2.5 Model Results

Snapshots of electric field strength and optical emissions calculated with the model for two different input currents are shown in Figures 2.5 and 2.6. The calculation was run on a 350 km$\times$110 km grid with $\sim$1 km$\times$1 km resolution. The time step of 16 ns was constrained by the conductivity at the upper altitude limit (see Figure 1.1) rather than by the grid size and the usual Courant condition [e.g., Taflove, 1995, p. 46 and 49]. The region below 50 km was assumed to be free space.

Figure: Cross section of electric field and ${\rm N}_2(1{\rm P})$ optical emissions for EMP. The color scale indicates magnitude of the electric field, while red contours show intensity of optical emissions. The contour values are at $2,4,6,... \;\times10^6$ photons cm$^{-3}$ s$^{-1}$. Time is measured from the onset of the return stroke current. The inset box shows the injected model lightning current and its integral over time.

Figure: Cross section of electric field and ${\rm N}_2(1{\rm P})$ optical emissions for QE. The format is similar to Figure 2.5 but the input current is much slower.

Figure 2.5 shows snapshots at various times over the course of $\sim$1 ms following the onset of a short-duration vertical lightning return stroke current flowing between 10 km altitude and the ground. The color scale shows only the magnitude of the vector electric field, while the red contour lines indicate the intensity of optical emissions in the ${\rm N}_2(1{\rm P})$ band. There are two regions of strong electric field evident in the simulation results. One is visible directly overlying the model lightning (at zero radius) and is due to the static field of the electric charge configuration modified by the return stroke current. The other region, evident as two outward-expanding arcs (or shells in three dimensions), is in this simulation much stronger above 60 km altitude than the static field. This is the radiated lightning electromagnetic pulse, and it is strong as a result of the fast-changing input current. The model current rises linearly over 30 $\mu$s to 120 kA and then decays exponentially with a time constant of 60 $\mu$s. Peaks in the electric field (proportional to the rate of change of the current moment) due to both the rise and fall of the current are visible, and an enhancement in the second such peak, especially at 0.470 ms, is evidence of constructive interference with the ionospheric reflection of the first half of the pulse.

Optical emissions from ${\rm N}_2(1{\rm P})$ are seen to occur in an expanding annulus that is instantaneously narrow both in vertical and radial thickness. The radial extent is on the order of $c(30 \mu{\rm s}+60 \mu{\rm s}) \simeq$30 km, while the photon emission rate exceeds ${\ensuremath{2\!\times\!10^{6}}}$ cm$^{-3}$s$^{-1}$ between 82 and 96 km altitude. The flash extends outward as far as 250 km in less than 1 ms, implying a horizontal extent of $>$500 km. Optical flashes produced in this way are indeed the phenomenon discussed earlier and called elves [Fukunishi et al., 1996b], and modeled optical signatures as they would be seen by instrumentation below the discharge are calculated from these results and discussed in Section 4.1.1.

The radiated electromagnetic pulse and its reflections continue to propagate in the Earth-ionosphere cavity and can be measured at large distances, as discussed in Section 3.1.

Figure 2.6 shows snapshots from a calculation using the same model with a slower input lightning current. The electric field strength and the optical emissions are now dominated by those due to the static electric field resulting from the changed charge configuration. In this case the model lightning current rises linearly over 300 $\mu$s to 100 kA and then decays exponentially with a time constant of 600 $\mu$s. The current flows between the ground and a distribution of charge located at 10 km altitude and having a Gaussian shape with half-peak width and half-peak height of $\sim$6 km. Three-dimensional lightning measurements indicate that a disc of charge at 7 to 8 km altitude would be a more realistic distribution [Stanley, 2000, p. 80]. However, these details do not greatly affect the electric field distribution at high altitudes and thus the outcome of the model, which in the quasielectrostatic case is sensitive primarily only to the total charge moment change [Pasko et al., 1999b].

The three inset boxes in Figure 2.6 show that the source current continues to flow throughout the simulated period, and well after the onset of the optical emissions. Optical emissions exceeding an intensity of ${\ensuremath{2\!\times\!10^{6}}}$ cm$^{-3}$s$^{-1}$ extend more than 80 km horizontally, and persist in the region overlying the causative lightning current for 1 ms or longer. The emitting region also descends in altitude noticeably over time as a result of enhanced ionization, which causes increased conductivity and a reduced relaxation time $\index{electric field!relaxation time ($\tau_{\rm E}$)}\ensuremath{\tau_{\rm E}}$, effectively `expelling' the electric field. These optical emissions correspond to the diffuse upper region of sprites [Pasko et al., 1998a; Pasko et al., 1997b] and their form as seen from the ground is discussed in detail in Section 5.1.

2.5.1 Ionization changes in the lower ionosphere

Figure 2.7: Model cross sections of ionization enhancement due to elves (EMP case) and the diffuse portion of sprites (QE case) 2 ms after the lightning stroke. The line shows the shape of the disturbed region deduced by Johnson et al. [1999]. The effect of dissociative attachment is evident in the dark band below each bright region.

