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Diurnal variation of TRMM/LIS lightning flash radiances.


This study provides, for the first time, an analysis of the climatological diurnal variations in the lightning flash radiance data product [epsilon] from the Tropical Rainfall Measuring Mission Lightning Imaging Sensor (TRMM/LIS). The [epsilon] values over 13 years (2002-14), and over a global scale (~38[degrees]S-38[degrees]N), reveal novel and remarkably consistent regional and seasonal patterns as a function of the local solar time (LST). In particular, the diurnal variation of [epsilon] (over both continental and oceanic regions) is characterized by a monotonie increase from late afternoon (~2000 LST), attaining a maximum around 0900 LST, followed by a decreasing trend. The continental (oceanic) [epsilon] values reach a broader minimum spanning from ~1500 to 1900 LST (~1800 to 2000). The relative diurnal amplitude variation in continental [epsilon] is about 45%, compared to about 15% for oceanic. [epsilon] This study confirms that the results are not affected by diurnal biases associated with instrument detection or other statistical artifacts. Notable agreement is shown between the diurnal variations of [epsilon] and the global-scale (~38[degrees]S-38[degrees]N) mesoscale convective system areal extent. Comparisons with recently published diurnal variations of cloud-to-ground lightning peak current over the United States also exhibit a marked similarity. Given the novelty of these findings, a few tentative hypotheses about the underlying physical mechanism(s) are discussed.


The diurnal variation of lightning imaging sensor flash radiance, in context with storm areal extent, is examined with special attention to the National Climate Assessment (NCA)

The diurnal variation of thunderstorm frequency has been documented since the early 1920s from Wilson (1921), Whipple (1929), and the famous Carnegie curve (Israel 1971; Wallace 1975; Williams and Heckman 1993), and more recently by regional (e.g., see Lopez and Holle 1986; Orville and Huffines 2001; Rudlosky and Fuelberg 2010; Chronis 2012; Nastos et al. 2013; Chronis et al. 2015b) and global ground-based lightning location networks (Lay et al. 2007; Hutchins et al. 2014).

In the late 1990s, the Lightning Imaging Sensor (LIS; Boccippio et al. 2000b; Koshak et al. 2000; Christian et al. 2003) on board the Tropical Rainfall Measuring Mission (TRMM) satellite revealed two important lightning-related facts. First, far more lightning occurs over land, and, second, the diurnal variation of the lightning flash count follows the sun ( 1999/essd10jun99_2/). The regional and worldwide ground-based lightning observations of Pinto et al. (1996), Orville and Huffines (2001), Orville et al. (2011), Chronis (2012), Villarini and Smith (2013), Virts et al. (2013), Hutchins et al. (2014), and Holle (2014), along with the references therein, are consistent with these findings. In particular, these studies demonstrate that most lightning over land occurs in the afternoon, a phenomenon that is likely driven by the diurnal continental radiative forcing (Williams and Heckman 1993; Chen and Houze 1997; Williams and Stanfill 2002; Chronis et al. 2015b). Regional departures from this pattern of behavior are present when the nocturnal boundary layer plays a decisive role in the convective instability (Wallace 1975; Balling 1985; Easterling and Robinson 1985; Lopez and Holle 1986; Williams et al. 2000). For example, continental storms over the U.S. Great Plains exhibit a propensity for nocturnal lightning activity, in the presence of mesoscale convective systems (MCSs; MacGorman and Morgenstern 1998; Zajac and Rutledge 2001; Orville and Huffines 2001).

Conversely, the diurnal variation in oceanic lightning activity is less pronounced than the continental activity (see Bailey et al. 2007), with preference for late night/early morning storms (e.g., see Orville and Huffines 2001; Lay et al. 2007; Hutchins et al. 2014). The mechanisms leading to this variation are discussed in detail in works by Chen and Houze (1997), Dai (2001), Yang and Smith (2006), Liu et al. (2008), and Nesbitt and Zipser (2003). Deviations from this typical oceanic behavior are noted where strong influences by land-sea-breeze circulation are at play (e.g., the Gulf of Mexico; Virts et al. 2015).

Surprisingly, the majority of studies have repeatedly addressed the diurnal response of the continental and oceanic lightning flash frequency (i.e., counts), overlooking other available lightning flash-related information. For instance, since 1998, LIS has been reporting a proxy for lightning flash energy [the "flash radiance data product," discussed in Koshak (2010), and which we indicate here by the symbol e]. Despite that [epsilon] has proven valuable to advanced applications in atmospheric chemistry modeling (Koshak et al. 2014b), still very little is known in terms of its diurnal variation (Beirle et al. 2014).

