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Explosions

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• Passive Fireballs
  • initial burst
  • pressure equilibrium
  • final dissipation

Basically there are three important phases to a passive explosion.

1. The "Sedov solution" expansion, where external pressure and density have neglible influence on the fireball. It might as well be a ball of hot gas expanding freely in empty space.
2. The expansion slows when the energy density of the fireball is comparable to that of the surrounding air. A pressure balance is eventually achieved. For militarily useful yields, this may be only a fraction of a second after the Sedov stage.
3. The fireball evolves as a hot bubble, rising buoyantly in the air. Roiling mushroom cloud etc. It's essentially the same as when someone releases gas in the bath, except that this bubble is incandescently hot and gradually loses energy due to its radiation.

Note that stage 2 does not occur if there is no external medium able to exert a pressure on the fireball, eg. in deep space.

initial burst

[Yet to be written.]

pressure equilibrium

The initial expansion of an explosion follows the power-law rule of the Sedov solution. This behaviour ends when the fireball is affected by the pressure of the ambient medium, such as the planetary atmosphere in the case of explosions on the ground. Ultimately the fireball slows and stops expanding at a point where its internal pressure matches the external pressure. A measurement of the volume of the fireball at this isobaric stage gives an estimate of its heat energy content.

The pressure of a gas is p=nkT, where n is the number of particles per unit volume, k is Boltzmann's constant, and T is the temperature. The internal energy per unit volume is u=p/(γ-1), where γ is the adiabatic index, a constant depending on the nature of the gas particles. For monatomic gas or plasma, γ=5/3; for the principally diatomic mix of the Earth's atmosphere, γ=1.4. The total heat content of a spherical fireball of radius r or diameter D is therefore E = (4π/3)r³p/(γ-1) = (π/6)D³p/(γ-1).

Assuming that the explosion expanded from a well defined point, with negligible mixing with external material or energy sources, then the estimated fireball energy is a lower limit estimate of the yield of the weapon that created the explosion. The remainder of the energy may heat or transmute the non-gaseous debris, or may propagate away in other forms: sound and shock waves, emitted light, neutrinos (in the case of nuclear explosions), gravity waves (for very violent stellar events).

Note that there must be an ambient medium in order for the fireball's expansion to halt in a temporary condition of pressure equilibrium. Such a medium may be air, or perhaps the pressure of a background magnetic field (a complicated topic). In open space, the Sedov solution (with its inherent deceleration in expansion) proceeds until the excess heat of the fireball is entirely radiated away, or until some other external influence comes into play.

Heat content of spherical fireballs in air of 1atm pressure (=101325Pa), for a given diameter at the stage of maximal expansion. For atmospheres where the air pressure is higher or lower than Earth standard, multiply the heat value proportionally.

max diameter (m)	heat content (J) if γ=1.4	heat content (J) if γ=5/3	max diameter (m)	heat content (J) if γ=1.4	heat content (J) if γ=5/3
1	1.33 x 105	7.96 x 104	100	1.33 x 1011	7.96 x 1010
3	3.58 x 106	2.15 x 106	300	3.58 x 1012	2.15 x 1012
5	1.66 x 107	9.95 x 106	500	1.66 x 1013	9.95 x 1012
10	1.33 x 108	7.96 x 107	1000	1.33 x 1014	7.96 x 1013
30	3.58 x 109	2.15 x 109	3000	3.58 x 1015	2.15 x 1015
50	1.66 x 1010	9.95 x 109	5000	1.66 x 1016	9.95 x 1015

final dissipation

Once the fireball is in pressure equilibrium with its surroundings, several processes compete to ensure its eventual elimination. Heat is lost due to the thermal glow of the fireball, which principally depends on the radiative area and its effective temperature. Fluid instabilities cause the surface of the fireball may break up and mix with colder ambient gas. Buoyancy effects cause the fireball to rise like a bubble through the dense medium, roiling and mixing as it ascends, possibly proceeding through a mushroom-cloud stage with an updraft below it. A firey mushroom cloud rises transonically, ie. a large fraction of the speed of sound in the surrounding air. The ascent is slower for a milder mushroom cloud where the density almost matches the ambient air. In practice, if we observe a mushroom cloud that is incandescently hot (as they often appear in energetic battles) we can assume that it is the highly buoyant type, and we can estimate its size by assuming that the rate of rise is about 80% of the sound speed. (Sound speed in the breathable air varies, but is roughly 330m/s.)

• Transition from the Sedov-Taylor blast wave solution up to the sound wave, C. de Izarra, J. Caillard & O. Vallée, Modern Physics Letters B, Vol. 16, No. 3 (2002), 69-72.

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