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Since the movement of the air is induced by buoyancy, i. e. there's a constant force acting on the air, so I would expect the velocity to increase during ascent, much like an object falling down due to gravity accelerates.

But wouldn't a velocity gradient tear the column apart along the axis? Or would it just lead to a corresponding pressure gradient and a deformation of the column (thinner at the top)?

If there is acceleration, there's probably a terminal velocity, just like with objects falling down through the atmosphere. What would that be limited by?

There's two effects that I would like to neglect for the purpose of this question, if possible, the dissipation of the warmer air into surrounding air and the cooling of the air as it rises. I suspect with a high enough $\Delta T$ and a column that is wide enough and not too tall, we can ignore those (for a bit, anyway)?

And just for context, what brought me to this was the idea of having hot air rising through a chimney to act as a pump via a constriction in the chimney that creates a Venturi effect. I was wondering whether it would be better to put that constriction further up in the chimney, so I get a higher pressure difference because of a higher velocity.

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I think it's better to understand what air actually does. It does not rise in a column. It rises in a toroidal vortex or "mushroom cloud". What's more, there is in-flow of air beneath it, and out-flow above, both rotating by coriolis effect (angular momentum). Its energy and momentum are the energy and momentum of that complex. –  Mike Dunlavey Feb 20 '13 at 15:14
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You are correct that since there is a buoyant force, you should expect acceleration. However, since air is not very dense, it quickly reaches its own terminal velocity as dissipative forces proportional in some way to the velocity of the flow. For falling objects, these dissipative forces increase as the square of the velocity. The same generally appears to be true for bubbles rising through fluids, which more nearly approximates your scenario of a rising column of air. Note, however, that the faster the column of air is rising, the higher its Reynolds number. As the Reynolds number increases, the flow becomes turbulent, and the dynamics can change significantly. Once the flow becomes turbulent, it will begin mixing much more rapidly with the surrounding air, and it will no longer be possible to neglect the cooling of the rising air.

If you are neglecting the fact that air becomes less dense with altitude, then there is not a velocity divergence (i.e., the air is not moving faster at the top than the bottom) because viscosity imposes a terminal velocity, and so the air is not rarefied as you propose.

(Also note that if your flow velocity approaches the speed of sound, you can start build some significant pressure gradients, but that's a much more complex problem, and unlikely to arise through simple air warming.)

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