How Does Hot Air Rise and Affect Weather Systems?

[DrGreg]

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.

It is understood that the box itself must be too heavy to rise due to buoyancy. If the hot air is initially under pressure when you drill the hole, air will escape at the top or bottom simply due to the excess pressure.

Consider what happens if you drill two small holes, one at the top and one at the bottom. If there is excess pressure, then at first hot air will escape from both holes due to pressure. But after the pressure has equalised, yes, more hot air will escape from the top, but at the bottom cold air will rush into the box. How does your theory explain that?

 

[michelcolman]

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.

Then why does my barbecue have an air hole at the bottom? If I open that hole, the coals start burning a lot more intensely and you can actually feel the air current through the hole. So I don't think a flame close to the surface would give more convection, on the contrary. Air below the flame seems to give a significantly faster convection.

For a different example, it is a well-known phenomenon that hot air will tend to build up in a large bubble at the surface and then suddenly let go. This, too, seems to indicate boyancy is lower if the pocket is touching the ground.

(although I can't quite come up with an explanation for that yet)

 

[Dadface]

It is understood that the box itself must be too heavy to rise due to buoyancy. If the hot air is initially under pressure when you drill the hole, air will escape at the top or bottom simply due to the excess pressure.

Consider what happens if you drill two small holes, one at the top and one at the bottom. If there is excess pressure, then at first hot air will escape from both holes due to pressure. But after the pressure has equalised, yes, more hot air will escape from the top, but at the bottom cold air will rush into the box. How does your theory explain that?

Precisely ,hot air rises and cooler air comes into take its place.I am not disputing that but I am stating that the surroundings have an effect on the net movement of hot air which is upwards.The hot air which initially comes out of the lower hole can diffuse sideways beyond the extremities of the base of the box and then this also can rise.Suppose the box was in an atmosphere which for the purposes of this analysis can be considered as being totally remote from the Earth's surface.Can you explain in what direction the net movement will now be?

Then why does my barbecue have an air hole at the bottom? If I open that hole, the coals start burning a lot more intensely and you can actually feel the air current through the hole. So I don't think a flame close to the surface would give more convection, on the contrary. Air below the flame seems to give a significantly faster convection.

For a different example, it is a well-known phenomenon that hot air will tend to build up in a large bubble at the surface and then suddenly let go. This, too, seems to indicate boyancy is lower if the pocket is touching the ground.

(although I can't quite come up with an explanation for that yet)

To answer your question we need to refer to the theory of combustion and perhaps this is irrelevant to the main point being discussed.

 

[Borek]

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

If it happens inside of the volume, there is no need for boundary - nor need for the cold air around...

 

[michelcolman]

If it happens inside of the volume, there is no need for boundary - nor need for the cold air around...

Not so fast. The cold air around creates the pressure gradient. Away from the pocket of hot air, this pressure gradient is exactly sufficient to keep the cold air stable. A hot air pocket, inserted anywhere in this cold air, will necessarily take the same pressure gradient because the pressure can even out horizontally. This pressure gradient, everywhere inside the volume, makes the hot air particles go up like I explained.

 

[michelcolman]

To answer your question we need to refer to the theory of combustion and perhaps this is irrelevant to the main point being discussed.

No need to explain, just clarify that you believe a flame will cause more updraft if it is burning just above the surface of the Earth instead of being higher up with plenty of fresh air underneath? Maybe I misunderstood, but that's what I thought you said. And it seems contrary to my experience, although I could be wrong.

 

[Dadface]

No need to explain, just clarify that you believe a flame will cause more updraft if it is burning just above the surface of the Earth instead of being higher up with plenty of fresh air underneath? Maybe I misunderstood, but that's what I thought you said. And it seems contrary to my experience, although I could be wrong.

For reasons I tried to explain in my earlier posts I think that the higher up the flame the greater the directionality of the hot air i.e less will go up and more travel in other directions.

 

[Lok]

Remove the Earth and leave just the hot and cold air behind - which way is up and which way is down? From this, you could probably say it has something to do with gravity (I might be wrong).

The hotter air would still push the colder air out of it's place because of faster expansion.

 

[rcgldr]

One assumption being made here is that it's the same original heated air molecules that travel upwards. As mentioned heat is trasnferred from the hotter molecules to the colder molcules, increasing the density of the previously hotter molecules, and decreasing the density of the colder molecules.

