendency for the molecules to migrate in a particular direction except for random brownian motion which is a rather slow process, even for molecules traveling faster than the speed of sound. Also, the sharp temperature gradient will smooth out because of the random nature of the collisions but this, too, is a slow process and the important thing is that it does not impose a net tendency on the entire group of molecules. So the "boundary" stays more or less in place.

Now gravity is added to the mix. Apart from the pressures on the sides of the cubes (momentum per unit square of cold molecules flying towards the hot ones and vice versa), a little bit of extra pressure differential is needed to compensate for the weight of the molecules.

If you just look at a cold column of air, a pressure differential will automatically develop so that each individual molecule gets slightly more pushes from below than from above, compensating for its weight. Otherwise all molecules would just tend to fall down and the pressure gradient would automatically develop.

Of course that doesn't mean each molecule stays exactly where it is, only that there's no net tendency for all the molecules to come crashing down.

Away from the cube, the pressure at an altitude just below that of the cube is just enough to support cold air molecules above. In other words, the molecules together are on average giving just enough extra collisions to compensate for the weight of the individual molecules above.

The pressure underneath the cube must obviously be the same because otherwise a horizontal wind would arise (higher energy molecules getting less collisions from the low pressure side and therefore moving that way).

But the hot molecules need less kinetic energy to compensate for their weight, since there aren't as many of them. The kinetic energy due to temperature is already compensated for (the hot molecules are more energetic but there are less of them, so the same pressure is required to keep them in place) but the extra pressure gradient due to gravity exceeds what is needed to counteract their weight. Each hot molecule, on average, is getting slightly more than its fair portion of anti-gravity-pushes, so they all tend to be pushed up.

Now this push is one that has a definite average value (unlike the average value of the pressures on the side which cancel out to zero), so the hot molecules will transmit their extra energy to the molecules above them as well (and meanwhile keep getting more energy from below). This "information" (extra momentum in a definite direction, up) travels at the speed of sound so that very quickly the top of the cube starts moving as well.

At the top of the cube a similar situation occurs: pressure above is lower in exactly the right proportion to roughly keep cold air in place (in fact the gradient works by giving more push from below and less push from above), so the top wants to go up just like the bottom.

So indeed, the whole pocket goes up as a whole! Apart from the diffusion at the edges that always occurs, it does more or less seem to be a collection of molecules going up, unlike the lenticular cloud and the pocket of young people outside Florida. I had posted that latter possibility as such, a possibility, not a theory, and it turned out to be mostly not the case. The actual molecules are going up, not just the distribution of molecules that happen to be hot at anyone time.

I know what you are all going to say now: "see, we were right all along, the cube acts like a single object, the macroscopic laws are correct, I'm glad you saw the light" but the point is that the phenomenon is now explained on a molecular level, which many of you said was impossible (even "nonsensical") to do.

 

[russ_watters]

The average speed of the individual molecules inside a gas is actually around 1.5 times the speed of sound, I looked it up. Of course they don't move very far at that speed, but that's just what I meant when I said that.

It really almost sounds like you are being purposely argumentative here. What we are discussing is the average velocity of the molecules, which is the bulk fluid velocity. The average speed affects only the diffusion rate.

Just like they impart momentum sideways on the side boundary, yet there you say it is happening incredibly slowly. And the bottom boundary should be going down too, then. In other words, you just described why hot air expands, not why it goes up. But at least we're finally talking about molecules.

The reasons why it expands and goes up are the same. They are both a function/product of buoyancy.

People keep saying that the speed of the boundary is so slow that it can be treated like the surface of a ballon, much less than the speed of the pocket as a whole. That's what I meant with "faster than the boundary". I meant the speed of the boundary relative to the pocket.

No. Obviously the boundary isn't moving faster than the bulk velocity. You're misunderstanding what people are saying. In terms of deviation from that average, the boundary isn't moving, it is just getting less defined. It is spreading out (and in).

