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Just as Ozone concentration peaks at around 20 km in the stratosphere, having a distribution around higher and lowers layers of the atmosphere, does the same happen for carbon dioxide or other greenhouse gases?

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They do, but not to the extreme that $\ce{O3}$ does. For reference, there is $\ce{O3}$ near the ground, but the $\ce{O3}$ in the stratosphere is amplified by the Chapman cycle that forms it.

Generally, greenhouse gases have the highest concentration in layers where they are generated. Let's think about a few (not all) of the greenhouse gases, their sources, and some relevant processes that keep them confined to their layer.

  • $\ce{CO2}$ - emitted by combustion and respiration. Because $\ce{CO2}$ is a mostly stable molecule, it can probably travel pretty high into the atmosphere before it falls down due to the molecular weight. But due to where it's generated it'd still be more concentrated near the ground.

  • Water vapor (yes, it is a greenhouse gas) - most prominent in the surface layer, but generally decreases with height. Generally caps out due to phase transition (it becomes liquid water and ice), but some can enter into the stratosphere.

  • Methane - agricultural and biological sources. Like water, it is most prominent in the surface layer. However, it isn't as bound by phase transition as water vapor is. It does interact with other chemicals in the atmosphere, such as the hydroxyl radical.

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