VLF wave intensity in the plasmasphere due to tropospheric lightning

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH: SPACE PHYSICS, VOL. 118, , doi: /jgra.50217, 2013 VLF wave intensity in the plasmasphere due to tropospheric lightning J. J. Colman 1 and M. J. Starks 1 Received 25 January 2013; revised 27 February 2013; accepted 7 March 2013; published 1 July [1] A climatology of VLF (very low frequency) wave intensity from lightning in the plasmasphere is constructed. Starting from Optical Transient Detector/Lightning Imaging Sensor (OTD/LIS) lightning data representing , a climatology of strikes is assembled with 1 1 latitude-longitude spatial resolution, averaged into 2 h bins for each month of the year. Assuming a linear relationship between optical flash rate and VLF power flux, and that the VLF amplitude drops off as one over distance, a proxy for VLF power is developed. A typical lightning spectrum is applied and the values are scaled by appropriate transionospheric absorptions for each time and place. These values are mapped along geomagnetic field lines in order to compare them to E-field spectral densities measured by the DEMETER satellite between 2005 and An overview of the DEMETER survey mode data is presented which leads to the best scaling of the lightning VLF climatology in LEO (low earth orbit). The resulting data set represents a monthly, 2-hour, solar minimum climatology of VLF wave intensity from lightning in LEO. Finally, the E-field spectral densities are converted to Poynting flux, mapped to the plasmasphere, and converted to B-field spectral densities. Good overall agreement is found with previous observations and estimates. This new climatology is expected to have a significant impact on calculations of pitch-angle diffusion for relativistic electrons in the inner radiation belt. Citation: Colman, J. J., and M. J. Starks (2013), VLF wave intensity in the plasmasphere due to tropospheric lightning, J. Geophys. Res. Space Physics, 118, , doi: /jgra Introduction [2] Very low frequency (VLF) radiation plays an important role in the plasma dynamics of the inner magnetosphere. The energetic electron distribution function in the region is generally thought to be controlled by wave particle interactions and Coulomb scattering [e.g., Thorne, 2010]. Despite decades of study, it has proven difficult to show conclusively what the dominant sources and sinks are for energetic electrons in various parts of the magnetosphere, in part because of a lack of understanding about the distribution of waves involved. [3] Theoretical treatments of energetic electron lifetimes within the radiation belts rely not only on empirical estimates of the electron distribution functions and lifetimes but also on estimates of the wave energy thought to exist there. In seminal work by Abel and Thorne [1998a, 1998b, 1999], estimates were made for the wave intensity, occurrence rate, and propagation angle of all waves then thought to contribute to particle lifetimes. These included plasmaspheric hiss, lightninggenerated whistlers (LGW), and VLF transmitter signals. 1 Space Vehicle Directorate, Air Force Research Laboratory, Kirtland AFB, New Mexico, USA. Corresponding author: J. J. Colman, Space Vehicle Directorate, Air Force Research Laboratory, Kirtland AFB, NM, USA. (jonah.colman@kirtland.af.mil) American Geophysical Union. All Rights Reserved /13/ /jgra Utilizing bounce-averaged quasi-linear pitch angle diffusion coefficients, the equilibrium distribution functions and precipitation lifetimes calculated were found to compare favorably to observations of an electron population generated in the Starfish high altitude nuclear test. VLF transmitters were thought to play a dominant role in the inner radiation belt. More recently, the Air Force Research Laboratory conducted an extensive validation of VLF intensity from transmitters in the plasmasphere comparing data from five different satellites to the output of widely used models for ground VLF injection and propagation. Discrepancies on the order of 20 db were found across the board [e.g., Starks et al., 2008, 2009]. These differences suggest that other assumptions and conclusions made by Abel and Thorne [1998a] should be revisited. [4] Lightning-induced electron precipitation (LEP) events associated with LGW have long been detected from rockets [Rycroft, 1973], satellites [Voss et al., 1984], and from the ground [Helliwell et al., 1973]. Inan et al. [2007] observed large regions associated with thunderstorms containing enhanced precipitating electron flux and suggested that they are produced and maintained by lightning. These regions showed up as large-scale enhancements to the overall background electron density, and not discrete impulsive events. The observations emphasize that it is important to understand the overall climatology of LGW and how they couple into, and are transported within, the magnetosphere, as well as from the perspective of individual LEP events. This broader perspective was also shown to be useful in the work 4471