Suppose you make a model about the location of old people and young people over several centuries. For example, there are large "pockets" of old people in Florida. Would you then say that young people by definition stay inside the boundary around the young pockets? Since the boundary of the pocket is a clear word with a clear meaning? Or might you consider the fact that young people get old while new young people are born, so there's a constant migraton to florida even though the pockets stay at the same place?

That's gibberish. It has nothing whatsoever to do with physics.

My point exactly. People would argue that the cloud is obviously stationary...

No, people wouldn't - or perhaps more correctly, no, you wouldn't. Not even you would, except that you are trying to argue against reality.

 

[russ_watters]

Q: Is the term buoyancy physically meaningful at the molecular level.

Answer this question and you can finish this thread off once and for all (The answer is no).

Fluid dynamics discussed here is for a continuum. Pop open any fluids book and it will tell you this in the first chapter.

Cyrus, I'm an engineer too, but I've been in enough arguments with physicists here to know that they often prefer statistical molecular models to bulk fluid transfer models for such things.

 

[Dadface]

That doesn't explain why hot air tends to go up, then.

It does if the heat source is close to the earth.It is easy to demonstrate,for example,that a flame close to the Earth's surface gives rise to a convection current where there is a bulk upward movement.If the same flame was placed at a higher altitude then the upward movement would be reduced.

Collisions with what? Obviously, the number of times hot molecules collide with cold molecules must be exactly equal to the number of times cold molecules collide with hot molecules!

Yes collisions with other molecules.Consider a hot and a cold molecule within a gas containing numerous other molecules.The mean free path of each molecule would be the same at any instant but the hotter molecule will be moving faster and in a given time will undergo a greater number of collisions.By random walk the displacement of the hotter molecule will be greater.

 

[michelcolman]

It really almost sounds like you are being purposely argumentative here. What we are discussing is the average velocity of the molecules, which is the bulk fluid velocity.

I'm not sure whether I used "speed" or "velocity" in my earlier posts, but never mind.

No, people wouldn't - or perhaps more correctly, no, you wouldn't. Not even you would, except that you are trying to argue against reality.

I really do have to explain everything, don't I?

While I was considering possibilities where the microscopic behavior of the molecules was less intuitive than the common macroscopic description, people were telling me off for not just "seeing" that the air goes up. After all, you can see condensation in air going up etc.

So I replied that you can also "see" a stationary lenticular cloud (condensation, too), so the fact that you "see" something does not mean it is so.

Obviously I don't expect you to think the molecules in a lenticular cloud is stationary, i was COMPARING the discussion about pockets of warm air with a hypothetical discussion about lenticular clouds where, using the exact same arguments ("you can see it") you would be wrong.

This was merely to show that the argument itself was wrong, which only meant I was looking for a more convincing argument.

Now let it rest, it's not important.

 

[michelcolman]

That's gibberish. It has nothing whatsoever to do with physics.

No, only with science in general.

You said the boundary around hot molecules by definition means that all the molecules stay inside. I gave an example of a boundary in a different context (the boundary around young or old people) that ends up containing different people after a while.

I was just pondering the possibility that the "pocket" was just the collection of molecules that happened to be hot at any particular time and, as the "pocket" went up, this merely meant molecules at the bottom had cooled down while molecules at the top had warmed up.

This turned out not te be the case, but it could have been a possibility and you shot it down by saying the molecules can't get out by definition.

So I gave the example of pockets of young and old people. It was not gibberish. But I don't know why I'm wasting time explaining the concept of metaphors or any other linguistic feature instead of doing something useful.

 

[michelcolman]

Cyrus, I'm an engineer too, but I've been in enough arguments with physicists here to know that they often prefer statistical molecular models to bulk fluid transfer models for such things.

EDIT: Oops, I misread that one, he was actually making my point rather than the opposite, so disregard what I wrote below.

