I've got a contradiction in my understandings of convection and Brownian motion. It seems to me that the assertion that gas molecules would be statistically unbiased in their direction (Brownian motion) fails to explain the macroscopic behavior of unbounded lighter molecules. For instance: why does a flame point upwards? I understand that heated gases are less dense than colder gases, of similar chemical composition, but why don't they just expand in all directions? Doesn't the Brownian model require hotter molecules to just vibrate faster, but still move in an unbiased direction?

I can see why helium in a balloon would rise in an air atmosphere. Basically the pressure differential between the top and bottom of the balloon results in a net upwards force on the balloon which is lighter than the net hydrostatic pressure difference across the height of the vessel. This behavior of rising lighter gases goes funny when the container disappears and the molecules are allowed to diffuse.

Can a heat and mass transfer professor help me with this conundrum?