Can an atmosphere be thicker at higher altitudes?
Can a world exist where the atmosphere is actually thicker at some altitude than at sea level? If so, what mechanisms would be involved?
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In general, I think the answer is "no"; buoyancy simply doesn't work that way, and you'd need to somehow circumvent buoyancy.
What might be plausible is to have a low spot in your terrain that is completely surrounded by higher terrain (maybe a caldera?) which, for some reason, is filled with a higher density substance. If this substance is unable to escape the basin, you might have higher pressure in the basin than at sea level, but this would be a very localized thing.
We do actually (sort of) have examples of this phenomena here on Earth. We usually call them "lakes"... although what they are full of isn't technically "atmosphere".
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The density profile of a planet's atmosphere arises from two laws of physics: hydrostatic equilibrium and the ideal gas law. Put together, they require that the density $\rho(z)$ be a function of the form $$\rho(z)=\rho_0e^{-z/H}$$ where $H$ is the scale height, determined by the planet's surface gravity, composition, and temperature. The scale height is inversely proportional to temperature; on Earth, temperature increases and decreases in different layers of the atmosphere, but not at a quick enough rate to cause the density to decrease with height at any altitude.
A dense layer of the atmosphere could briefly form if there was extremely rapid cooling at a particular altitude. This could be maintained only by a complete lack of heat transfer via convection or radiation from nearby layers, which might require a thick band of optically thick clouds. However, presumably the atmosphere would eventually bounce back into equilibrium as the heat would be transferred back to the cooler layer.
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