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Green house gases typically have higher lifetimes than aerosols and their precursors (e.g. SO2) so, as I understand it, GHGs usually spread more or less evenly across continents, no matter where they are emitted from, whereas aerosols' concentrations are much more heterogeneous and higher near the sources. As a result, all things being equal, big emitters' GHGs affect other countries more in part because big emitters have "aerosol shields", so to speak, since aerosols and their precursors are often emitted alongside GHGs and have a cooling effect. Am I right?

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Your general line of thinking is correct. However, here are a few things that make the whole concept somewhat more complicated:

  1. It is widely accepted that the general tendency of "global dimming" (due to increased aerosol emissions) has been reversed above most regions since the 1980s-1990s, i.e. there has been "global brightening" [see Wild (2012) or IPCC (2013)]. If we accept that dimming mainly has a cooling effect, then despite big emitters having this "shield" as you nicely put it, the trend during recent decades is a weakening of this "shield", i.e. a warming effect. That is: there is less warming compared to a hypothetical absence of heavy load of aerosols above these regions, but there is more warming compared to their own past conditions (and as is known, the most worrying aspect of global warming is not the high absolute temperatures, but their wildly unnatural and quickly increasing trend).
  2. The effect of dimming and brightening, due to increasing and decreasing aerosols in the atmosphere respectively, is not always cooling and warming respectively. The type of aerosols responsible for these trends plays an important role. Reflecting aerosols in general have a cooling effect, because they allow less solar radiation to enter the Earth's energy balance, but absorbing aerosols (e.g. black carbon) do not necessarily have such a cooling effect: the solar radiation blocked by them does not reach the planetary surface, but is nevertheless absorbed by the atmosphere, and thus contributes to the energy balance (simply put, the energy that you do not get on the ground, you still get in the atmosphere and thus you have no cooling effect). [See NASA (2010)]
  3. While generally, as you said, aerosols don't spread evenly over the planet, it has been hypothesized that there is a saturation aspect in their dimming effect. I.e. when large concentrations are present an increase has a small effect, while over places with low concentrations small increases have a comparatively larger dimming effect. The hypothesis is that, because aerosols cause dimming not only directly, but also indirectly by acting as nuclei for cloud formation, and because cloud characteristics are more heavily affected by increases of such nuclei when their absolute levels are low than when they are already high (i.e. there is a logarithmic relation), small increases in aerosol levels over low-pollution areas can have a significant dimming effect. [See again Wild (2012)]
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