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Or they could decide to negotiate an end to the Korean war. Which would mean accepting the regime as legitimate. Which in your opinion is the uglier decision ?If I had to take a gander at what the US would actually do in a real war... they would assign different targets to different forces. The guided missile destroyers might be tasked with taking out missile silos, while conventional artillery and bombers and jet attackers and attack helicopters would all be concentrating on NK artillery. If we did any nuke warfare at all, it would be first strike tactical nukes on artillery positions so as to wipe them all out as fast as possible. It would be the goal of any first strike plan to save the city of Seoul. It would be the goal of any war strategy to save as many civilian lives on the Korean peninsula as possible. I don't think there's any value in using our nuclear subs in a real war; but if North Korea were able to hit a population center with one of their nukes; then it would be immediate retaliation in kind... that's deterrence, that's what our subs do.
The bitch about this situation isn't the nuclear threat, the nuclear threat can be contained, what would be very hard to stop is their chemical warfare. North Korea has pretty old conventional warfare, some of it is ancient, but they have around 4500 mlrs (multiple launch rocket system... the same as they were doing target practice with, yesterday) and they can fire their chemical weapons with their artillery... or their ballistic missiles. The artillery has about 25 miles range, the scud missiles about 40 miles range. So we know where they would all be... but we would want to wipe it all out before they could fire that into the streets of Seoul, so maybe a bunch of those MOAB bombs? or tactical nukes? or both? What ever gets missed in a preemptive strike will be immediately redressed by our artillery along the border, and our bombers and attack jets and attack helicopters.
The sooner we can take out their offensive capabilities, the less civilians die on both sides. I don't think South Korea will invade (they could)... I think China will do that, China would replace Kim with a different puppet and eliminate the Nuke program.
Hopefully, low civilian casualties. Lots of stuff goes wrong in war... you never know what to expect. What I don't expect is for Kim to hand over his WMD... the only way to eliminate those will be through force.