The bomb would shower the local magma, rock and dirt with radioactive particles while a tremendous amount of pressure is being created in the chamber very quickly. There's only one place for this magma, rock and pressure to go: Out.
Some of the erupting material will be lava, but that isn't the dangerous bit. The deadliness comes from the plumes of ash. For example, the Icelandic volcano Eyjafjallajökull virtually brought global air travel to a standstill with plumes of fine dust. In this case, the dust traveled all the way to Italy, Russia and towards the East Coast of America. In our nuclear bomb example, this dust would also be radioactive. And the power of the chain reaction in the bomb would push more ash into the environment for a longer period of time, causing long-term contamination hazards.
The estimated ash cloud from Eyjafjallajökull. Bourrichon/Wikimedia Commons. (CC BY-SA 3.0)
But what would happen if Tsar Bomba was set off in the chamber of the world's most famous supervolcano: the Yellowstone supervolcano? This supervolcano has no conical mountain over it signaling its existence. Instead, it's an enormous bubble of magma below the surface of the park. The eruptive power of this volcano would dwarf the explosive power of the bomb. The bomb would set off a supervolcano eruption with no problem, but equally just drilling into the chamber would do the trick. Andrews postulated to IFLScience about the global catastrophe that would follow the detonation of a nuclear bomb in Yellowstone.
"If you buried [a nuke] in Yellowstone, deep enough and it was powerful enough to crack that magma chamber open ... you'll just cause an extinction event. That's when the nuke wouldn't be the problem," commented Andrews. The ruinous power of the volcano would dominate the comparatively puny power of the bomb.
The lava from this supervolcanic eruption would likely be contained within the confines of the park, however the lava isn't humanity's biggest problem. It's the dust that is blown into the atmosphere. Plumes of cold ash would spew out from the ground for many hundreds of kilometers around. It would ruin crops and air traffic, resulting in isolation, health problems from the dust and food shortages. On top of this, the dust would be radioactive.
A prismatic spring in Yellowstone National Park. Lorcel/Shutterstock.
Were Tsar bomba to explode in Yellowstone, then the closest you could be to the blast waves before you sustained an injury is around 160 kilometers (100 miles). This radius isn't the apocalyptic destruction that one might expect. However, like Eyjafjallajökull, it's the clouds of dust that could be swept all across America that would do the real damage.