Moreover, since the Carnegie curve offers insight into the state of the atmosphere's electrical circuit and coupling with the climate system (Williams et al. 1999; Williams 2005; Rycroft et al. 2008; Blakeslee et al. 2014), the diurnal variation of e is likely also an important indicator of climate variation. That is, the raw lightning flash count alone is only one possible indicator of climate variation; however, given the variable lightning flash currents and channel lengths, the energy of a lightning flash can vary by many orders of magnitude, hence providing a more accurate picture of the interrelationship between lightning and climate. In addition, it is important to note that [epsilon] is directly linked to lightning nitrogen oxide (LN[O.sub.x]) production, which in turn directly influences greenhouse gas, ozone concentration, and hence air quality and climate (Koshak et al. 2014a). Finally, since the variation in [epsilon] depends on the variation of the flash energetics, it provides direct insight into the microphysical and kinematical state of the thunder cell and its ability to separate charge. Therefore, any climate variations that change the characteristics of thunder cells will likely have an associated impact on the types and characteristics of the lightning produced, which in turn affects the statistics of the observed values of e. Hence, one important intent of this work is to establish a baseline for the diurnal variation of [epsilon] across a global scale, with coverage across many diverse regions and seasons. This sets the stage for assessing changes in [epsilon] obtained from important follow-on space-based lightning imagers [namely, the future International Space Station Lightning Imaging Sensor (ISS/LIS; Blakeslee and Koshak 2016), and the Geostationary Operational Environmental Satellite-R (GOES-R, now GOES-16) Geostationary Lightning Mapper (GLM; Goodman et al. 2013)]. In particular, because diurnal variations in [epsilon] will be evaluated over the continental United States (CONUS) up to ~38[degrees]N, this work represents an important continuation of, and a unique contribution to, the National Climate Assessment (NCA) lightning--climate study given in Koshak et al. (2015).

This paper is organized as follows: The next section discusses the data and methodologies employed. Next, we provide the diurnal variation results and also examine additional possible influences or biases (e.g., geographical and seasonal effects, instrument detection biases, sample size biases). We then discuss these findings and explore a few candidate physical mechanisms that might be important in explaining the observed diurnal variations. Moreover, it is shown that the diurnal variation in [epsilon] is remarkably similar to the diurnal variation of the cloud-to-ground peak current over the United States. Finally, our conclusions are outlined.

DATA AND METHODS. LIS is a nadir-staring optical imager that employs a wide (~80[degrees] x 80[degrees]) field-of- view lens system that focuses the image on a high-speed 128 x 128 pixel-array charge-coupled device (CCD). LIS detects lightning during both the daytime and nighttime. During the daytime, the solar-lit cloud swamps the lightning signal. Hence, several filtering methods are employed to improve the signal-to-noise ratio in order to make the daytime detection of lightning possible. The first of these filtering methods is spatial filtering; that is, the CCD array size mentioned above coupled with the low-Earth orbital altitude of LIS results in a ~4 x 4 [km.sup.2] (nadir) to 6 x 6 [km.sup.2] (perimeter) pixel footprint that roughly matches a typical thunder-cell cloud-top size. That is, if a much larger pixel footprint were employed, the chance for swamping the signal with solar reflection from non-lightning-producing clouds would increase. Second, spectral filtering is used. The lens system contains a narrowband (~1 nm) interference filter that operates in the near-IR at 777.4 nm, a prominent oxygen emission multiplet within the lightning spectrum. Third, the LIS employs temporal filtering. A diffuse cloud-top lightning optical pulse is typically on the order of ~400-[micro]s width at half maximum. Hence, the LIS employs a CCD frame time of ~2 ms, which was the technology capability at the time, and reasonably matches the lightning pulse width. Finally, the fourth filtering method for improving the signal-to-noise ratio is to subtract a running average of the LIS background radiance (denoted here by BG) at a pixel from the current pixel BG value. If this residual at a pixel exceeds a specified threshold, then an optical "event" is said to occur [here, pixel-level event is the vernacular used to define a component of the full optical flash emission; see Mach et al. (2007)]. The LIS Real Time Event Processor (RTEP) carries out this processing. Note that the BG exhibits large variations due to the diurnal solar zenith angle and, of course, also depends on cloud albedo. The instrument threshold settings are specified in accordance with the expected range in BG values; that is, higher BG values are associated with higher thresholds in order to minimize lightning detection false alarm rates due to photon shot noise (i.e., a type of electronic noise resulting from random temporal fluctuations of photos hitting an optical device). The reader is referred to Christian et al. (1989), Koshak et al. (2000), and Boccippio (2002) for more details and discussions of the LIS instrument characteristics summarized here.

