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The global cooling hypothesis of the 1970s included the theory that aerosols lead to global dimming. That is, that pollutants reduces the amount that the Sun warms the Earth, causing the Earth's surface to cool.

My understanding is that the theory itself was sound, though in the end the effects of greenhouse gases dominated the trend. Hence, climate change takes the form of global warming.

Setting aside the obvious biological effects from aerosol pollution (e.g. less plant growth), how much of that pollution would be physically required to offset global warming? Are the effects even in the same ballpark?

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I found a decent review article by Zhang et al. (2015) covering various geo-engineering proposals that is worth reading. As for atmospheric pollution, they report that pollution should be injected at controlled altitudes into the stratosphere, where it is longer-lived. Using $\text{SO}_2$, apparently around $2-7 \text{Tg}$ of stratospheric injection would be needed, which is substantially lower than the global $\text{SO}_2$ emissions (e.g. Lee et al. (2011) report annual emissions around $50\text{Tg}$ $\text{S}$, probably the figure has decreased a bit since then).

So, as for your question, it is not about simply polluting more but about controlledly injecting pollutants into the correct atmospheric layers.

References

Lee et al., SO2 emissions and lifetimes: Estimates from inverse modeling using in situ and global, space‐based (SCIAMACHY and OMI) observations, Journal of Geophysical Research Atmospheres 116 (2011), https://doi.org/10.1029/2010JD014758

Zhang et al., Review of geoengineering approaches to mitigating climate change, Journal of Cleaner Production 103 (2015), pp. 898-907, https://doi.org/10.1016/j.jclepro.2014.09.076, available at http://www.homepages.ed.ac.uk/shs/Climatechange/Geo-politics/Comparisons.pdf

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How much? A lot more than you could afford, and probably more than the government could afford. The most effective climate dimming agent is said to be sulphur dioxide,sprayed into the upper atmosphere, where it reflects the sun's rays back into space. Smoke and dust particles also have a dimming effect if there is enough of them. We get both of these effects when there is a massive volcanic eruption like Pinatubo in the 1980s, or better still, Tambora in 1815.

The trouble is that the effect doesn't last very long. Pinatubo had a very slight effect for about a year, Tambora (the most massive eruption in historical times) a more dramatic effect for several years, but global warming persists for centuries. The dust eventually falls to earth, the SO2 becomes acid rain and wreaks havoc with buildings and non-resistant structures on the ground. If you were an eccentric billionaire and wanted to do it, you wouldn't be allowed to. It wouldn't achieve the effect you wanted in any case, but if you went ahead regardless and the expense of the project didn't bankrupt you, the law suits by people damaged by your pollution would.

As you say, pollution inhibits plant growth, which is the opposite of what we need because plant growth reduces atmospheric CO2.

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