Charge Moment Analysis

The charge moment is defined in this study to be the product of the charge magnitude and its average height above ground. It was shown in Chapter 2 that the charge moment magnitude is the fundamentally important parameter in conventional breakdown models of sprite initiation. The quantity being measured is the moment due to the removal of charge to ground by a CG discharge. This corresponds to a ``charge moment change''. Since the electric field produced by intracloud charge transfers will decrease with range at a rate greater than cloud-to-ground transfers ($r^{-4}$ versus $r^{-3}$, respectively), the electric field below the base of the ionosphere will primarily be produced by the charge moment change of the CG, provided also that the interstroke interval is much longer than the local relaxation time.

The average charge moment of -CG strokes has been known for nearly a century. Wilson (1925) stated that the mean charge moment of lightning discharges was 50 C$\cdot$km. Pierce (1955) measured the charge moment change of -CG strokes to be typically about 30 C$\cdot$km while Brook et al. (1962) measured an average charge moment change of about 22 C$\cdot$km per stroke with an average charge transfer of about 2.6 C. Krehbiel et al. (1979) measured -CG stroke charge transfers between 1 C (or less) and 20 C in four multi-stroke flashes in New Mexico. They found that the negative charge was removed from altitudes of 4.5-6 km above the ground, which was at 1.8 km MSL. The charge moment changes for the strokes was between $<$5 C$\cdot$km and $\simeq$120 C$\cdot$km. Numerous studies indicate that the negative charge region in storms is centered roughly on the altitude of the $-$15$^{\circ}$C isotherm (Krehbiel, 1986), which is at $\simeq$7 km MSL during the midlatitude summer.

Typical +CG parameters were determined more recently. Berger (1972) measured the current at instrumented towers struck by lightning. He found that +CGs sustained high currents for a much longer period of time than their -CG counterparts and that 95% of the +CGs reached a peak current within 200 $\mu$s compared to only 18 $\mu$s for 95% of the -CGs (Fisher, 1988). The +CG charge transfers measured by Berger (1972) were as large as 450 C.

Krehbiel (1981) used multiple station electric field measurements at KSC to determine the altitude and quantity of charge removed by three different +CGs. The +CGs were associated with horizontally extensive discharges in the stratiform region of a large storm system. Krehbiel (1981) determined that the +CGs removed about 14-45 C of charge to ground within the initial 3 ms from average altitudes of 6.7-10.8 km MSL ($\pm$2 km) and with average currents of 5-18 kA. The +CG charge moment changes were 130-430 C$\cdot$km within 3 ms after the stroke onsets. These +CG charge moments were much larger than the charge moment changes typically attained by -CGs.

Brook et al. (1982) were able to locate and measure the charge transferred to ground by several +CGs produced by winter storms in Japan. They found +CG charge transfers as large as 300 C in only 4 ms with a correspondingly large charge moment change of $\simeq$900 C$\cdot$km. Brook et al. (1982) found that all the +CGs analyzed for a couple of storms exceeded 300 C$\cdot$km in only a few milliseconds, while the +CGs from another couple of storms were 100 C$\cdot$km or less on the same time scale. More recently, charge moment estimates of +CGs have been obtained by remote ELF measurements (Cummer and Inan, 1997; Bell et al., 1998), which are in general agreement with the above measurements.

The charge moment change will be estimated for all of the +CG waveforms as well as some -CG waveforms which were successfully acquired within the 50 minute time interval between 01:40 UT and 02:30 UT on June 22nd, 1997. This time interval corresponds to the sprite-producing phase of the MCS. An electric field change ($\Delta$E) threshold of $\pm$2 V/m was used as a minimum requirement to minimize errors due to the $\pm$0.3 V/m noise level of digitized slow-antenna data.

A ``positive-z'' convention will be used for the electric field observations. Thus, a net increase in the negative charge overhead would produce an increase in the electric field. However, the ``negative-z'' convention will be adopted for the peak-current and charge moment measurements since it is more commonly used in the literature for such measurements. Since a +CG lowers positive charge to ground, the current and charge moment change of a +CG will be indicated as positive.