The primary LIS data product that we shall investigate is the so-called flash radiance. Spatially adjacent optical events within a 2-ms frame define an optical "group." A 5.5-km spatial constraint and a 330-ms temporal constraint are employed to decide which groups belong to the same optical "flash" (Mach et al. 2007). The flash radiance data product [epsilon] is actually a spectral energy density (in J [m.sup.-2] [ster.sup.-1] [micro][m.sup.- 1]; i.e., the LIS instrument is integrated over the duration of the flash so that the time unit vanishes). We will, however, loosely keep referring to it in its common nomenclature as a flash radiance. Within the LIS processing algorithm, the value of [epsilon] is obtained in an expedient/convenient manner by simply summing up the individual event "radiances" within a flash. This "shortcut" approach means that the value of each flash radiance should technically be reduced by roughly a factor of [DELTA][omega]/[DELTA][OMEGA] to give the true spectral energy density, where [DELTA][omega] is the typical pixel field of view and [DELTA][OMEGA] is the flash field of view, each subtended at the LIS; that is, the LIS flash radiance data product overestimates the true spectral energy density by approximately a factor of [DELTA][OMEGA]/ [DELTA][omega]. However, we are not concerned about this technical nuance, since in this study we are just analyzing the relative changes in the quantity [epsilon] that itself is proportional to the intercepted spectral flash energy [in J [m.sup.-2] [micro][m.sup.-1]; see the appendix to Koshak (2010) for additional details].

Overall, this study compiled ~15 million lightning flashes detected from individual LIS orbits during a 13-yr period spanning from 2002 to 2014. Years prior to 2002 are excluded to avoid possible complications from the TRMM orbital boost that occurred during August 2001 (Liu et al. 2008), but 2015 was also excluded as a result of intermittent LIS operation before its final decommission.

Each flash is grouped into hourly bins of local solar time (LST). Note that LST is computed from each flash's longitude. All flashes are further grouped as being continental or oceanic based on a 5 km x 5 km landmask. The flash count in each 1-h LST bin is reported as the summation (i.e., total hourly flash count), whereas [epsilon] is reported as an hourly average. The analysis also examines the average BG values associated with each event in a flash, in order to determine if any instrument-threshold-related effects bias the diurnal variations of [epsilon] (see the section below on possible biases due to LIS threshold settings).

RESULTS. Figure la illustrates the continental and oceanic total flash count diurnal (in LST) variation in absolute units. Figure lb reports the same as Fig. la but in relative units (i.e., normalized by the respective maximum, in percent). Figures 1a, b are in agreement with the discussion in the opening section of this paper in that they support the continental lightnings preference for the late afternoon (~1500-1700 LST). This diurnal behavior is well documented by a multitude of space-based or ground-based lightning observations (references herein). The oceanic flash count, when compared to the respective continental count in absolute units, exhibits little diurnal variability (Fig. la). However, when the count is plotted in relative units (i.e., as a percent of the maximum value), the oceanic lightning activity during the late night/early morning hours becomes much more evident (Fig. 1b) and lies in agreement with previously published results (as discussed in the introductory section of this paper).

We report on the continental and oceanic [epsilon] diurnal variations in absolute (Fig. 1c) and relative units (Fig. 1d). The first observation to be gleaned from Fig. 1c is that oceanic flashes exhibit larger [epsilon] values than the respective continental flashes. This oceanic-continental contrast in e is documented by Boccippio et al. (2000a, b) and Beirle et al. (2014), and this analysis further confirms that the contrast exists throughout the entire course of the day (Figs. 1c,d). In addition to LIS's optical emission, the aforementioned continental--oceanic contrast in the cloud-to-ground lightning flash peak current has been documented by several authors (e.g., see Lyons et al. 1998; Orville and Huffines 2001; Hutchins et al. 2014; Cooray et al. 2014, and references therein) and more recently studied for its physical origin by Chronis et al. (2016).

A key finding of this study pertains to the continental and oceanic e diurnal variations shown in Figs. 1c,d. These exhibit a monotonie increase from ~2000 LST, reaching a maximum around 0900 LST, and a reduction thereafter (Fig. 1c). The continental [epsilon] has a broad minimum spanning between ~1500 and 1900 LST, whereas the oceanic [epsilon] gently decreases and does not reach a minimum until around 2000 LST (Fig. 1c). In terms of relative e units (i.e., normalized by the respective [epsilon] maximum), the total continental diurnal variation is ~45% compared to ~15% for the oceanic (see Fig. Id).

The current literature does not provide any background information in terms of the [epsilon] diurnal variation shown in Figs. 1c,d. The following subsections test whether the diurnal variations illustrated in Fig. 1 are due to any regional and/or seasonal effects or other statistical artifacts.

Sensitivity to regional and seasonal parsing. Regional parsing. The LIS global-scale (i.e., ~38[degrees]S-38[degrees]N) total flash counts and e dataset employed in Fig. 1 are parsed into four regions (Fig. 2) encompassing part of the Americas (0[degrees]-35[degrees]N, 120[degrees]-45[degrees]W; region 1), Africa (15[degrees]S-15[degrees]N, 20[degrees]W-45[degrees]E; region 2), Southeast Asia (10[degrees]S-20[degrees]N, 90[degrees]-140[degrees]E; region 3), and Australia (35[degrees]-16[degrees]S, 100[degrees]-160[degrees]E; region 4). Note that for region 1 we have excluded latitudes south of-0[degrees] to avoid possible effects from the South Atlantic anomaly (SSA; Buechler et al. 2012). For each of the four regions the same analysis that led to Fig. 1 is repeated.