That doesn't mean they don't want to understand the microscopic behavior at all. It's just a matter of convenience.

Didn't you derive the ideal gas law from molecular motions in one of your physics courses? I know I did, and I found it quite interesting.

Also, macroscopic models often have to introduce strange and underexplained concepts (boundary layer, etc...) to explain certain phenomena which later turn out to actually be easier to explain at a microscopic level.

There was even a post on one of my earlier threads, about how a wing works, which explained pretty much the whole thing (boundary layer, Coanda effect, etc...) by considering the air to be billiard balls. All of the observed phenomena (yes, before you start typing away again, the boundary layer does exist) were explained through the random collisions of molecules. (lots of molecules, lots of collisions).

Computers are not powerful enough for this kind of simulations (and won't be for a long time) but this would actually be the most accurate model for aerodynamics.

 

[vin300]

Yes collisions with other molecules.Consider a hot and a cold molecule within a gas containing numerous other molecules.The mean free path of each molecule would be the same at any instant but the hotter molecule will be moving faster and in a given time will undergo a greater number of collisions.By random walk the displacement of the hotter molecule will be greater.

Good one. Since the hotter molecules undergo more collisions with the surroundings they lose more heat to the surroundings which intuitively means it expands and cold air comes in.I'm not sure of the mean free path being same everywhere at all instants.Lesser dense air has more free path of its molecules.And remember the hotter air does not remain hot so you cannot say that about the displacement of a single molecule.

 

[Dadface]

Good one. Since the hotter molecules undergo more collisions with the surroundings they lose more heat to the surroundings which intuitively means it expands and cold air comes in.I'm not sure of the mean free path being same everywhere at all instants.Lesser dense air has more free path of its molecules.And remember the hotter air does not remain hot so you cannot say that about the displacement of a single molecule.

Also,don't forget that since the collisions are elastic any kinetic energy lost by one molecule during a collision is gained by the other molecule.

 

[vin300]

Also,don't forget that since the collisions are elastic any kinetic energy lost by one molecule during a collision is gained by the other molecule.

Yes the hotter air transfers energy to the colder and the latter becomes hot, that is what is meant by expansion of hot air and replacement by colder air.

 

[Borek]

That doesn't explain why hot air tends to go up, then.

It does if the heat source is close to the earth.

Trick is, hot air goes up even if it is far from the surface - so you will need different explanation for the same process depending on the distance from the surface.

It is easy to demonstrate,for example,that a flame close to the Earth's surface gives rise to a convection current where there is a bulk upward movement.If the same flame was placed at a higher altitude then the upward movement would be reduced.

Any proof for the latter statement?

 

[Borek]

the point is that the phenomenon is now explained on a molecular level

I admit I have not read through your explanation, but I suspect that if it is correct, it will work also for the baloon with rigid surface. At the scale we are talking about characteristic of the boundary is less important than its existence. That's why these pockets do exist.

 

[Dadface]

Trick is, hot air goes up even if it is far from the surface - so you will need different explanation for the same process depending on the distance from the surface.



Any proof for the latter statement?

Is it agreed that when air is heated the molecules move faster and the air tries to expand?If so do a thought experiment where a heat source is placed in a closed box which is suspended from strings.If a single hole is made in the top of the box the expanding air will move out of that hole.If the hole is at the bottom the air will move out of that hole.The same reasoning applies no matter where the hole is made.It can be concluded that the movement of the hot air is not determined only by gravity and the other factors but is also affected by the structure and geometry of the surroundings.Close to the ground the Earth itself forms a barrier which restricts the downward movement of the hot air but at a higher altitude the downward moving hot air has greater space in which to move.The upward moving air transfers kinetic energy by collision to other atmospheric molecules, as does the downward moving air, but the latter also transfers some energy to the Earth when it arrives there.If the Earth heats up as a result then energy can be transferred back to atmospheric molecules which can only move up or sideways but not down.