Table 3.1 shows charge moment ($QZ$) estimates based on the cumulative quasistatic electric field change at 1 and 4 ms after the onset of the return stroke. These time delays were chosen based on earlier measurements of 1.5 to 4 ms delays to sprite onset (Rairden and Mende, 1995). The CG was approximated by a dipole in the charge moment calculations. The error introduced by this approximation will be small ($<\!4\%$) since the ranges to the charges are much greater than their average heights, which were on the order of 7 km.

The $QZ$ values were calculated based on the range to the NLDN stroke as well as a range to the center of the flash as indicated by LDAR. The effect of ionospheric images as a function of range was also included in the calculations (see Appendix C).

Table 3.1: The cumulative charge moment change at 1 and 4 ms post return-stroke delays is calculated for a dipole at an LDAR-indicated range as well as the NLDN-indicated range. The strokes shown in bold produced sprites.

Charge Moment calculations for 1997, June 22, 1:40-2:30 UT
Time [UT]		LDAR		NLDN, $t\!=\!0$		$t = 1$ ms			$t = 4$ ms
H:MM	Seconds	Range	${\sim}R_d$	Range	I$_{peak}$	$\Delta$E	QZ [C$\cdot$km]		$\Delta$E	QZ [C$\cdot$km]
		[km]	[km]	[km]	[kA]	[V/m]	LDAR	NLDN	[V/m]	LDAR	NLDN
1:42	57.4092	82.9	15.0	71.1	50.2	6.2	265	154	10.0	478	265
1:51	2.7627	86.1	13.4	82.1	61.3	4.3	213	179	7.1	393	325
1:55	47.9570	83.8	11.5	83.1	85.5	6.3	281	272	8.2	408	393
1:56	36.9219	-	-	50.4	-46.1	-3.1	-	-24	-3.3	-	-26
1:58	19.0034	-	-	50.5	-51.6	-2.8	-	-22	-2.7	-	-21
	19.0557	-	-	50.4	-47.8	-3.2	-	-24	-3.1	-	-25
	19.1108	55.1	3.4	53.1	-13.7	-2.5	-25	-22	-2.2	-23	-20
1:59	38.4795	50.5	1.0	49.5	-37.3	-3.3	-25	-24	-3.3	-26	-25
	38.9351	50.4	2.2	49.8	-40.5	-3.5	-27	-26	-3.7	-29	-28
2:00	13.9844	50.0	1.8	49.1	-30.1	-2.5	-19	-18	-2.8	-22	-20
2:03	54.7979	47.4	2.2	45.4	-27.7	-5.0	-32	-27	-5.9	-38	-33
2:04	51.2842	51.4	1.0	47.3	-44.9	-4.0	-33	-25	-4.1	-34	-26
	51.6587	-	-	47.1	-28.7	-3.9	-	-24	-5.7	-	-36
	51.7808	-	-	47.3	-19.0	-3.0	-	-19	-3.6	-	-23
2:06	21.3560	-	-	47.2	-49.6	-5.2	-	-33	-5.8	-	-37
	21.4175	49.9	1.3	48.6	-17.4	-4.1	-31	-29	-4.7	-36	-33
2:07	20.8521	49.0	0.9	47.5	-35.3	-3.4	-24	-22	-3.9	-28	-26
	20.8853	48.9	0.8	48.0	-20.5	-2.3	-16	-15	-2.1	-15	-14
	20.9297	48.9	0.8	47.2	-38.9	-3.9	-27	-24	-3.6	-26	-23
	20.9976	48.9	0.8	48.7	-17.3	-2.2	-15	-15	-2.1	-15	-15
	21.1348	49.6	1.3	47.8	-24.8	-2.7	-20	-18	-3.4	-25	-22
2:15	19.4189	71.1	9.9	52.4	78.7	15.3	381	136	25.7	683	234
	19.5112	71.4	14.6	74.6	29.4	3.7	93	109	7.0	187	222
	19.5156	71.4	14.6	75.6	4.9	2.5	63	77	3.2	86	108
2:16	40.9277	-	-	55.3	-33.2	-3.1	-	-32	-3.5	-	-38
2:19	50.7617	-	-	57.4	-60.1	-4.4	-	-53	-5.1	-	-63
	50.8408	56.7	2.0	56.6	-20.0	-2.1	-23	-23	-2.5	-29	-29
	51.1572	58.8	3.3	56.9	-18.7	-2.1	-27	-25	-3.0	-41	-36
2:22	22.0957	-	-	53.2	-98.3	-6.2	-	-57	-6.8	-	-65
	22.1514	59.2	1.0	56.1	-54.1	-3.5	-46	-38	-3.7	-50	-42
	22.1904	59.2	1.8	56.1	-27.1	-2.2	-29	-24	-2.8	-38	-31
2:24	15.5098	-	-	60.9	-64.6	-4.2	-	-62	-4.8	-	-72