As highlighted in Fig. 1a, all regions exhibit a total flash diurnal variation, typical of the so-called continental convective chimneys (see Williams and Satori 2004), with either a narrower (e.g., region 2) or broader (e.g., region 4) afternoon maximum (Fig. 3a). For regional differences in continental lightning activity differences, see Williams et al. (2002). The oceanic total flash counts of regions 1-4 (Fig. 3b) also closely follow the diurnal flash counts shown in Fig. la, in further agreement with the discussion in the opening section of this paper.

As far as the continental [epsilon] diurnal variation is concerned, all regions (Fig. 3c) exhibit an obvious consistency with the e diurnal variation shown in Fig. 1c. In terms of the oceanic [epsilon] diurnal variation, regions 1, 3, and 4 (Fig. 3d) also highly agree with the findings in Fig. 1d (-0900 LST); however, region 2 (i.e., Africa) exhibits a diurnal [epsilon] maximum around 1300 LST that is approxiately four hours later than the typical 0900 LST result (Fig. 3d). Region 4 (i.e., Australia) exhibits the highest [epsilon] maximum for both continental and oceanic flashes (Figs. 3c,d). In contrast, region 2 (i.e., Africa) exhibits the lowest e maximum for both continental and oceanic flashes (Figs. 3c,d).

Seasonal parsing. The original dataset employed in Fig. 1 is now parsed into the winter [December-February (DJF)], spring [March-May (MAM)], summer [June-August (JJA)], and fall [September-November (SON)] seasons and the diurnal variations of total flashes and [epsilon] are reported in Fig. 4.

As far as the continental/oceanic total flashes are concerned (Figs. 4a,b), we observe no deviation from the diurnal variation of the unparsed dataset (e.g., Figs. 1a,b). In terms of [epsilon], all seasons consistently exhibit a continental [epsilon] diurnal maximum around 0900 LST and a broader minimum (~1500-1900 LST; Fig. 4c), in high agreement with the unparsed dataset (Fig. 1c). For the oceanic [epsilon] diurnal variation (Fig. 4d), all seasons exhibit a diurnal maximum (minimum) around 0900 LST (~2000 LST), also in line with the unparsed dataset shown in Fig. 1d. The winter (DJF) season demonstrates the highest [epsilon] values for continental and oceanic flashes.

Possible biases due to LIS threshold settings. As mentioned in the earlier section dealing with our data and methods, the LIS instrument threshold for lightning detection is set higher for larger BG values in order to minimize the false alarm rate due to photon shot noise. Therefore, it is important to determine if any [epsilon] diurnal increases (decreases) can be attributed to respective increases (decreases) in BG.

Figure 5a illustrates a bell-shaped diurnal BG variation, which simply confirms that the brightest cloud scenes occur at solar zenith (maximum BG values ~1200 LST, both continental and oceanic BG). To facilitate the comparison, Fig. 5b is a copy of Fig. Id, which showed the continental and oceanic [epsilon] diurnal variations. Figures 5c,d serve as the means of investigating whether a relationship/function between e and BG can be established throughout daytime hours (i.e., BG > 0). Note that in Fig. 5 all variables are normalized by their respective maximum values and, hence, are reported as percentages.

The scatterplot in Fig. 5c between [epsilon] and BG exhibits three different trends: a positive linear trend from ~0500 to 0900 LST, a negative linear trend from 0900 to 1300 LST, and a rather invariant trend between ~1400 and 1800 LST. The term "invariant" refers to a statistically insignificant regression slope (i.e., [epsilon] = slope x BG, where the p value of the slope is greater than ~0.5 and thus statistically insignificant). Although both [epsilon] and BG increase from about 0400 to 0800 LST, [epsilon] decreases substantially, while BG continues increasing toward its peak. In addition, [epsilon] increases in the same manner prior to 0400 LST (i.e., before the BG values become greater than zero). In fact, ~20% of the continental [epsilon] diurnal variation occurs during night-time hours (i.e., BG = 0; Fig. 5c). Therefore, the diurnal variation of [epsilon] appears to be a real (natural) variation and not an artifact of the LIS instrument thresholding methodology.

Interestingly, had we tested the same hypothesis for the oceanic [epsilon], the findings in Fig. 5d would have indicated that 1) increasing BG values during ~0600-1100 LST relate to increasing [epsilon] values and 2) decreasing BG values during ~1300-1700 LST relate to decreasing [epsilon] values. In other words, had the oceanic [epsilon] been exclusively considered, the evidence would have suggested that the oceanic [epsilon] diurnal variation might be partly attributed to diurnal BG biases. However, it would be unreasonable to suggest that such bias is selective for oceanic and not continental [epsilon]; hence, the linearity between the oceanic BG and [epsilon] diurnal time series observed in Fig. 5d is circumstantial. LIS decay through the several years of operation could further constitute a possible introduction of erroneous results; nonetheless, Buechler et al.'s (2012) deep convective cloud (DCC) analyses have demonstrated that the BG values have been stable during 1998-2010.