What is is traveling upwards is a bubble of heat and a somewhat changing set of molecules, not the original set of molecules that initiated a thermal at near ground level. Rather than dissappating in all directions, a heat bubble (thermal) travels upwards and in a circular motion (like a tornado). Cold air tends to flow downwards around the perimeter of the thermal, while hot air rises up internally. At the bottom part of a thermal, the surrounding air is drawn inwards into the thermal, and the thermal expands in diameter with altitude.

http://www.thermikwolke.de/thermals/pdf/page25.pdf

http://www.thermikwolke.de/thermals/pdf/page30.pdf

http://www.rcsoaring.com/docs/thermals_2006.pdf


wiki link:
http://en.wikipedia.org/wiki/Thermal

 

[jachyra]

For reasons I tried to explain in my earlier posts I think that the higher up the flame the greater the directionality of the hot air i.e less will go up and more travel in other directions.

I understand what you are trying to say. Thanks for you earlier post. It helped me look at the phenomenon in a different perspective.

 

[jachyra]

Jeff Reid,

Great links! Really helps to show what Borek and the others have been trying to explain since the beginning.

 

[vin300]

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".

Good.Now bring to light your theory of convection without pressure difference :zzz:

 

[michelcolman]

One assumption being made here is that it's the same original heated air molecules that travel upwards. As mentioned heat is trasnferred from the hotter molecules to the colder molcules, increasing the density of the previously hotter molecules, and decreasing the density of the colder molecules.

What is is traveling upwards is a bubble of heat and a somewhat changing set of molecules, not the original set of molecules that initiated a thermal at near ground level. Rather than dissappating in all directions, a heat bubble (thermal) travels upwards and in a circular motion (like a tornado). Cold air tends to flow downwards around the perimeter of the thermal, while hot air rises up internally. At the bottom part of a thermal, the surrounding air is drawn inwards into the thermal, and the thermal expands in diameter with altitude.

http://www.thermikwolke.de/thermals/pdf/page25.pdf

http://www.thermikwolke.de/thermals/pdf/page30.pdf

http://www.rcsoaring.com/docs/thermals_2006.pdf


wiki link:
http://en.wikipedia.org/wiki/Thermal

Too bad the links only contain the word "molecule" once (in the rcsoaring one), and even there it's in a different context. So all the links are macroscopic, while the point of the discussion is what happens at a microscopic level.

You did say something interesting in your own text, though. You say that individual molecules are not traveling up, but rather their momentum is transferred up and the bubble ends up containing different molecules?

I can see how this is happening at the boundaries (momentum is always being exchanged there, and molecules move through the boundary in different directions all the time) but for the vast majority of molecules, I thought they should simply move up with the bubble. Am I wrong there? Where did you find this information?

 

[michelcolman]

Jeff Reid,

Great links! Really helps to show what Borek and the others have been trying to explain since the beginning.

The links are interesting, but a bit pointless in this discussion (unless I missed some interesting part of one of them).

Yes, hot air rises, and you can describe the whole thing by looking at densities and pressures, but that's all an approximation, a very useful approximation, but an approximation nonetheless.

I wanted to know what was happening at a microscopic level, with individual molecules being pushed around. Just like I learned how to derive the ideal gas law that way, I now wanted to see convection explained that way too.

Since few people seemed to even understand what the question was, I finally came up with a theory of my own, which nobody has really commented on yet (except by saying "I admit I haven't read it yet")

Too bad, because it's really not far away from the macroscopic view! If the other posters would just take the time to read it, they would find it corresponds pretty much to the macroscopic theories they love so much, only in slightly more detail.

Executive summary of my theory: a pressure gradient has built up in the atmosphere (cold air) which is just right to keep cold air stable in the field of gravity. Each molecule gets slightly more collisions from below which, for a cold molecule, is just enough to support its weight (on average! they are all still moving around randomly, some get more pushes than others, they just don't get a net tendency in any particular direction). Inside the hot bubble, the same pressure gradient exists but the total extra momentum transfers from below are now divided over fewer hot molecules. This means every individual hot molecule (on average) gets slightly more than its fair share of gravity-canceling momentum. The sum of all pushes exceeds its weight, so it goes up. Of course this is only a general tendency, superimposed on their own brownian motion zigzagging faster than the speed of sound, but since it has the same average sign everywhere, it does tend to move the whole bubble up as a whole.

There, was it really that complicated/impossible/"nonsencical"?!