2 and one for the southern hemisphere). Utilizing the apex radius (R A ), we can equivalently define an invariant apex latitude, l A = cos 1 (1/ (R A /R e )), where R e is the radius of the earth. Figure 1. A schematic showing the organizational framework being followed in this work. by Gemelos et al. [2009], which demonstrated seasonality in electron precipitation consistent with lightning and inferred that lightning is a significant contributor to the formation of the slot region. [5] It is important to categorize the intensities and origins of VLF radiation in the inner magnetosphere primarily in order to understand the sources and sinks of energetic electrons in the radiation belts. The radiation belts contribute to reducing satellite lifetimes in orbit and are a serious hazard to human health during space travel. Precipitating electrons from the radiation belts have a significant impact on ionospheric plasma and are thought to affect the chemistry of the middle atmosphere [e.g., Thorne, 1977; Rozanov et al., 2005]. Previous work constructing climatologies of VLF radiation from satellite data have shown their utility in enhancing our understanding of the radiation belt environment [e.g., Parrot, 1990; Abel and Thorne, 1998a; André et al., 2002; Meredith et al., 2007]. They can help us discriminate terrestrial/non-terrestrial and anthropogenic/ non-anthropogenic sources of wave energy, and therefore contributions to radiation belt particle lifetimes. In addition, understanding the contribution of lightning to VLF intensity in low earth orbit (LEO) may help separate and characterize the global propagation characteristics of VLF radiation both within the earth-ionosphere waveguide (EIW) and through the ionosphere. 2. Organizational Framework [6] The intent of this study is to construct a climatology of whistler wave energy from lightning in LEO where observations are more frequent and propagate those values into the magnetosphere where they are less so. Previously constructed climatologies of lightning are considered as a source of VLF energy, which can propagate within the EIW. The VLF energy in the EIW can then leak out and undergo trans-ionospheric attenuation. A schematic for this approach is shown in Figure 1. [7] In this study, the model will be scaled to measurements of VLF E-field spectral density taken by the DEMETER satellite. Field lines are traced utilizing the 2011 version of the International Geomagnetic Reference Field (IGRF) and organized around an apex coordinate system [VanZandt et al., 1972]. These follow field lines and consist of the apex radius, apex longitude, and height above geoid (which specifies two points on the field line, one for the northern 3. Wave Data [8] The wave data used in this study were obtained from the Instrument Champ Electrique (ICE) experiment [Berthelier et al., 2006] aboard the French micro-satellite DEMETER. Spectral amplitudes in the ELF/VLF range are employed, from 10 Hz up to 20 khz at a resolution of Hz, collected between January 2005 and December The stated sensitivity and dynamic range in the VLF are 0.05 mv/m/ Hz and >80 db, respectively, with an instrumental noise of mv/m/ Hz. Only survey mode data are used here where the electric field component is perpendicular to the orbital plane (sensor component: E12). DEMETER s orbit is quasi-sun-synchronous at 660 km and so measurements are taken near two local times, around 1030 and 2230 for downward and upward orbits, respectively. We use the VLF electric spectrum data type (APID 1132, Type 0). In this mode, waveforms are collected for s, the power spectrum is computed onboard, and 40 spectra are averaged together giving a time resolution of s. This data product is available nearly continuously during the DEMETER mission. [9] In order to investigate the data in a climatological sense, they are averaged into 1 1 geographic latitude and longitude spatial bins for each half orbit for each month of the year. The data are also averaged over five frequency bins to give a final spectral resolution of Hz. The bulk of the spatial locations then have observations making up their averages (corresponding to individual spectra), thus giving us some confidence that they are representative and explaining the binning dimensions employed. Figure 2 shows some representative spatial plots of the data in units of log(mv 2 /m 2 /Hz) overlaid on a global map