Possible sample-size-related biases. Another bias that could possibly introduce numerical artifacts is the sample size of the diurnal total flash count. For instance, a hypothesis could be framed around the fact that [epsilon] peaks during a diurnal period when the respective total flashes are minimum (~0900 LST; e.g., see Figs. 1a,c). To facilitate this comparison, we combine the diurnal total flash counts and e averages into a single scatterplot, where the respective hourly bins are also annotated (Fig. 6). We observe that the total flashes are monotonically reduced between ~2100 and 0900 LST, reaching their minimum when e is maximum (~0900 LST; Fig. 6a). Two statistical tests for significant differences in the [epsilon] means are performed (Students and Wilcox). The results yield that the [epsilon] means at 2100 and 0900 LST are different at the 99.99% level (p value < 0.001). These statistical tests are repeated for intermediate diurnal periods (e.g., 2000 and 1000, 1900 and 1100 LST, and so forth) and in all cases the results also yielded significantly different *** means at the 99.99% level. Moreover, if the sampled total flash counts were biasing the diurnal [epsilon] variation, then this would also be evident in the oceanic e results. However, from Fig. 6b we observe no consistent relationship between the two variables during any diurnal period.

DISCUSSION. An exten sive part of the analysis (presented in the previous three subsections) was dedicated to examining the consistency of the results shown in Fig. 1. Even though the diurnal variation in [epsilon] was examined for 32 different cases (i.e., four regions, four seasons, and two surfaces: continental and oceanic), only the oceanic part of region 2 (Africa) has a diurnal maximum (-1200 LST) in [epsilon] that is somewhat of an outlier. Therefore, it would be fair to state that the diurnal variation in [epsilon] is regionally and seasonally robust but it is also free of significant biases related to the LIS threshold setting methodology, or sample size. Hence, the findings in this paper are significant because they reveal a real (i.e., natural) diurnal variation in a specific lightning flash characteristic ([epsilon]) that has not been previously documented, and which begs a preliminary physical explanation.

Relationship between e and Ip. Works by Guo and Krider (1982), Idone and Orville (1985), and Gomes and Cooray (1998) found a linear relationship between the light intensity and the current I of laboratory-induced spark discharges. More recently, Wang et al. (2005) documented that current and light signals at the bottom of the rocket-triggered lightning channel exhibit a linear relationship along the rising portions of their waveforms (i.e., respective peaks). This relationship disappears and the light emission amplitude decreases much faster than the respective current (Wang et al. 2005). A relationship between lightning peak current and lightning strokes detected by the World Wide Lightning Location Network (WWLLN) has also been documented in Hutchins et al. (2012).

In light of the abovementioned Laboratory-demonstrated physical ties between [epsilon] [i.e., optical ** brightness of a flash; see the appendix to Koshak (2010)] and lightning current, the findings in the recent study by Chronis et al. (2015b) might further corroborate this relationship on a diurnal scale. The study was based on the National Lightning Detection Network (NLDN; Cummins et al. 1998; Cummins and Murphy 2009). Unlike spaceborne sensors, ground networks detect that the ground wave of the electromagnetic field radiated from the cloud-to-ground (CG) lightning flash. These peak radiation fields are positively correlated with the so-called first CG return stroke peak current (Ip, also shown in Hutchins et al. 2012). Chronis et al. (2015b) reported a consistent Ip quasi-sinusoidal diurnal (in LST) variation peaking around 0900 LST and attaining a minimum at ~1900-2000 LST over the CONUS for the period 2001-10. Although the values of Ip are not complicated by cloud optical thickness as is [epsilon], the retrieval of Ip from measurements of radiated electric fields is affected by both modeling assumptions and measurement errors (Rachidi and Thottapillil 1993). For the purposes of Fig. 7 the diurnal variation in [epsilon] is recalculated for 30[degrees]-50[degrees]N, 120[degrees]-70[degrees]W (roughly the CONUS) from the unparsed LIS dataset. Despite the Ip and [epsilon] being retrieved from disparate observational platforms (i.e., ground-based versus spaceborne), Fig. 7 confirms the physical ties between Ip and e on the diurnal scale. Figure 7b highlights a strong diurnal linearity (linear correlation coefficient ~0.8) between [epsilon] and Ip, but also temporal agreement in terms of the respective diurnal maxima and minima while their differences lie in their relative diurnal variation (Fig. 7a; ~25% vs 45%).

Given the regional limitation of the previous [epsilon]-Ip covariation study, we cannot assume that these diurnal similarities will be further evident on a global scale. For instance, Chronis et al. (2015b) did not compute the Ip diurnal variation over the oceans adjacent to the CONUS given the expected bias in Ip due to the reduction in the NLDN detection efficiency (Nag et al. 2011). Nevertheless, the previous comparison between [epsilon] versus Ip in Fig. 7, viewed in context with the findings in Fig. 6, has offered this study a distinct advantage that is discussed next.