 

[Dark Horse]

Hi,
I had the same thought, 'why does hot air rise', or to put it another way 'why does a more energetic gas molecule rise when it essentially has the same mass as a cold molecule'
This is what I postulated

Air gets less dense as you increase in altitude.
If you heat up a bit of air the average velocity of the individual gas molecules increases, so the pressure of that patch of air increases and in an attempt to equalize with the surrounding air expands. Since the pressure above this parcel of hot air is ever so slightly less than the pressure below this parcel of air there are less molecular collisions from above so the net effect is the pocket of hot air moves upwards; you can feel the draft caused by the mass of air moving.

Watching food dye circulate in a glass of water would sort of confirm that it is the same molecules which are moving rather than just the energy being transferred to another group of adjacent molecules or some form of diffusion is dominant.

It would be an interesting experiment to see how convection currents flow in a centrifuged medium.


cheers
Martin

 

[rcgldr]

Inside the hot bubble, the same pressure gradient exists but the total extra momentum transfers from below are now divided over fewer hot molecules.

Except the hot bubble doesn't rise if the surrounding air is at the same temperature. The hot bubble needs the surrounding air to be cooler (and denser) in order for the hot bubble to rise.

 

[michelcolman]

Except the hot bubble doesn't rise if the surrounding air is at the same temperature. The hot bubble needs the surrounding air to be cooler (and denser) in order for the hot bubble to rise.

The cold air created the pressure gradient. Without the cold air there would be no excessive pressure gradient so the hot air would not rise.

 

[mikelepore]

Here's my hypothesis. You know that toy with the five stainless steel balls in a row that have nearly elastic collisions with each other - I don't know what it's called - Sprott's book 'Physics Demonstrations' just calls it the colliding balls toy ... I think the group of molecules in the hot air do that with each other for a while, exchanging momentum among themselves while they stay segregated in a group to some extent. This allows them as a group to be subjected to the larger pressure below them and the lesser pressure above them.

 

[Dark Horse]

I am going to have to revise my answer to 'I don't know'

Pressure = number of molecular collisions per area * average momentum exchange per collision.

so in a hot body of air each molecule has a statistically higher energy so exerts more force on the surrounding molecules and expands until the number of collisions has decreased to the point where the forces are in equilibrium. Sounds ok so far, agrees with observation that hot gasses expand and cold gasses contract.
Previously I said the hot air rose because there were statistically more collisions from below than from above due to minute changes in are density due to gravity but that is true for cold air as well as hot air and I don't observe cold air rising hence I must revise my answer to 'I don't know'

 

[rcgldr]

I'll take a shot at this. The higher temperature in a thermal decreases it's density compared to the surrounding cooler air. Then it's a buoyancy effect. Ignoring dynamic pressure effects, the total pressure = static pressure + ρ g h, where ρ is density, g is the force of gravity, and h is height. The pressure decreases with altitude, and for any given volume of air, the weight of that volume of air is opposed by an equal and upwards force on that volume of air (otherwise the air would be accelerating upwards or downwards). If that volume of cooler air is replaced by hotter air with lower density, than the upwards force is more than the weight of the volume of the hotter air, and that hotter air is pushed upwards.

http://en.wikipedia.org/wiki/Buoyancy

 

[the_awesome]

I'm a noob. Just want to ask. Air is made up of different gases - water vapor, nitrogen, carbon dioxide, etc. So why not look into the effects of each separate molecule, and the characteristics of those elements?
Could it be, that the different temperature effects of each element while being gaseous, caused them to separate into their own elements, which gives the appearance that it is hot and cold air?

For example:
Oxygen density is 1.429 g/L, while nitrogen density is 1.251 g/L.
Oxygen heat capacity is 29.378 J·mol−1·K−1, nitrogen is 29.124 J·mol−1·K−1.
Therefore, with the same amount of energy, nitrogen's temperature will be higher than that of oxygen. Since nitrogen has a lower density, it will rise and be on-top of oxygen. Since it will be warmer than oxygen - due to its specific heat capacity - it will give the effect that warmer air is on top of colder air.

Note that I'm a noob, but to me this is logic.

 

[Borek]

Could it be, that the different temperature effects of each element while being gaseous, caused them to separate into their own elements, which gives the appearance that it is hot and cold air?

This can be easily check experimentally - and separation is not observed.

Similar idea was planned to be used for separation of isotopes, see cryogenic distillation.

 

[the_awesome]

This can be easily check experimentally - and separation is not observed.

Similar idea was planned to be used for separation of isotopes, see cryogenic distillation.

