Prediction of Lightning Inception by Large Ice Particles and Extensive Air Showers

We derive that lightning can start if the electric field is 15% of the breakdown field, and if elongated ice particles of 6 cm length and 100 free electrons per ${\mathrm{cm}}^3$ are present. This is one particular example set from a parameter range that we discuss as well. Our simulations include the permittivity $\epsilon (\omega )$ of ice. 100 free electrons per ${\mathrm{cm}}^3$ exist at 5.5 km altitude in air showers created by cosmic particles of at least $5\times {10}^{15}\mathrm{eV}$. If the electric field zone is 3 m high and $0.2{\mathrm{km}}^2$ in the horizontal direction, at least one discharge per minute can be triggered. The size distribution of the ice particles is crucial for our argument; more detailed measurements would be desirable.

Figure 1

Conditions of discharge inception at 5.5 km altitude. The lines of constant hydrometeor length $\ell$ show when a discharge can start. The axes show the reduced thundercloud field $E_{\text{bg}}/E_k$ and the shape parameter $R/\ell$. The red diamond indicates the case in Fig. 2. The dashed curve indicates the optimal $R/\ell$ ratio for given $E_{\text{bg}}/E_k$. The ellipsoids are drawn to scale to illustrate the hydrometeor shape and length.

Figure 2

Streamer below an icy hydrometeor of length $\ell =6\mathrm{cm}$ and curvature radius $R=0.4\mathrm{mm}$ in a background field of $E_{\text{bg}}=0.15E_k$ at 5.5 km altitude. Left: the electron density (green to yellow) with the hydrometeor (blue). Right: The electric field strength. Both figures zoom into $r\le 2\mathrm{mm}$ and $-0.35\mathrm{cm}\le z\le 0.15\mathrm{cm}$ after 46 ns of simulation.

Figure 3

Sketch of the model ingredients; not to scale. In the event box (blue) at an altitude of 5.5 km we assume an electric field $E\ge 2.7\mathrm{kV}{\mathrm{cm}}^{-1}$, a density $n_{\mathrm{HM}}\ge {10}^{-7}{\mathrm{cm}}^{-3}$ of hydrometeors of correct shape and size, and at least one extensive air shower creating a thermalized free electron density $n_e\ge 100{\mathrm{cm}}^{-3}$ in a core (red) of radius $\ge 100\mathrm{cm}$. The height $h$ and horizontal area $A$ of the event box are determined by shower distribution and inception frequency.