of continental outlines. The high latitude cutoffs are due to DEMETER not collecting data in that region while the white pixels near the equator are due to being below the range of the colorbar, which is set just below the stated instrument sensitivity. Figure 2 reveals significant diurnal and seasonal variation. While the striations represent orbits sampling a storm and imply that we have not fully sampled the event distribution, the fact that we are resolving features such as continental outlines and seasonal cycles imply that we are very close. The overall spatial and temporal features of the averaged DEMETER data are constant from year to year. 4. Optical Lightning Data [10] The optical lightning flash data used here were collected by the Optical Transient Detector (OTD) and its follow-on the Lightning Imaging Sensor (LIS). Inter-calibrated and merged data products were downloaded from nasa.gov/data/index.html in December From the HRMC_COM_FR and LRADC_COM_SMFR2 data sets, a climatology was constructed with 1 spatial resolution and 2 h temporal resolution for each monthly value. Figure 3a shows the global monthly averages and Figure 3b the 4472

3 Figure 2. Maps showing average VLF power spectral density measured by the ICE instrument aboard DEMETER between 2005 and 2009, 4990 Hz and 5088 Hz, (a) downward orbits in January, (b) downward orbits in July, (c) upward orbits in January, and (d) upward orbits in July. DEMETER did not collect VLF data at high magnetic latitudes; values below the range of the colorbar are also plotted in white. global annual average diurnal cycle of the combined climatology. The two vertical lines in Figure 3b represent the average local time of DEMETER overpasses. Both Figures 3a and 3b show the canonical average of 44 5 lightning flashes occurring around the globe every second and other expected climatological features [Christian et al., 2003]. [11] Figure 3a also shows the monthly averages separated into hemispheres, and the dominance of the northern hemisphere on the overall global cycle is clearly evident, resulting in an August peak. Figure 3b also shows the diurnal cycle broken into the major land mass areas along with surrounding oceans, and there are some variations in relative amplitude with time. The region-to-region Figure 3. (a) Global averages of flashes/second are plotted versus month over an entire annual cycle (black), the geographic northern hemispheric values are plotted in blue and the southern hemispheric values are plotted in red; (b) the global annual averages in flashes/second are plotted versus local time over an entire diurnal cycle, also shown are six spatial subsets corresponding to the major global land masses. 4473

4 Figure 4. Average VLF power spectral density measured by the ICE instrument aboard DEMETER between 2005 and 2009 plotted as a function of frequency: (a) downward orbits and abs(l A ) below 48, (b) upward orbits and abs(l A ) below 54. differences shown here are consistent with those shown at a higher temporal resolution in Mach et al. [2011] (also from OTD/LIS data). 5. Interpreting the Wave Data [12] In order to constrain our initial model with measurements of VLF spectral density taken by the DEMETER satellite, the contribution of lightning to the DEMETER data set must be categorized. Lightning occurs predominantly over land, in the afternoon/evening, and during summer. These are characteristics associated with terrestrial meteorology and are not expected to be shared by non-terrestrial VLF sources observed in LEO. In addition, VLF from lightning propagates up through the ionosphere before being detected. In general, there is more absorption at low magnetic latitudes, during the day, and at higher frequencies. These relationships are typically characterized by power laws [Helliwell, 1965].It can be clearly seen in Figure 2a that the minimum VLF spectral densities at DEMETER altitudes occur along the magnetic equator and not the geographic equator. [13] Globally, lightning peaks in July and so we expect VLF power from lightning to also show a peak in July. Looking for hemispheric trends is complicated by ducted whistler radiation appearing in the data at the magneticconjugate location. Even in the absence of a duct whistler propagation is strongly affected by the geomagnetic field and plasmapause location such that power may reach the conjugate hemisphere at similar invariant latitude (l A ). As a first look, Figure 4a shows a plot of log(mv 2 /m 2 /Hz) versus frequency for daytime orbits and l A below 48, Figure 4b shows nighttime orbits and l A below 54, the different months are represented by colors. Higher l A has been excluded to avoid the strong emissions associated with auroral