Many months ago I saw a picture that was taken many years ago of an explosion, possibly in the Atlanta area. If I recall the explosion was caused by fuel in railroad cars. However, the explosion formed a distinctly mushroom shaped cloud (caused by Rayleigh-Taylor instability) and I remember reading that any explosion (specifically meaning it does not have to be a nuclear explosion) can form a mushroom cloud if the temperatures involved are above a certain limit. However I forgot what the name of this limit is and I have been unable to find it through searching. I seem to recall it had a rather Russian sounding name. Can anybody tell me what it is?
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$\begingroup$ I am not at all certain that temperature the the right metric; I would expect the critical quantity to be energy (assuming a point-like source on the scale of mushroom cloud formation). $\endgroup$– dmckee --- ex-moderator kittenCommented Nov 25, 2013 at 17:02
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$\begingroup$ So was the limit whose name escapes me bogus then? Or am I misremembering what the critical parameter was? $\endgroup$– MichaelCommented Nov 25, 2013 at 17:06
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$\begingroup$ I could believe the critical parameter could be described in a number of ways, and you can switch amongst them using assumed parameters for your particular situation (density of your medium, value of $g$, etc.). $\endgroup$– user10851Commented Nov 25, 2013 at 17:19
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1$\begingroup$ I would agree with @dmckee, it should be more about energy deposition. Which if people make some assumptions (calorically perfect air for example, which is an okay assumption when $T$ is really high since all energy modes are populated, provided one picks a $c_v$ that is appropriate) then it could be correlated to a temperature since $T = e_{int}/c_v$. So it's possible there is a critical temperature, but it's more likely due to an assumption that related critical energy deposition to a temperature. $\endgroup$– tpg2114Commented Nov 25, 2013 at 19:44
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1 Answer
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Before looking for the critical parameters, let's review how mushroom clouds work. A large amount of energy is released at the source of the explosion, effectively a point source for our discussions. That large energy release causes a blast wave that propagates outward and leaves behind a core of very hot, high density gas. If the blast wave was strong, the pressure in the core is much lower than atmospheric and a return wave comes back to the core and the waves collide at the point source (see this page for example of pressure trace). This causes a new outward directed shock wave and again a returning wave and so on until pressure relaxes back to atmospheric. The colliding of the returning waves can induce an initial upward velocity in the pocket of hot gas.
Let's look at that initial upward velocity as small. A mushroom cloud stem is formed by the hot gas at the point source rising upwards due to buoyant convection. As the gas rises, it begins to cool and the surrounding atmosphere is also cooling. The plume stops rising higher when it reaches a level of the atmosphere where the hot gases have cooled enough to match the surrounding temperature. This is why higher energy explosions will generate taller mushroom clouds. But there is still an upward momentum flux which effectively "hits a wall," causing the upward momentum to transfer to radial momentum. This causes the plume to mushroom out.
So we're effectively looking at Rayleigh-Taylor instability due to buoyancy in the atmosphere. In addition to the usual suspect non-dimensional parameters in fluid dynamics, there are 2 that are of particular interest. The Atwood number and the Froude number.
The Atwood number is a ratio of densities used to study hydrodynamic instability. The number is defined as:
$$ A = \frac{\rho_h - \rho_l}{\rho_h + \rho_l} $$
where $h$ indicates heavy and $l$ indicates light. There are critical values of this to form Rayleigh-Taylor instability in various fluids and is specific to the fluids considered. If we consider a perfect gas such that $\rho = P/RT$ then the Atwood number can be written as: $$ A = \frac{P_e/T_e - P_a/T_a}{P_e/T_e + P_a/T_a} $$ where now $e$ indicates explosive and $a$ indicates atmosphere. If we normalize things by $P_a$ and $T_a$ (only in case we're given explosive performances as ratios to atmospheric conditions) we get: $$ A = \frac{(P_e/P_a)(T_a/T_e) - 1}{(P_e/P_a)(T_a/T_e) + 1} $$
In this way, it is possible to tie a critical number back to temperature of the explosive. But it's not quite as simple because it's also tied to the pressure (and the relation between pressure and temperature depends on the explosive considered).
The Froude number when studying turbulent plumes is typically the densimetric version (because of the Boussinesq approximation) but we can't make that assumption here because the initial gas densities are quite large. If we want to look beyond the initial energy deposition and look at "later" times, then the Boussinesq approximation is valid. The Froude number can be thought of as the ratio between the upward force and the gravitational force. So we would expect mushroom clouds to require a large Froude number near the point source (approaching 1 at the top of the cloud where it flattens out and is in balance). It can also be thought of as kinetic energy to potential energy. If we look at it this way, we get:
$$ F = \frac{u^2}{gh} $$
where $u$ is the upward velocity at a point, $g$ is acceleration due to gravity, and $h$ is the height. If we consider the internal energy of the explosive as $e_{int}$ and assume it to be released entirely to kinetic energy (and assume things are calorically perfect so $e_{int} = c_v T$), we get:
$$ F = \frac{c_v T}{gh} $$
Of course, this means that $F$ is a function of height so any "explosion" will generate a plume, but it may only be an inch tall if there isn't much energy released :) But, it does allow one to determine a height at which $F = 1$ given an initial temperature -- ie. the height at which the cap of the mushroom forms.
So my guess is there is not a single, critical number. Initial Froude numbers are probably in the 20-30 range and Atwood numbers are probably around 0.75-0.95 but I can't find any definitive sources on those figures. I also can't find any other number at the moment, but I will keep looking.