Flash radiance [epsilon] in a capacitor model. A simple capacitor model can be invoked to provide some insight into the strong covariance found between [epsilon] and Ip (Fig. 7). The top plate of the capacitor represents the horizontal extent of the upper positive thundercloud charge (+Q) and the bottom plate the negative thundercloud charge (-Q). The area of each plate is A, and d is the vertical separation distance between the plates. The electric field E between the capacitor plates represents the vertical thundercloud electric field. The capacitance is given by C = Q/V = [[epsilon].sub.0] A/d, where V is the voltage across the capacitor plates that represents the thundercloud electric potential between the charge centers (i.e., V = Ed) and [[epsilon].sub.0], the permittivity of free space. Since the thundercloud is located above the conducting Earth, it is more realistic to envision the thundercloud capacitor to be sitting above a conducting plane rather than to just view the thundercloud as an isolated capacitor.

When this model improvement is made, the electrostatic boundary conditions are met by replacing the conducting plane with image plates (i.e., the standard method of images is invoked). Next, we take each plate to be a circular disk of radius a, with a constant charge density [sigma] = [+ or -]Q/A = [+ or -] Q/([pi][a.sup.2]) for the positive (top) and negative (bottom) plates, respectively.

Hence, as the plate area increases, so too does Q in order for [sigma] to remain constant. The electric potential due to one charged plate a distance z along the circular disk axis is given by [sigma][[([a.sup.2] + [z.sup.2]).sup.1/2] - [absolute value of z]/(2[[epsilon].sub.0]). Carrying out the method of images for all four plates (i.e., the two real plates and the two image plates), one obtains by superposition the voltage across the capacitor V as


where [z.sub.N] and [z.sub.P] are the altitude of the negative and positive charged plates, respectively, and hence d = [z.sub.P] - [z.sub.N] (i.e., the thundercloud depth). Note that V = [[phi].sub.N] - [[phi].sub.P], where [[phi].sub.N] is the electric potential of the negative plate with respect to the ground and [[phi].sub.P] is the potential of the positive plate with respect to the ground (again, each found from the method of images). Plots of [[phi].sub.N], [[phi].sub.P], V, and E = V/d are provided in Fig. 8 as a function of increasing plate area (i.e., circular plate radius increasing from 10 to 2,000 m); the input values are Q = 40 C, [z.sub.N] = 7 km, and [z.sub.P] = 10 km.

The potential [[phi].sub.N] plays a fundamental role in setting a limit on Ip (Stolzenburg and Marshall 2008). Analytic models for CG lightning leaders involving long and thin conductors show that the charge per unit length [lambda] along the CG leader channel as it extends to the ground is proportional to [[phi].sub.N] (Kasemir 1965; Mazur and Ruhnke 1998; Williams and Heckman 2012). The return stroke peak current Ip represents the neutralization of the deposited CG leader charge and is given by Ip = [lambda][upsilon], where [upsilon] is the return stroke speed. Hence, a larger magnitude of [[phi].sub.N] (or equivalently a larger magnitude of V; see Fig. 8) implies a larger magnitude of A and, hence, a larger magnitude of Ip. Finally, the larger magnitude of Ip implies more channel brightness (i.e., larger [epsilon]). Therefore, these simple considerations lead to the expectation that Ip and [epsilon] should vary together, as is indeed found in Fig. 7.

Effect of flash rate. Assuming that the thundercloud capacitor's geometrical characteristics are fixed (i.e., A and d are constant), it can be argued that frequent lightning discharges limit the growth of the electric field E, which in turn limits the value of the potential [[phi].sub.N] and thereby reduces the values of Ip and hence [epsilon] as discussed above. Hence, one would expect the flash rate and [epsilon] to have an inverse relationship. An important observation gleaned from Fig. 6a is that the total flash count and [epsilon] indeed follow an inverse relationship; however, it should be noted that this is true only during ~2200-0900 LST.

In contrast, this relationship is asymptotic during ~1600-2100 LST, a diurnal period during which the total flash count is reduced by almost 50% (Fig. 6a), while [epsilon] varies by less than ~3% (Figs. 1,6a).

Oceanic flashes cast more doubt on a simple inverse relationship, in that Fig. 6b reveals that no relationship during any diurnal period can be established between the two variables (linear correlation coefficient -0; Fig. 6b). Evidently, the collective results of Fig. 6 strongly advocate that the diurnal total flash count variation cannot unequivocally explain the respective [epsilon] variations throughout the entire 24-h day.