processes. [14] There is a clear annual cycle evident in Figure 4 with maximum power in July and August as expected for a lightning driven process. This annual variation is seen in both daytime and nighttime orbits with similar amplitude. The nighttime measurements show higher values by an order of magnitude or more, as expected given the larger occurrence of lightning (e.g., Figure 3), the lack of a strong D-layer, and the smaller electron column densities expected during DEME- TER upward orbits. Both plots show a broad peak around 3 5 khz consistent with lightning VLF emissions that have undergone frequency-dependent EIW transport followed by absorption in the ionosphere. The sharp peaks in the spectra are due to VLF transmitters. Below 2 khz, the nighttime spectra show a minimum near 1.6 khz that shows a solar based seasonality, consistent with the EIW cutoff discussed by, e.g., Cummer, [2000] and Toledo-Redondo et al. [2012]. The other nighttime minimum below 1 khz is most likely related to the proton cyclotron frequency in the ionosphere below which whistlers are said not to propagate. The proton cyclotron frequency is proportional to the magnitude of the magnetic field and so varies spatially, but is typically around 600 Hz in regions of interest. The daytime features below 1 khz show little annual cycle, consistent with hiss-like emissions unrelated to lightning [e.g., Bortnik et al., 2009]. [15] In order to investigate the overall spectral, spatial, and temporal correspondence of the DEMETER VLF data and lightning in more detail, Figures 5 and 6 show color plots of VLF spectral density as a function of longitude and frequency. The plots are separated into daytime and nighttime orbits as well as magnetic hemisphere and season, with Figure 5 showing July and Figure 6 showing January. In these figures, VLF transmitters appear as horizontal bands, with NWC being most prominent at a frequency of 19.8 khz and a longitude of 114 (it also shows the spectral broadening observed previously, [e.g., Parrot et al., 2007]). The hiss like emission below 1 khz seen in Figure 4 shows up at all longitudes during the day. Otherwise the most evident features show qualitative agreement with the spatial and temporal variation associated with lightning. More VLF spectral density is present during nighttime orbits, and in the northern hemisphere summer. There are major features associated with landmasses in their summer hemispheres (and their conjugates) consistent with Figure 3. Figure 6a (January daytime orbits) shows the lowest magnitudes and very little in the way of systematic patterns, with the exception of the hiss band already mentioned. In fact, a large fraction of the observations in Figure 6a manifest the effects of instrumental sensitivity. Along with Figure 2, Figures

5 Figure 5. Average DEMETER VLF power spectral density for July plotted as a function of frequency and longitude: (a) downward orbits with l A between 15 and 50, (b) upward orbits with l A between 15 and 50, (c) downward orbits with l A between 15 and 50, and (d) upward orbits with l A between 15 and 50. In each figure, the average local LHR is also plotted as described in the text. and 6 make it clear that the broad peak at 5 khz seen in Figure 4 is strongly correlated to major landmasses and so to lightning. This is also consistent with the seasonal variation shown there and the spectral content of lightning. [16] Also depicted in Figures 5 and 6 are the averaged lower hybrid resonance frequencies (LHR) calculated from averaged plasma compositions estimated with the 2011 version of the International Reference Ionosphere (IRI) combined with averaged geomagnetic field values from the IGRF. The calculation of the LHR frequency follows that described in Brice and Smith [1965], and six ions were included in the calculation. [17] The nighttime peak near 15 khz shown in Figure 4b also correlates with the landmasses and shows clear frequency dependence as a function of longitude. Figures 5 and 6 present an explanation for the higher frequency waves showing an annual cycle in Figure 4. They are consistent with the LHR hiss described by Brice and Smith [1965] and observed innumerable times since. In that work, they are described as being generated in the immediate vicinity of the satellite, associated with non-ducted whistler propagation from the opposite hemisphere, and exhibiting a lower cutoff at the LHR. This lower cutoff reflects the fact that the LHR itself defines a cutoff frequency for propagation transverse to the ambient magnetic field. Additional discussion of relevant LHR observations and theoretical treatments can be found in, e.g., Kimura, [1966], Bell and Ngo, [1990], and more recently Shklyar