Is the issue pressure or density? Why would the pressure be higher at one point then another?
The following assumes the burner is not operating, and it hasn't been recently operating. Temperature, pressure, and density gradients are at equilibrium.
My image is that the interior and exterior temperature, pressure, and density are the same at the bottom. There is no physical barrier to separate interior from exterior, and they are at equilibrium.
At the balloon top, the air is at its highest temperature, highest pressure, and lowest density. The higher T dominates over the higher P, and thus the lower density.
The pressure/height in the balloon analogy would be an inverted pressure/depth in water situation. Greater height results in greater pressure like greater depth results in greater pressure.
No, I don't have the physics to back this up, and I'm not going to pursue such. I have nothing further to say beyond "I may be wrong" - Let a real physicist take it from here.
Moose
Added by edit (Gas laws):
Boyle's law - WikipediaCharles's law - Wikipedia
Edited by Minnemooseus, : See above.