Effect of storm area. From the same capacitor model, a larger horizontal extent (i.e., larger capacitor plate area A) yields a larger Ip given o is constant (see Fig. 8 and the associated discussion). The results presented by Nesbitt and Zipser (2003) are particularly useful for this examination for two reasons. The first is that their study compiled a database of the areal extent of storms observed from sensors on board the same satellite as LIS (TRMM). The second is that the Nesbitt and Zipser (2003) analysis also was performed as a function of LST (as done here). In particular, Nesbitt and Zipser (2003) documented the climatological (December 1997-November 2000) diurnal precipitation features of convective storms as retrieved from TRMM's microwave ice scattering intensity and areal (i.e., horizontal) extent. The largest of the identified storm groups in their study were the MCSs. We deem that the MCS might be a key component in the observed e diurnal variation for the following reasons. The MCSs have a unique dynamical organization that allows them to develop and propagate, even when the main source of atmospheric instability on a diurnal scale (i.e., the sun) is absent (see Houze 2004). The latter might provide clues relevant to the [epsilon]-increasing trend observed from the late night hours until -0900 LST (see Figs. 1c,d). Moreover, the MCS areal extent diurnal variation exhibits a marked similarity with the continental and oceanic [epsilon] results shown in Figs. 1c,d (see Figs. 4, 5c in Nesbitt and Zipser 2003). In particular the diurnal area variations for both continental and oceanic MCSs exhibit a distinct maximum around 0900 LST followed by a decreasing trend, minimizing around 1500-1600 LST (continental) and -2000 LST (oceanic). The mechanisms Controlling the MCS maturity and decay diurnal phases are discussed in Nesbitt and Zipser (2003) and references therein. Figure 9 illustrates the diurnal variation of the MCS areal extent (from Figs. 4, 5c in Nesbitt and Zipser 2003) and the respective [epsilon] (i.e., from Figs. 1c,d) in relative (Figs. 9a,b) and absolute units (Figs. 9c,d). The noteworthy covariation between the two variables is not only evident from the linear correlation coefficient (-0.9; Figs. 8d, 9c) or the temporal coincidence between the respective diurnal maxima-minima, but also exemplified in terms of the relative diurnal variation. For instance, the continental MCS areal extent (-55%; Fig. 9a) and [epsilon] (-45%; Fig. 9a) relative diurnal variation are in close agreement, but in even closer agreement lie the respective oceanic values (-15%; Fig. 9b).

In light of the physical linkages between e and A, as well as their diurnal similarity demonstrated in Fig. 9, the MCS areal extent hypothesis is certainly worth pondering as a preliminary physical explanation. Despite that not all the storms are MCSs, the processes that dictate the diurnal areal extent are not only restricted to the MCS storm type but also apply for smaller-scale storms. This is shown to be true for the continental storms in Nesbitt and Zipser (2003, see their Fig. 4c) and is also suggested in Chen and Houze (1997, see their Fig. 18 and references therein) as a more general characteristic diurnal storm evolution.

CONCLUSIONS. Since the Carnegie observations, lightning's diurnal variation has been established as one of its robust characteristics. However, during the past three decades, and despite the wealth of available lightning flash information, studies addressing variables other than flash counts are few and far between. Based on 13 years' worth of LIS observations, this study has revealed a previously overlooked, but very interesting, diurnal variation related to the flash radiance. The diurnal variation of the LIS flash radiance data product, [epsilon], exhibits a consistent increase from late afternoon (~2000 LST) until ~0900 LST and a decreasing trend reaching a broader (continental) or narrower (oceanic) minimum spanning between 1500 and 1900 LST. The overall continental (oceanic) relative variation is ~45% (~15%).

The diurnal variation in [epsilon] was initially interrogated for regional and seasonal consistency as well as statistical biases. We found that the diurnal variation is consistent on seasonal, regional, and global scales, and does not appear to be associated with any artifacts (e.g., LIS threshold setting biases or sample size biases). In addition to documenting these novel findings and vetting potential biases, this study discussed some plausible arguments that might be important in explaining the findings herein. The physical ties and similarities to the Ip over the CONUS, but also the Ip's independence from cloud optical thickness, strengthens the hypothesis in favor of a physical mechanism that truly reflects the diurnal variations of flash energetics. Follow-on studies based on lightning energetics from, for example, the World Wide Lightning Location Network (Hutchins et al. 2012, 2013; Virts et al. 2013) or peak current from, for example, Vaisala's Global Lightning Dataset (Said et al. 2013) are expected to shed more light on the diurnal variations highlighted herein.

Moreover, using a simple electrical capacitor analog for a thundercloud (expanded to include a conducting plane Earth), it was shown that one should expect good covariance between e and Ip, which is indeed what we found empirically (see Figs. 7, 8). Also, the diurnal variation of the total flash count (i.e., a limiting factor to E) explains to a certain extent the diurnal variation of [epsilon] over land; however, it does not explain the diurnal variation of [epsilon] over the ocean. Based on the same capacitance model, we further hypothesized that the thundercloud horizontal areal extent might explain the observed diurnal variation of [epsilon] (Fig. 9a). Data from previous studies support this hypothesis based on MCS storm types.

