Use of tactical nuclear weapons in hypothetical Second Korean War?

Something I’ve been thinking about for the past week is whether the United States would choose to use tactical/battlefield nuclear weapons in the event of a North Korean invasion of South Korea. This hypothetical war could occur at any time after the end of the first Korean War, and let’s assume that North Korea uses sheer numbers to overrun US/RoK troops and push them closer to defeat/stalemate by capturing and holding a vast portion of the South. Now obviously using a nuclear weapon to slow the North Korean advance would eliminate a portion of the KPA’s remaining first rate troops at such a stage of the war, but I doubt the RoK would let US nuclear weapons fall on their own soil, not to mention the DPRK retaliating with their own nuclear arsenal. Thoughts?
 
The US doesn't really have any battlefield nuclear weapons anymore, all the nuclear artillery shells and tactical missiles went at the end of the cold war. They currently have a variety of air deliverable B61 free fall weapons they could also use cruise missiles with a W80 as a battlefield weapon but it lacks the immediacy of a nuclear artillery shell.. The UK reduced the number of warheads on their Trident missiles including some with a single (implied) for use as a tactical weapon after they scrapped the WE 177, of course that also had the advantage of increasing range.

If you are the SK government and it looks like you are going to get overrun then use of nuclear weapons to break up and degrade follow on forces would be a rational decision.
 
The US doesn't really have any battlefield nuclear weapons anymore, all the nuclear artillery shells and tactical missiles went at the end of the cold war. They currently have a variety of air deliverable B61 free fall weapons they could also use cruise missiles with a W80 as a battlefield weapon but it lacks the immediacy of a nuclear artillery shell.. The UK reduced the number of warheads on their Trident missiles including some with a single (implied) for use as a tactical weapon after they scrapped the WE 177, of course that also had the advantage of increasing range.

If you are the SK government and it looks like you are going to get overrun then use of nuclear weapons to break up and degrade follow on forces would be a rational decision.
The war itself doesn’t have to be present day. I was thinking any point following the first Korean War, including the time in which we fielded nuclear artillery/tactical nuclear weapons. Also, I’m pretty skeptical as to how eager the RoK would be to allow a nuclear weapon be used on their own soil, as their citizens may not be so thrilled about the escalation to nuclear warfare.
 
From the perspective in my service this would have been the most likely in the 1950s, and fairly likely up through the 1970s. Very likely in all cases were this part of a larger Soviet vs West war.
 
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