Above: Figure 2-23 on p. 2-59 of Dolan's Capabilities of Nuclear Weapons, DNA-EM-1, 1972, showing the rapid decay of the peak overpressure with increasing distance from a 1 kt nuclear surface burst:
R (feet) - P (psi)
25 - 300,000
40 - 60,000
70 - 10,000
150 - 1,000
400 - 70
1,000 - 10
20,000 - 0.1
The curve, based on Brode's theoretical calculations with programs that include both hydrodynamic motion and radiation flow, can be represented by the simple equation:
P (psi) = (1.7 x 1010 /R3.4) + (7.0 x 106 /R2) + (1,700 /R),
where R is distance in feet. The R3.4 fall in pressure at the smallest distances differs from the simple theoretical R3 prediction for the fall in overpressure due to dispersal of energy over the increasing mass of engulfed ambient air (this mass is proportional to R3) because the shock front is losing energy by radiating thermal radiation at the highest overpressures, which causes an additional fall in peak overpressure with distance. Scaling to other explosion yields is done by multiplying the distances by the cube-root of the total kiloton yield.
Dolan gives also a free air burst curve in Figure 2-2 on p. 2-7, which can be obtained by scaling the surface burst peak overpressure curve to a yield of about 0.565 kt, implying that surface bursts have an effective yield (due to reflection of blast wave energy into a hemispherical region) of 1.77 times the free air burst yield. Hence, the distance for any given pressure in a surface burst extends about 1.771/3 = 1.21 times as far as in a free air burst in sea level air. For a perfectly rigid surface, an effective yield increase factor of 2 would be expected since the same amount of blast energy for any radius would be concentrated in a hemisphere with only half the volume of the sphere for that distance. A reflection factor of 1.77 therefore implies that only 100(1 - 1.77/2) = 11.5% of the blast energy in a surface burst is permanently absorbed by the ground in the cratering, ground shock, and soil heating (fallout formation) processes. If the initial blast energy is 50% of the total yield in a free air burst, then in a surface burst it will be reduced to 44%. A discussion of blast theory and some test data is given in an earlier post linked here.
The history of the precursor is discussed in earlier blog post about Glasstone and Dolan. The billowing of thermally-raised smoke and dust in the blast wave of the TRINITY test (100 feet over dark desert soil) in 1945 should have been the suggested a modification of the blast by dust loading of the air in that region, but the first film of the precursor shock wave was obtained on the DOG shot of TUMBLER-SNAPPER in the Nevada in 1952. Dark coloured (brown) desert sand, consisting of crystals of silica, was exploded or 'popcorned' into hot dust by thermal radiation exposures of 11-19 cal/cm2 for yields of 35 kt to 1.4 Mt; a similar effect on lighter coloured (grey-white) coral sand required 15-27 cal/cm2. This formed a cloud of hot dust-laden air several metres high over the ground, which caused the blast wave to speed up and change in characteristics. The density of the dust added to the air increased the blast wind or dynamic pressure (which is directly proportional to the density), while the added momentum increased the duration of the blast winds, greatly increasing damage to structures and vehicles by the 'sandstorm effect' of the air-blasted dust cloud. The peak overpressure is somewhat reduced by the upward refraction of energy due to the temperature-height profile in the precursor region.
In 1953, the precursor effect was demonstrated by a comparison of damage from the ENCORE and GRABLE shots. The second test was at lower altitude so the thermal radiation was able to popcorn the desert effectively, creating far greater dynamic pressure effects than ENCORE at the same overpressures for drag effects on jeeps, trucks, and other dynamic-pressure sensitive targets. At subsequent tests in Nevada, selected areas around ground zero were flooded to form shallow lakes, while other areas were coated with asphalt, concrete, grass and other surfaces to investigate precursor development as a function of the reflective and physical nature of the surface. Precursors were noted at higher overpressures over coral sand, including surface bursts of over 30 kt yield (so that the fireball at thermal maximum is high enough to irradiate the ground with sufficient thermal energy to cause popcorning). Dolan's Capabilities of Nuclear Weapons, DNA-EM-1, 1972, p. 2-81, states that dust blast precursors will occur over dark city asphalt for burst altitudes below 800W1/3 feet, for W kilotons total yield, and for bursts over dark desert sand precursors will occur for burst altitudes below 650W1/3 feet. These formulae are valid for yields of 1-50 kt where observations are available (for other yields consideration must be given as to whether there is sufficient thermal exposure in the time before blast arrival for a dust layer to be produced).
Above: some typical qualitative precursor blast waveforms for overpressure and dynamic pressure, taken from Dolan's DNA-EM-1, 1972, which on pages 2-81 to 2-89 includes a detailed predictive system to indicate the shape of the precursor waveforms as a function of yield, height of burst and distance from ground zero. This was later developed into a quantitative precursor waveform prediction system in the late 1990s. At very high overpressures, the blast arrival is so soon after that detonation that very little of the thermal radiation has been emitted by the fireball, so there has been little development of a precursor in the available time. Therefore, the precursor develops gradually as the shock travels outward into areas which have been irradiated for longer times after burst, where enough thermal radiation has been emitted to cause a hot dust layer ahead of the shock wave. At long distances, the blast wave runs out of the dust layer because it encounters a region where the thermal radiation exposure has simply not been strong enough to 'popcorn' the sand or to 'smoke' the asphalt or grass. When this happens, the precursor encounters cooler air which makes it slow down, allowing the main blast wave (still travelling through air warmed by the precursor) to catch up and merge with the precursor, forming an ideal shaped blast wave once again.