It is premature to give a final verdict for the exact causes of the diurnal variation in [epsilon], but this paper has provided some important insights that are seemingly in line with simple modeling results and observations. Within this context this postulate can be questioned because of the uncertainties in the [epsilon] observations and diverse storm types, but also the employed oversimplifications of the thundercloud processes via a simple parallel plate capacitor model, which might disregard important but currently elusive mechanism(s) also contributing to the observed [epsilon] diurnal variation. Nevertheless, the findings herein raise various and important implications. Given the intimate relationship of lightning flash energetics and the production of nitrogen oxides (Koshak et al. 2014b), these results are pertinent to air quality modeling and atmospheric chemistry (e.g., Koshak et al. 2014a). In addition, the continental-oceanic, regional, and seasonal [epsilon] contrasts shown herein (see Figs. 1,3,4) could be examined from the viewpoint of a physical mechanism that regulates the flash energetics on a regional or seasonal scale across the convective spectrum, including flash radiance storm-scale observations and possible applications as a proxy for a storm's severity and updraft intensity (e.g., see Schultz et al. 2011; Chronis et al. 2015a). Within the context that flash radiance is an important component in modeling storms from a capacitor perspective, we argue that the findings herein might also raise additional implications for studies on the global electric circuit (Rycroft et al. 2008; Williams 2009; Chronis 2009; Hutchins et al. 2014; Blakeslee et al. 2014). We believe that the science community has just begun to understand how flash energetics can be studied independently but also in conjunction within the same framework as traditional lightning-related research. To this end, the GLM and ISS/LIS missions are expected to significantly augment the already available plethora of lightning flash observations and hence further explore several of the claims made by this contribution.

ACKNOWLEDGMENTS. This work was jointly supported under NASA Program Announcement NNH14ZDA001NTNCA (Climate Indicators and Data Products for Future National Climate Assessments; Dr. Jack Kaye and Dr. Lucia Tsaoussi, NASA Headquarters) and through a NOAANASA Interdepartmental Purchase Request Economy Act Order (NA15AANEG0143) for the NOAA GOES-R Geostationary Lightning Mapper Visiting Scientist Risk Reduction program under the guidance of Dr. Steve Goodman, senior (chief) scientist for the NOAA GOES-R Systems Program. The authors would also like to extend their appreciation to Mr. Dennis Buechler, Dr. Kenneth Cummins, and Dr. Rich Blakeslee for their insightful comments.


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AFFILIATIONS: Chronis--University of Alabama in Huntsville, Huntsville, Alabama; Koshak--NASA Marshall Space Flight Center, Huntsville, Alabama

CORRESPONDING AUTHOR: Themistoklis Chronis,

The abstract for this article can be found in this issue, following the table of contents.

DOI: 10.1175/8AMS-D-16-0041.1

Caption: Fig. 1. (a) Total LST hourly continental (solid) and oceanic (dashed) flash counts in absolute units, (b) As in (a), but in relative units (%). (c) Continental (solid) and oceanic (dashed) e in LST hourly averages, (d) As in (c), but for relative units. The x axis is shown in LST.

Caption: Fig. 2. Regions 1-4 for the regional parsed LIS dataset.

Caption: Fig. 3. As in Fig. I, but the four selected regions for (a) continental and (b) oceanic flash counts in absolute units and (c) continental and (d) oceanic [epsilon].

Caption: Fig. 4. As in Fig. I, but the four seasons for (a) continental and (b) oceanic flash counts in absolute units and (c) continental and (d) oceanic [epsilon].

Caption: Fig. 5. Diurnal variation in relative units for continental (solid) and oceanic (dashed) (a) BGR and (b) e. (c) Scatterplot showing continental hourly BGR (x axis) and e (y axis) LST hourly averages in relative units, (d) As in (c), but for oceanic results.

Caption: Fig. 6. (a) Scatterplot between continental LST hourly flash counts (x axis) and e (y axis) LST hourly averages in absolute units, (b) As in (a), but for oceanic results.

Caption: Fig. 7. (a) Diurnal variation in relative units between hourly e (solid) and Ip (dashed) LST hourly averages over the CONUS, (b) Scatterplot showing results for the same variables.

Caption: Fig. 8. The numeric results of a simple capacitor model for (top) [[phi].sub.N] (red), [[phi].sub.P] (blue), and V (green), and (bottom) E = V/d.

Caption: Fig. 9. (a) Diurnal variation in relative units for continental (solid) [epsilon] and MCS areal extent LST hourly averages, (b) As in (a), but for oceanic results, (c) Scatterplot showing differences between continental MCS areal extent and [epsilon] hourly averages in absolute units, (d) As in (c), but for oceanic results.
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Title Annotation:Tropical Rainfall Measuring Mission Lightning Imaging Sensor
Author:Chronis, Themistoklis; Koshak, William J.
Publication:Bulletin of the American Meteorological Society
Article Type:Report
Date:Jul 1, 2017
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