I don't really understand the site. So yeah, whatever lol.
Another question:
Methane and carbon dioxide has a higher density than most gases. And yet, when it is being produced at coal plants, etc, you see it given off - and it rises deep into the atmosphere. So therefore, it is because heat as lowered the density - making it lighter than normal air molecules such a oxygen and nitrogen.

 

[Dark Horse]

The classical reason for why hot air rises is
A volume of hot air expands there for it has less density that the surrounding air and therefor rises. The question here is how does this work on a molecular level since hot molecules weigh the same as cold molecules (ignoring E=MC2 stuff)

(which then poses another question, how does a molecule of say N2 store heat energy?, How does He store heat energy? Please don't digress, open another forum if you want to answer this one)

 

[mikelepore]

which then poses another question, how does a molecule of say N2 store heat energy?, How does He store heat energy?

The temperature of a substance is a measure of the average kinetic energy of the individual molecules, which have a statistical distribution. Heating a gas means to give many of its molecules greater speeds.

 

[the_awesome]

Since we imagine gas particles as moving in a straight line (until collisions
happen - in which case momentum is conserved), then we have to assume that at
any given time, there will be gas particles moving "down" instead of "up". If
there are no other forces acting on the gas, then a gas would simply disperse in
all directions independent of its relative temperature.

However, if we factor in the effect of gravity on these particles, then we have to
imagine that each particle not only has its kinetic energy (a vector containing
speed and direction) (KE) but is also affected by gravitational force (a vector
that is always pointing towards the center of the Earth) (g). If we follow this
reasoning, then a particle would have a higher "downward" (KE + g) vector than an
"upward" vector (KE - g).

As such, only particles with high KE will tend to be moving up. Even at collisions,
the resultant motion will favor the downward motion since this is the direction of
g. But since, on average, hotter particles will have higher KE's, then those tend to
move up.

As air (or any other medium) is heated, its molecules vibrate more rapidly, and move
over a greater distance. The net result is that the average spacing between molecules
will increase, and therefore the density of the material will decrease (same mass in a
greater volume = less density)
Why does the hot air rise? Technically it does not.
It is passively displaced upwards as the cooler and DENSER surrounding air moves
downwards, propelled by gravity. Hot air can only rise if there is cooler air to
push it upwards.
The motion and collisions of individual molecules has very little to do with the
process, as temperature is a product of the AVERAGE motion of the molecules in a
sample of air.

You will not find an explanation in terms of individual molecules because
there is not one. Convection is a collective phenomenon. The masses of
fluid that move convectively are VERY LARGE compared to the sizes of
the individual molecules and to the mean free path of the molecules.

At the level of individual molecules, energy is transferred in
collisions between molecules and in movement of molecules from one place
to another. This energy transfer is conduction, of course. In real
fluids, molecules do not go very far before running into each other. So
molecular transport through the bulk fluid (diffusion) is slow over long
distances. Consequently, masses of hot or cold fluid really do tend to
travel as groups, and the model of contained masses is appropriate.

If molecules did not interact, if they truly acted as ideal gas point
masses, hot fluid would not spontaneously rise. One mass of gas would
not exert a pressure force on another. Fast-moving (hot) molecules
would simply spread through the available space very quickly, and that
would be that. The speeds (related to temperatures) of gas molecules at
high altitudes would be lower than at low altitudes, because of their
gravitational potential energy (projectiles slow down as gravity pulls
against them). So you would still observe a "lapse rate," (temperature
decreasing as altitude increases) but hot air (or other fluid) would not
spontaneously rise unless it were contained.

In real fluids, that containment is provided by the molecules getting
physically in the way of each other.

http://www.Newton.dep.anl.gov/askasci/chem07/chem07217.htm"

I have a theory that contrary to popular belief hot air does not rise, (at least of its own accord) but that it only appears to do so because it is forced out of the way by colder, denser air. Does anyone agree?

You may be right. The force that they respond to is gravity. Cold air, being denser, will, in a fluid medium, be drawn closer to the earth, displacing warmer, less dense air.

http://uk.answers.yahoo.com/question/index?qid=20061110052203AAyQKx9"

 

[Dark Horse]

---------quote-----------
"However, if we factor in the effect of gravity on these particles, then we have to
imagine that each particle not only has its kinetic energy (a vector containing
speed and direction) (KE) but is also affected by gravitational force (a vector
that is always pointing towards the center of the Earth) (g). If we follow this
reasoning, then a particle would have a higher "downward" (KE + g) vector than an
"upward" vector (KE - g).

