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There would be heat diffusion, of course, but heat diffusion occurs even without the exchange of fluid parcels between each environment.

We do know that cold air tends to be denser than warm air, and that pressure differences drive the exchange of fluid parcels. But let's assume that there is no pressure difference for the sake of this question.

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I would like you to precise some point : you want to know the rate of air exchange without any air displacement due to pressure difference? If you want to know what happens during a cold winter and the diffusion of heat through walls (which block any air flow), the heat flux will be proportional to the temperature gradient (temperature difference divided by wall thickness). An interesting phenomena occurs when you suddenly open your door to the cold outdoor air : as you said, cold air is denser than warm air, so two layers of warm and cold air separated by a vertical boundary is not a stable configuration. The two layers will mix dynamically through a Rayleigh-Taylor instability.

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Is this really a Rayleigh-Taylor instability? The classical example is the denser fluid on top. In your example there interface is vertical. – Bernhard Nov 2 '12 at 11:37
Thanks for the answer. :) Yes - I would like to know the rate of air exchange without any air displacement due to pressure difference. Interesting point about the Rayleigh-Taylor instability. :) – InquilineKea Nov 2 '12 at 12:38
According to the wikipedia definition the Rayleigh-Taylor configuration is just two fluid of different density separated by an interface that occur when the lighter fluid push the denser fluid, the Rayleigh–Bénard instability configuration is denser fluid in the top of lighter fluid. I guest in some sense it's can be a RT, for me it's look like frontal instability generating gravity current. – aberration Nov 2 '12 at 14:46
In Rayleigh-Bénard instability there is no denser fluid, simply a fluid that is heated up from below. I agree that it might look more like gravity current than RT in the end. Let's bring up another instability then : the kelvin-Helmoltz instability that mixes up those two layers haha – Hervé Elettro Nov 3 '12 at 11:13
In Rayleigh-Bénard instability the fluid at the bottom is heated, this have for effect to decrease the density of the fluid at the bottom, when the gradient become strong enough the instability take place. So yes the fluid have larger density at the top and lower density at the bottom. Using the Boussinesq approximation, the relation used between the density and the temperature is $\rho=\rho_0 \beta T$ with $\rho_0$ the characteristic density, $\beta$ the thermal coefficient (negatif) and $T$ the temperature fluctuation. – aberration Nov 3 '12 at 21:02

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