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
Scientific American:
"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.
Try the links in the MadSci Library for more information on Physics.