As such, only particles with high KE will tend to be moving up. Even at collisions,
the resultant motion will favor the downward motion since this is the direction of
g. But since, on average, hotter particles will have higher KE's, then those tend to
move up."
---------end quote---------

Isn't what you said also true of cold molecules?
Why would the hotter particles will have higher KE's, tend to move up?
Are you implying its some sort of escape velocity issue and the slower molecules just can't make it to the higher altitudes and the million or so collisions made every second is just noise in the macroscopic molecules trajectory?

 

[Cleonis]

I know a hot air balloon goes up because the density of the hot air inside is lower. But what about free air, not trapped in a balloon?

[...] why would this cause the air to rise, to such an extent that it even draws surface winds that fill the gap?

(Admittedly I have not read up on the thread, so I may be duplicating here an explanation already given.)

Let me consider the simplest setup I can think of where the considered phenomenon occurs. In an enclosed room a volume of air is present. On one side of the room there is a heater. The output of the heater is conductive; the air that is in contact with the heater is heated up. Outside the room the temperature is lower so that heat is conducted out of the room.

Then a circulation will commence. The heated air does not have enough weight to counterbalance the denser air on the unheated side of the room. As a consequence the air mass in the room will barrel roll. Since heat is conducted away from the room the heat unbalance persists, so the flow pattern persists.


I believe this translates to large scale motion in the atmosphere.
In different areas air mass is heated differently. The expanded air in warmed areas does not have enough weight to counterbalance the denser air of colder areas, and a pattern of convection flow will arise.

When air mass rises it is always in the process of being _pushed_ up. Moving up is always going against gravity. Surface winds are not drawn in; inflow of surface winds is the cause of less dense air mass being pushed up.


Cleonis

 

[turbo]

When air mass rises it is always in the process of being _pushed_ up. Moving up is always going against gravity.
Cleonis

This notion is WRONG. Warm air is not being "pushed up". It rises because masses of warm air are more buoyant because it is less dense. Do you have some alternative physics in which an envelope of hydrogen gas is being "pushed up"? Please don't mislead young people who might come here to learn.

 

[Dark Horse]

In a macroscopic context, where you talk about air pressures, densities, volumes convection is pretty easy to understand; air heats up, it expands, is less dense than its surroundings and hence rises.
But the question is how does this work at the molecular level?
Hot molecules weigh the same as cold molecules hence why does a hot molecule rise?

cheers

 

[the_awesome]

In a macroscopic context, where you talk about air pressures, densities, volumes convection is pretty easy to understand; air heats up, it expands, is less dense than its surroundings and hence rises.
But the question is how does this work at the molecular level?
Hot molecules weigh the same as cold molecules hence why does a hot molecule rise?

cheers

You will not find an explanation in terms of individual molecules because
there is not one.

 

[rcgldr]

Warm air is not being "pushed up". It rises because masses of warm air are more buoyant because it is less dense.

If "pushed up" means an upwards force, than it's a correct explanation. From wiki:

Any object, wholly or partly immersed in a fluid, is buoyed up by a force equal to the weight of the fluid displaced by the object

http://en.wikipedia.org/wiki/Buoyancy

For the surrounding gas or liquid the pressure at any point (ignoring dynamic pressure effects related to net velocity) is equal to static pressure + density x gravity x height, where gravity is a negative number.

In the case of warm air, the pressure component related to height is less because of the lower density, but the surrounding air will compress the warmer air's static pressure component with height.

The thing to note is that the static pressure component isn't directional, but the gravitaional pressure component is directional, it decreases with height. For the surrrounding air, the downwards force of gravity is equally opposed by the upwards pressure differenital force and that air doesn't move. However in the case of the warmer, lower density air, the downwards force from gravity is exceeded by the upwards pressure differential force, and that volume of air is "pushed upwards" by the imbalance in the forces.

At the molecular level, in the surrounding air, the pressure decreases with height, but the movement of the molecules remains random so there is no net movement. In the volume of lower density hotter air, the movment of the molecules is less random there is a net upwards component of velocity. The cold air directly under a thermal bubble is moving upwards along with the bubble, and cold air surrounding the bubble is moving downwards to fill in what would otherwise be void created by the upwards movement of the lower density bubble.

There's some initial state where a thermal is released and accelerates, but quickly reaches some relatively low vertical velocity. I don't know what causes thermal inversions or how thermal layers get trapped near the ground and get released as bubbles.

