A Fossilized Energy Distribution of Lightning

When lightning strikes soil, it may generate a cylindrical tube of glass known as a fulgurite. The morphology of a fulgurite is ultimately a consequence of the energy of the lightning strike that formed it, and hence fulgurites may be useful in elucidating the energy distribution frequency of cloud-to-ground lightning. Fulgurites from sand mines in Polk County, Florida, USA were collected and analyzed to determine morphologic properties. Here we show that the energy per unit length of lightning strikes within quartz sand has a geometric mean of ~1.0 MJ/m, and that the distribution is lognormal with respect to energy per length and frequency. Energy per length is determined from fulgurites as a function of diameter, and frequency is determined both by cumulative number and by cumulative length. This distribution parallels those determined for a number of lightning parameters measured in actual atmospheric discharge events, such as charge transferred, voltage, and action integral. This methodology suggests a potential useful pathway for elucidating lightning energy and damage potential of strikes.

The model fulgurite energy and length was calculated every third of a standard deviation, starting with the highest energy fulgurites and summing the length from these of each lower energy fulgurite.
To these fulgurites we applied a simple fracturing model. This model, which was derived from the diameter-length correlation of the fulgurites ( Figure S1) Max length (cm) = 2.3105 x diameter (cm) + 3.613 provides an artificial maximum length of each fulgurite fragment. If the fulgurite or fragments were longer than this maximum length, the fulgurite was assumed to break in two, yielding two smaller fragments. Note that this "max length" is better described as an average length for these fulgurites, but will be used in the current model as a maximum to provide the number of fragments expected for each fulgurite. We acknowledge this to be a simplification of the actual fracturing process.
Each fulgurite will generate a total of where the length over max length was rounded up or set to 1 if the length was smaller than the max length. Thus each fulgurite is assumed to generate a specific number of fragments that is a multiple of 2. With this data, a cumulative number vs. energy per length could be constructed.
Finally, the energy per unit length is multiplied by the fragment length to give a total energy preserved within each fragment. The energy vs. cumulative number could then be constructed. Subsequently, the max length of each of the 20 fulgurites is determined using the empirical relationship above, and the number of (equivalent) fragments is determined from the initial 1 m of length. For the 0.05 MJ/m fulgurite (0.025 cm diameter), we expect each fragment to be smaller than 3.7 cm, and hence there will be 32 fragments each 3.125 cm long. A 1.95 MJ/m fulgurite will have a maximum length of 5.9 cm, and hence there will also be 32 fragments each 3.125 cm long (as opposed to 16 fragments 6.25 cm long). Each fulgurite hence provides 32 fragments in this scenario, and the corresponding distribution is akin to the cumulative length distribution.
Finally, by multiplying each fulgurite fragment's length by its energy per length, a total energy preserved in each fragment can be determined. Comparing these total energies to the cumulative number yields the distribution provided as Figure S2-F.
These results of these simulations can then be compared to the fulgurite data ( Figure S2).

Compositional Data.
A sample XRD of the quartz sand is provided as Figure S3. This diffractogram was acquired on a BTX Olympus desktop XRD for 2 hours (400 scans), and is comparable to XRD patterns from quartz from King County, Washington, USA, from the rruff.info website (http://rruff.info/quartz/R040031). 1 Raman of both the quartz sand and of the fulgurite demonstrated only SiO2 and lechatelierite glass. Example Raman spectra of fulgurite quartz, lechatelierite, and the rruff.info quartz sample are provided as Figure S4. 1 These Raman data were acquired on an EnWave Sense Raman microscope over 30 seconds scanning time with a 785 nm laser.
Additionally, a single fulgurite was crushed and analyzed by ICP-MS (a ThermoFinnegan Element2 ICP-MS) following established techniques (e.g., Haynes et al. 2010). 2 These data show almost pure SiO2 (Table S1). Diameter vs length Figure S2. Comparison between fulgurites and modeled fulgurite distributions. A) Fulgurite energy per unit length vs. cumulative length (identical to Figure 3B but with a linear scale). B) Fulgurite energy per unit length vs. cumulative number (identical to Figure 3A but with a linear scale). C) Fulgurite energy vs. cumulative number (identical to Figure 3C but with a linear scale). D) Modeled fulgurite distribution in terms of energy per unit length vs. cumulative length assuming a normal distribution of energy per length, and all fulgurites forming with constant length. E) As D), but plotting energy per length against cumulative number, after applying fractures. F) As D), but plotting energy preserve per fragment against cumulative number. G)-I) Y-axes are the same as the graphs vertically above. Modeled distributions are based here on energy per unit length following a normal distribution, and length following a normal distribution (the average energies correspond to the longest lengths). J)-L) As before, but assuming a lognormal distribution, with the highest energy per length corresponding to the longest length.