Figure 2.7 compares the ionization changes produced in elves and in the diffuse upper portion of sprites. The central minimum in the EMP case is due to the radiation pattern of a vertical dipole, and it suggests that even when elves and sprites occur together, the extra ionization in elves is not likely to affect significantly the breakdown processes in sprites occuring overhead the causative CG. On the other hand, it would not be surprising for the large ionization enhancements evident in the QE case to affect the formation of any subsequent streamer breakdown. Indeed, observations in Section 5.1 show a correlation betweeen the tops of the columnar features and the curved lower boundary of the observed diffuse region. The shape of the spatial distribution of the optical emissions is also discussed further in Section 5.1.

Also, it is interesting to note in this regard that at the later times shown in Figure 2.6, the spatial resolution of the model becomes inadequate to resolve the large gradients evident in electric field (and conductivity) which arise at the lower boundary of the region of enhanced ionization. This behavior is indicative of the need for a prohibitively higher spatial resolution in order to reproduce streamer behavior.

2.5.2 Early/fast VLF perturbations

Johnson et al. [1999] determined that the lateral extent of the ionospheric disturbance responsible for so-called ``early/fast'' VLF perturbations was  90$\pm$30 km, suggesting that clusters of ionization columns in sprites were not the cause. Instead, the authors suggest that a quiescent (rather than transient) heating (rather than ionization) mechanism [Inan et al., 1996a] could explain the observations. However, as shown in Figure 2.7, the diffuse upper region of sprites may produce significant ionization enhancements with a horizontal scale of $\sim$80 km and at an altitude of 70 to 85 km, below the nighttime VLF reflection height and where the time scale for relaxation of electron density enhancements is 10 to 100 s [Glukhov et al., 1992]. These characteristics qualify the diffuse sprite region as a candidate for a cause of at least some of the VLF scattering events as resolved by Johnson et al. [1999].

In addition, ``post-onset peaks'' lasting $\sim$1 s may speculatively be ascribed [Inan et al., 1996d] to the heating and ionization change evident in Figure 2.7 at lower altitude (down to 70 km), where the three-body electron attachment time scale is $<$10 s [Glukhov et al., 1992]. However, this would predict a broader VLF scattering pattern for the post-onset peak portion of the event, and may also require a temporary lowering of the VLF reflection height [Wait and Spies, 1964].

VLF early/fast events are observed with both $+$CG and $-$CGs, and do not correlate well with lightning return stroke peak current, as reported by NLDN. This, in part, led Inan et al. [1996a] to propose a less exotic cause than electrical breakdown. However, many lightning discharges of both polarities may produce significant charge moment changes on 0.5 ms time scale and may produce sprite halos but no further sprite breakdown, for which additional charge moment changes, possibly accumulating over some milliseconds, may be necessary. Many of these events may be invisible when integrated on a 17 ms video field. As mentioned by Inan et al. [1996a], a combination of quiescent, EMP, and QE effects is likely necessary to explain all observed VLF early/fast events.

2.5.3 Multiple events

In view of the relatively high lightning rate in many thunderstorms, it is of interest to evaluate possible cumulative effects of electromagnetic pulses from successive lightning strokes. As mentioned in Section 1.3, such a study is also motivated by (1) the common occurrence of several return strokes in the same location within a few hundred milliseconds (a cloud-to-ground ``flash'') and (2) by the possibility of elves affecting a large enough area that consecutive cloud-to-ground flashes (separated by tens of seconds) in a large storm system may heat the same region of the ionosphere.

Figure 2.8: Effect of multiple lightning strokes on ionization in elves. Blue regions show net electron depletion, while red shows ionization enhancements. Unlike in Figure 2.7, the axes are not to scale.

Figure 2.8 shows the electron density changes resulting from 1, 3, and 10 successive cloud-to-ground strokes with 2 ms separation. The color scale shows net attachment in blue and net ionization in red. Changes to the electron density are locally as high as a factor of 8 increase and a factor of 3 decrease after 10 strokes. The electron density enhancement peaks at a radius of $\sim$80 km [Rowland, 1998] and the attachment peaks in the vicinity overlying the lightning.

Figure 2.9: Multiple-stroke ionization changes area-averaged over $r<150$ km. In (a), the differential changes of Figure 2.8 are put into the context of absolute electron densities in the lower ionosphere. (b) The same data but presented in the format used by Taranenko et al. [1993a].

Figure 2.9 presents the same data but averaged over the 300 km diameter region overlying the causative lightning. This is relevant for consideration of the large-scale effects of an intense and distributed lightning storm such as a mesoscale convective system. Figure 2.9(b) shows normalized electron density changes and provides direct comparison with the one-dimensional results of Taranenko et al. [1993a], though using a different model lightning current. The altitude distribution of electron density enhancements and depletions agrees well for a single return stroke. For multiple successive strokes, however, the two dimensional model used here shows that the altitude of peak enhancement drops with successive strokes by $\sim$2 km after 10 strokes.

Figure: Multiple-stroke ${\rm N}_2(1{\rm P})$ emissions area-averaged over $r<150$ km.

Figure 2.10 shows a similar area-averaged altitude scan of optical emissions. These are seen to decrease in intensity and mean altitude for multiple incident electromagnetic pulses.