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Zombie Apocalypse.
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Nuclear Armageddon.
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Alien Invasion.
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Leftist Totalitarian World Government.
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You don't understand.
There have been a combined total of 540,000 kt of nuclear explosions across 2500 tests. Sedan was a shallow buried 110 kt explosion that excavated 11,000,000 tons of material. There are currently around 3000 deployed nuclear weapons. Assuming that ALL detonations in an nuclear exchange were buried to maximize material excavated, (an insane assumption) they would excavate around 5x10^10 tons of material. That is on the same order of magnitude of the 1991 eruption of Mount Pinatubo, which ejected 1.1x10^10 tons of material.
Even if we assumed that 5 times the material gives a 5 times the reduction in temperature (it doesn't, see the Mount Tambora eruption of 1815), that is still only 2 degrees.
Nuclear winter is pure fear mongering. It does not exist.
I think you are severely underestimating the amount of ash and debris that would be put into the air from several dozen cities
I'm severely overestimating it. Most of the nukes would be airburst, and their fireballs would never make contact with the ground. The shockwaves are what cause the damage, not the radiation.
https://en.wikipedia.org/wiki/Sedan_(nuclear_test)
Nuclear winter has little to do with radiation and more to do with ash. It operates on a similar principle to the concept of the volcanic winter. You start nuking common western cities filled with skyscrapers you are going to end up with incredible amounts of debris being shoved into the atmosphere and the sheer amount of ash is going to be a problem
Air burst causes less ash. To maximize the amount of particulate you would bury the nuke somewhere around 50 to 150 feet underground. I already showed that even if you buried every single nuke and detonated them together (optimizing for the amount of material put into the atmosphere), it would still produce less particulate than a large volcanic eruption. You also seem to think that there is something magical about nuclear ash. There isn't. It is on the whole less toxic and less destructive, than a large volcano.