Propagation

The streamer which develops from the positive ion region has a net positive charge in its tip and is referred to as a positive streamer. A schematic illustration of a positive streamer is shown in Figure 2.10. Multiple electron avalanches are produced by free electrons in the high electric field region ahead of the streamer tip. The avalanches fill in the positive tip, but also leave behind positive ions. These positive ions become the new streamer tip and the avalanche process can continue.

Figure 2.10: Schematic diagram of a positive streamer. A net concentration of positive ions in the streamer tip creates a strong electric field which produces electron avalanches. As the electron avalanches propagate into the net positive charge region, they leave behind positive ions which become the new streamer tip, advancing the streamer forward. Adapted from Bazelyan and Raizer (1997) Figure 3.1.

The avalanche process and resulting streamer motion will be continuous provided there are a sufficient number of free electrons ahead of the streamer tip and a sufficiently large external electric field. At tropospheric altitudes, the free electrons are generated by photoionization. At the altitudes of sprites within the mesosphere, enough free electrons may already be present for stable streamer propagation (Raizer et al., 1998).

The minimum electric field, $E_+$, required for positive streamer propagation in air at standard temperature and pressure (STP) has been measured to be between 400 and 440 kV/m (Phelps, 1974; Allen and Ghaffar, 1995; Loeb and Meek, 1940), corresponding to an $E_+/N$ of 15-16 Td. This is $\sim$8 times less than the breakdown threshold, $E_k/N$, of $\simeq$123 Td (see Section 2.2.3).

Measurements of positive streamer propagation by Griffiths and Phelps (1976) show that $E_+/N$ is not a constant, but actually diminishes with decreasing air density. Their measurements indicate that $E_+/N$ decreases from $\simeq$15 Td at 1000 mb ($\simeq$0 km MSL) to $\simeq$9 Td at 300 mb ($\simeq$11 km MSL) and a constant temperature of 25$^{\circ}$C. Raizer et al. (1998) noted that at higher altitudes and lower atmospheric densities, the attachment of electrons to oxygen loses importance due to the lower collision frequencies and the $E_+/N$ value should approach that for molecular nitrogen (6 Td), a non-electronegative (non-electron-attaching) gas. They calculated that for a charge moment change of 350 C$\cdot$km, a positive streamer would propagate from an altitude of 80 km MSL down to 48 km MSL, where it would terminate. Triangulated measurements of sprites show that they often extend down to 50 km or lower (Sentman et al., 1995) while the charge moment changes of sprite-producing discharges have been measured to be $\gtrsim$300 C$\cdot$km (Huang et al., 1999; Cummer and Stanley, 1999). A charge moment analysis of several sprite-producing discharges is presented in Section 5.3.

Raizer et al. (1998) calculated that the velocity of the downward positive streamer propagation would maximize at $\simeq$1.2$\times$10$^7$ m/s about 4 ms after it was initiated at an altitude of 80 km under the influence of a 350 C$\cdot$km charge moment change. Actual measurements of maximum average positive streamer velocities in sprites will be presented in Chapter 5 and compared with theoretical predictions.

A schematic diagram of a negative streamer will differ from that of the positive streamer in that the tip of negative streamer will be negatively charged and the electron avalanches will point outward from the tip. As was discussed by Pasko et al. (2000), the minimum electric field, $E_{-}$, for negative streamer propagation is two to three times higher than for positive streamers and thus $-$CG sprites will tend to be less vertically extensive than $+$CG sprites.