Initiation Threshold vs. Height

The earliest estimate of the charge moment required to reach a critical field, $E_k$, at high altitudes was made by Wilson (1925). Wilson estimated that a charge moment of about $2800$ C$\cdot$km would be required to reach $E_k$ at 60 km altitude, and that a moment about $1/10$ of this ($\simeq\!280$ C$\cdot$km) would be required to reach $E_k$ at 80 km altitude. However, Wilson's insightful early model didn't take a couple of factors into account. He did not incorporate the effect of an ionosphere, which nearly doubles the observed electric field at high altitudes (see Figure 2.7). Wilson also estimated an air density at 60 km which was only 64$\%$ of the 1976 U.S. Standard Atmosphere (Section 2.3.1). Fortuitously, the effect of these approximations almost cancels.

A more recent estimate was made by Fernsler and Rowland (1996) which incorporated the effects of a realistic ionosphere height between $80\!-\!85$ km altitude. They estimated that the sprite initiation threshold would be $300$ C$\cdot$km at 80 km altitude and only $150$ C$\cdot$km at 85 km. They also estimated that a much larger charge moment change of $8000$ C$\cdot$km would be required to produce conventional breakdown at lower altitude during the daytime when the base of the ionosphere is lower (see Section 4.3). These charge moment estimates were based on the 1976 U.S. Standard Atmosphere and an assumption that $E_k/N\!=\!100~Td$.

Figure 2.9: The charge moment threshold for sprite initiation is plotted as a function of altitude between 70 and 81 km for 3 different parent discharge configurations.

The charge moment required to reach a breakdown condition of $E_k/N\!=\!123~Td$ (Section 2.2.3) is plotted in Figure 2.9 as a function of altitude. The height of the ionosphere, $Z_{ledge}$, was set to 81 km (Section 2.4.4). The moments were calculated based on electrostatics. Thus, the reduction in the electric field due to the relaxation time as well as the enhancement due to retardation effects Pasko et al. (1999) are not included.

Since the charge moment threshold depends somewhat on the source charge configuration (Section 2.5.2), the thresholds for different parent discharge geometries are plotted. As was discussed previously, a point charge at $15$ km altitude represents an extreme and unlikely example, and is plotted merely as a limiting value.

Figure 2.6 shows that the relaxation time exceeds $1$ ms for altitudes at and below $80$ km. For a uniformly charged disk of radius $R_d\!=\!10$ km and height $Z_d\!=\!10$ km, the charge moment threshold would be $\simeq\!360$ C$\cdot$km for breakdown at $Z\!=\!80$ km. If the disk's radius is increased to $R_d\!=\!20$ km and the height decreased to $Z_d\!=\!5$ km, the charge moment threshold criteria rises to $\simeq\!390$ C$\cdot$km. For breakdown at $Z\!=\!78$ km, the disk charge moment thresholds would be $\simeq\!490$ C$\cdot$km and $\simeq\!530$ C$\cdot$km, respectively. The initiation altitude of sprites will be determined in Chapter 5 and will be shown to be similar to the 78 km MSL altitude used here.

Pasko et al. (1997a) showed that discharges with charge moments exceeding the conventional breakdown threshold in a region below the base of the ionosphere would produce diffuse luminosity due to ionization within that region. Observations of these events will be presented in Chapter 5.