et al. [2010]. Figures 5 and 6 show that the LHR emissions are associated with lightning in the conjugate hemisphere, so that Figure 5 shows maxima at frequencies above the LHR in the southern hemisphere associated with lightning over North America and Euro/Asia in the northern hemispheric summer. Such features are harder to pick out in Figure 6 but are still extant, especially in the nighttime data where instrument sensitivity is not an issue. The longitudinal structure and shift in the predicted lower cutoff of the LHR hiss band between day and night is consistent with the calculated values. By necessity, we have presented LHR values averaged over a wide range of latitudes in these figures, binning the results into narrower latitudinal regions results in good agreement between the shifting LHR and the lower cutoff frequency (not shown). We do not expect the lower cutoff to be an absolute cutoff in any case since the figures also incorporate 0+ whistlers, which do not typically couple strongly into LHR waves. [18] The preceding analysis shows that many climatological features of the DEMETER VLF data set can be explained by lightning. This conclusion is consistent with previous work 4475

6 Figure 6. Average DEMETER VLF power spectral density for January plotted as a function of frequency and longitude: (a) downward orbits with l A between 15 and 50, (b) upward orbits with l A between 15 and 50, (c) downward orbits with l A between 15 and 50, and (d) upward orbits with l A between 15 and 50. In each figure, the average local LHR is also plotted as described in the text. by Nemec et al. [2010]. The spatial, annual, and diurnal features have all been investigated and along with spectral properties are found to be consistent with lightning. Theoretical predictions and satellite observations confirm that the wave vector of a 0+ LGW is near vertical (radial) within the ionosphere [Santolík et al., 2009] and the wave fields oscillate roughly perpendicular to that direction. Therefore, we expect the measured wave component perpendicular to the orbital plane to be a good representation of its intensity. Nemec et al. [2010] also concluded that DEMETER survey mode data represents a sufficient proxy for the intensity of electromagnetic waves. This data set then provides an excellent benchmark for a climatology of lightning generated VLF in LEO constructed by other means when limited to regions where LGW dominate. 6. Calculating Pseudopower at LEO [19] In order to construct a climatology of VLF radiation from lightning in low earth orbit, we must convert flashes/ km 2 /day in the troposphere to Watts/km 2 /Hz at LEO. As an intermediate step, we will calculate a pseudopower climatology, which will not attempt to quantify the absolute power associated with the lightning flash data. We define a unit of pseudopower as equivalent to the average power of a flash included in our lightning climatology. Our approach assumes that the number of flashes/km 2 is proportional to the amount of power/km 2 radiated in the VLF. This is equivalent to assuming that the strikes are incoherent and the distribution of power over all strikes is invariant in time and place. Initial studies of seasonal variation in the charge moment distribution functions for intense lightning indicate overall similarity to the total [Sato et al., 2008]. It is known that the diurnal cycle of flash number is quite different over oceans [Mach et al., 2011] and so might be the power distribution [Füllekrug et al., 2002]. However, most lightning occurs over land. While it would certainly be of interest to include a spatially and temporally dependent flash to power conversion factor, we consider that there is insufficient guidance in the literature to do so at this time. [20] Rocket and satellite observations show that sferics can couple to the whistler mode over distances of at least 1000 km from the flash [Holzworth et al., 1999; Chum et al., 2006; Santolík et al., 2009]. The ultimate cutoff is related to detector sensitivity and background noise levels rather than any physically based propagation effect as evidenced by the analysis of Jacobson et al. [2011]. In the most complete study of its kind, Fiser et al. [2010] analyzed 4476

