|MadSci Network: Physics|
You have asked a very difficult question. I've consulted a few texts, and
none of them provide a way to calculate the diameter of a mushroom cloud.
Let me start by quoting
"All atomic bombs produce a bulge and a stem, but the really huge, flat clouds--the ones that could be described only as mushrooms-- come from the very high-yield explosions caused by thermonuclear weapons (hydrogen bombs). The fireball from an H-bomb rises so high that it hits the tropopause, the boundary between the troposphere and the stratosphere. There is a strong temperature gradient at the tropopause, which prevents the two layers of the atmosphere from mixing much. The hot bubble of the fireball initially expands and rises. By the time the bubble has risen from sea level to the tropopause, it is no longer hot enough to break through the boundary. (In other words, the bubble encounters material that has more energy than it does, so it is no longer buoyant.) At that point, the fireball flattens out; it can no longer expand upward, so it expands to the side into an exaggerated mushroom cap. The same thing happens to big summer thundercloud when they rise up to the tropopause, producing a characteristic flattened-anvil shape."
This just tells you why we get mushroom clouds, not how large they are. I next consulted The Effects of Nuclear Weapons, a book published in 1957 by the US Department of Defense and the US Atomic Energy Commission (precursor to today's Department of Energy). This book provides a table detailing how quickly the cloud will rise for a given energy bomb, and this information (on page 40) on how that varies as a function of weapon energy: "The cloud from a 20-kiloton explosion will have risen about 1.5 miles [in 30 seconds] and that from a 1-megaton explosion about 7 miles [in 110 seconds]. Within about 10 minutes, the bottom of the mushroom head will have attained an altitude of 5-15 miles, according to the energy yield of the explosion. The top of the cloud will rise even higher."
I did find an equation that allows one to calculate the radius of the fireball: R(feet) = 230W^(2/5) where W is the bomb energy in kilotons of TNT equivalent. However, I was unable to find a relation between the size of the fireball and the size of the cloud.
The cloud formation is dependant on many things, not just the size of the bomb. Those factors include the local terrain (hilly, flat, mountainous), the soil type (dirt, sand, hard or soft rock), the weather (temperature and humidity) and the altitude.
I found an Australian website that gave a table of cloud size, but gave no supporting details; I can't vouch for its accuracy so I won't quote it.
I'm sorry for not being able to directly answer your question, but I hope this helps.
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