 

[ManDay]

I haven't read through the thread but I just wanted to give an answer:

The erratic movement of each molecule due to the fluids inner energy is not a matter here. THe point is that as a consequence of this movement a "hotter molecule" takes up more space than a cooler.

That said, speaking in terms of "billiard balls", where each ball represents one molecule, the hotter molecules are balls of the same mass, yet a larger diameter than the cooler ones.

Why would the bigger balls be pushed upwards? Good question, given that every molecule is exerted the same graviational force on, and honestly I can't explain with my limited knowledge of inter-molecular correlation. It has something to do with pressure (force/area) at last - that's all that I can certainly state. I could only try to give a vivid illustration, but I think, judging from your OP it will 1.) not really satisfy you 2.) be something you can probably think of yourself.

 

[michelcolman]

You will not find an explanation in terms of individual molecules because there is not one.

I gave one earlier in this thread.

I really don't see your problem. Molecules do behave, to a great degree of accuracy, like perfectly elastic billiard balls. You can derive the ideal gas law from the average velocities and masses of the individual molecules in exactly this way. There is absolutely no reason whatsoever why it would be impossible to describe the rise of hot air in the same way too.

Macroscopic explantations (buoyancy, etc...) are very useful and simple, but they will always remain an approximation. They work because statistical distributions of molecules do, for most intents and purposes, behave like continuous fluids (even though no such thing actually exists in reality). It is obviously much easier to just pretend air is a continuous medium rather than consider quadrillions of tiny molecules bouncing around at well over the speed of sound.

But that does NOT mean it is impossible to do. It is not as easy as it first looks, and you will need to use some statistical averaging, but is is possible! It has to be, after all, air IS a bunch of molecules and NOT a continuous medium.

 

[michelcolman]

There's some initial state where a thermal is released and accelerates, but quickly reaches some relatively low vertical velocity. I don't know what causes thermal inversions or how thermal layers get trapped near the ground and get released as bubbles.

In order for hot air to rise, it has to be surrounded (horizontally) by cold air. The cold air forms a pressure gradient, which is horizontally transmitted to the hot air so it takes the same gradient. For cold air, this pressure gradient is exactly what's required to keep the cold air in place (otherwise it would rise or descend, changing the pressure gradient). For the hot air, which is less dense, the pressure gradient exceeds the weight of the hot air, so it is pushed up by the differential pressure.

Thermal layers and inversions can remain trapped for a long time if they are not horizontally connected to surrounding cold air. For example, a large area of land that is heated simultaneously, an area surrounded by higher terrain, etc... In this case they simply create their own pressure gradient which is exactly adapted to their weight.

From a molecular point of view:
- pressure is simply the net effect of billions of collisions of molecules.
- a volume of hot air with the same density as cold air will naturally have more pressure since every molecule carries a lot more momentum, and the collisions will be more frequent too.
- hot air expands because fast molecules tend to push the slow ones away. The situation stabilises when the density of the hot air has decreased enough to make the pressure equal (less molecules times more momentum per molecule makes the same pressure). Since any molecule on the boundary now gets the same total amount of push momentum from both sides, it will not tend to go anywhere on average. Individual molecules will still move about randomly, of course, but there will no longer be a general tendency for expansion.
- The vertical pressure gradient means each cold air molecule is getting slightly more collisions (or slightly more energetic ones) from below than from above which, on average, is just enough to support its weight. Obviously individual molecules are still moving in all directions randomly, but there is no general tendency for all the cold air molecules to start moving in any particular direction.
- The hot air molecules, too, are getting slightly more pushes from below than from above. Only, the same amount of total momentum (all the collisions from below vs. above) is distributed over a smaller number of hot molecules due to the lower density. On average, the hot molecules get more than their fair share of momentum from below, more than what would be required to support their weight, so they start moving up. Once again, individual molecules may very well go in opposite directions, you just add a small upward vector to the random vectors of the individual molecules, so that on average they tend to move up.
- The random motions of the molecules are very fast (a bit more than the speed of sound) but they don't go very far between collisions, only a few nanometers at a time before dashing off in another direction again. Therefore, this random brownian motion does not tend to take individual molecules very far. That's why the small upward tendency turns out to be very noticeable and does move pretty much the entire volume of hot air up as if it was one physical object (even though it isn't really). In fact this is precisely why macroscopic approximations work so well.

Obviously, once the upward motion starts, there will be a pressure reduction below the bubble and an increase above, which will stabilise the speed of the bubble.