 

[jachyra]

If so do a thought experiment where a heat source is placed in a closed box which is suspended from strings.If a single hole is made in the top of the box the expanding air will move out of that hole..If the hole is at the bottom the air will move out of that hole.

If your box has negligible mass, then the entire box would also go up as the air comes out from the bottom hole. Which means your hot air is going up AND down =) I don't know if you can use this thought experiment.

 

[vin300]

Trick is, hot air goes up even if it is far from the surface - so you will need different explanation for the same process depending on the distance from the surface.



Any proof for the latter statement?

The density of air goes on decreasing with altitude, so does the specific heat.You need more heat to raise the temperature of the same volume of air so the temperature difference is less and the convection is less as implied by the formula that follows
The difference of pressure is thus lesser(BtW,g is also a bit lesser)
Air goes up because like static pressure in liquids the difference pressure is due to its depth in the atmosphere
Copied from wiki:
Convection is predicted using the rayleigh number
Ra=ρ0gαΔTL^3/κ μ
Ra=ΔρgL^3/ κ μ
Δρ is the difference in density between the two parcels of material that are mixing
ΔT is the temperature difference across the medium
ρ0 is the average density of the medium
α is the coefficient of volume expansion
κ is the thermal diffusivity
L is the characteristic length-scale of convection
μ is the dynamic viscosity.

The value of L is uncertain

 

[Borek]

The density of air goes on decreasing with altitude, so does the specific heat.You need more heat to raise the temperature of the same volume of air

But we are talking about the difference that have its source in the presence of the near surface, not in the differences that depend on the pressure/density. At least that's where the discussion started and I have not seen anyone stating "we are discussing different case and phenomena now".

 

[michelcolman]

I admit I have not read through your explanation, but I suspect that if it is correct, it will work also for the baloon with rigid surface. At the scale we are talking about characteristic of the boundary is less important than its existence. That's why these pockets do exist.

I am still refining my theory, and now realize that the pocket of hot air is slightly different from a balloon after all (although the behavior looks the same macroscopically).

For the balloon, the momentum is transferred at the surface of the balloon. More pressure (collisions of surrounding air molecules with the tissue of the balloon) below than above, gives boyancy to lift the balloon. Nobody cares what happens inside the balloon (or any other lighter than air object).

In the pocket of hot air, the transfer of momentum is NOT just happening at the boundary! (I only just realized this).

The surrounding area has a pressure gradient that is just enough to keep the cold air stable. A cold air molecule will, on average, receive slightly more collisions from below than from above and this will be just enough to compensate for its weight. That doesn't mean none of them come down: some will still come down, while others will go up, in random brownian motion, but there is no net average tendency. So you won't feel any up- or downdraft.

Inside the pocket of hot air (not just at the top and bottom), the same pressure gradient exists because pressure can even out sideways with the surrounding cold air. So each hot air molecule, too, is receiving slightly more push from below than from above. But this total extra push, summed over all the molecules at a certain altitude, is divided over a smaller number of hot air molecules in the layer above that altitude as compared with the number that would be in a similar slice of cold air. This means the hot ones get more than their fair share of upward gravity-canceling momentum and therefore, on average, all the hot air molecules will tend to go up.

Unlike the rigid balloon, the effect is NOT caused at the boundary but happens everywhere inside the volume of hot air.

Of course you are free to make a statistical consideration about the total momentum being transferred to the volume of hot air from all sides, and come to the correct conclusion that the hot air must be pushed up because otherwise it doesn't add up, but I think my analysis is more detailed.

(At least that's my theory, feel free to correct me if I'm wrong)

 

[Dadface]

If your box has negligible mass, then the entire box would also go up as the air comes out from the bottom hole. Which means your hot air is going up AND down =) I don't know if you can use this thought experiment.

Exactly, the hot air goes up and down,it travels in all directions outwards from the source.The point I am making is that the surroundings have an effect on the net movement of air.