The LDAR-based range was calculated from the average LDAR VHF source location for the time interval extending from 250 ms before the stroke to 10 ms afterwards. The pre-stroke time interval of 250 ms was chosen somewhat arbitrarily, but should give a reasonable approximation to the extent of the removed charge. If a longer time interval corresponding to the flash duration prior to the stroke had been used instead, then older channels which have decayed will unneccessarily bias the range towards the point of origin in a convective cell. If a much shorter time interval was used, then the horizontal development would likely be under-represented.

In order to reject outliers due to mislocation or the rare case of a coincident flash, points were rejected if they were more than three times the average VHF source distance from the calculated average VHF source location. The average VHF source location was then recalculated along with a new average distance, $R_d$, which corresponds to the approximate ``radius'' of the discharge. If there were fewer than 4 VHF sources used to calculate the final center location and radius $R_d$, the results were rejected as being statistically insignificant.

Table 3.1 supports the previous observations that $+$CGs usually transfer considerably more charge than $-$CGs on millisecond time scales. The charge moment changes are consistent with previous measurements discussed earlier. The $+$CG discharges also had a much greater horizontal extent (larger $R_d$) than the $-$CG discharges, as was previously observed by Krehbiel (1981). The larger charge removal of the $+$CGs relative to $-$CGs may simply be a result of the larger horizontal extent.

The sprite-producing +CGs, which are denoted by a bold font, produced larger NLDN range-based charge moments than the non-sprite-producing +CGs at 02:15:19.51 UT. It is not known if the 02:15:19.4189 UT +CG produced sprites since the video image was saturated by the +CG. However, no luminous persistence indicative of sprites was seen after saturation (see Section 3.10).

The +CG at 02:15:19.5112 was followed a little over 4 ms later by another +CG according to the NLDN. Whether or not this was a real +CG is not clear from the electric-field data (see Figure 3.26). In any case, the video record indicates that no sprites were produced by the +CG(s), in spite of a charge moment change well in excess of that typically observed for $-$CGs. This suggests based on the LDAR-derived charge moments at $t\!=\!4$ ms in Table 3.1 that the charge moment threshold for sprite initiation is probably between $\simeq\,$200 and $\simeq\,$400 C$\cdot$km. This estimate is consistent with a previous ELF measurement of at least 250 C$\cdot$km attained by sprite-producing discharges within the initial 5 ms.

A static field calculation such as in Figure 2.9 indicates that 400 C$\cdot$km would be sufficient for conventional breakdown at $Z\,\gtrsim\,$79 km. The relaxation time ($\tau _r$) should exceed 1 ms just above 80 km altitude and will be several milliseconds at 79 km altitude according to Figure 2.6. Thus, the charge moment measurements here provide some evidence that sprites may develop when a conventional breakdown threshold is reached. A more precise answer to this question requires that the time of sprite onset be measured in relationship to the parent discharge charge moment change. Such observations will be presented in Chapter 5.