7 Figure 7. Spatial plots of pseudopower scaled as described in the text, (a) January, 1100 h local time, 5 khz, 660 km, (b) July, 2300 h local time, 5 khz, 660 km. The units of the colorbar are log (pseudopower/km 2 /day) and values that fall below the scale indicated on the colorbar are plotted in white. ~30000 lightning whistler pairs and found that the maximum whistler amplitude at the satellite depends primarily on the proximity of the source lightning to the magnetic footprint of the detecting satellite. For distances in the range of km, they found that the mean whistler amplitude in LEO drops off in inverse proportion to the distance from the location of maximum amplitude. In light of these studies, we utilize an inverse square relationship between a flash location and the power it delivers to the ionosphere. A more complete treatment might transition from the inverse square treatment utilized here to the db/mm treatment more familiar to radio engineers at some distance from the flash [e.g., Watt, 1967]. However, such a treatment must also attempt to account for the coupling of the EIW mode radiation with k-vectors parallel to the earth s surface into the whistler mode with their k-vectors perpendicular. [21] Once we have a VLF power distribution in the EIW, we must approximate the attenuation of the wave energy in the ionosphere. Here, we use the Helliwell curves, which are calculated employing refractive indices from the quasilongitudinal approximation to Appleton s equation. The validity of Helliwell s approach is checked in some detail by Tao et al. [2010], who find reasonable agreement with a state of the art calculation. In employing the Helliwell curves, one must keep in mind that the quasi-longitudinal approximation may fail near the magnetic equator, and that the entire climatological variation is represented by one function of latitude. Our values are taken directly from Helliwell [1965, Figures 3 35]. We adopt a generic initial spectrum shape for lightning borrowed from Lauben et al. [2001], which shows a broad peak between 2 and 6 khz. [22] Both ducted and non-ducted propagation of whistler waves in the magnetosphere have been observed or inferred [e.g., Smith and Angerami, 1968]. Plasmaspheric observations of ducted whistlers are rare, but both mechanisms are thought to be important in different contexts. Ducted propagation follows field lines, so power input at a given l A shows up at the equivalent conjugate latitude with frequency dependent temporal dispersion and little damping. Non-ducted propagation is more complex and power at a given l A can show up at either a relatively higher or lower conjugate latitude depending on plasma gradients and wave frequency. It is notable that DEMETER records a lot of power near 2 5 khz in the conjugate region of the North American continent (Figure 2d). Non-ducted transport calculations (following Starks et al. [2008]) at these frequencies indicate that such waves should undergo magnetospheric reflection above DEMETER orbit altitudes. There is too little lightning below the assumed conjugate region to generate 0+ whistlers at the observed average spectral densities. So it seems that ducted transport has a major impact on the spatial distribution of the power observed. In the current treatment, only ducted propagation is explicitly calculated greatly simplifying interpretation. Pseudopower is assumed to remain within the flux tube defined by the geomagnetic field lines at the corners of each grid location. Figure 7 shows results of the ducted transport calculation applied to the EIW power distribution after Helliwell attenuation. Note that we have mapped the values from apex coordinates back to a 1 1 geocentric grid at the altitude in question and for ease of comparison with the DEMETER data scaled the data to have a maximum value of 100. These represent our relative pseudopower distributions at LEO. [23] Figures 7a and 7b show a marked resemblance to Figures 2a and 2d, respectively. In general, lightning locations are driven by storm locations, and while their occurrence can be described climatologically, specific storm and flash locations are stochastic events. So we do not expect a one to one correlation between DEMETER data and our model. Nevertheless, it seems that many of the large scale features of the DEMETER survey data can be explained on the basis of this analysis. 7. Scaling the Model to DEMETER [24] We will now consider how best to scale the pseudopower climatology to the DEMETER observations. We know that not all of the spectral density in the DEMETER data represents LGWs, and we know that many salient features of the transformation from sferic to LGW at LEO are being treated here in a less than ideal fashion. Nevertheless, we will assume that if the pseudopower climatology is 4477

8 Figure 8. (a) Log scaling factors determined as described in the text plotted as a function of month; (b) correlation coefficients between the data and model for abs(l A ) from Different frequencies are represented by color as indicated. representative of the contribution of LGW to VLF intensity at LEO, then they should scale linearly. That is, the pseudopower climatology is currently on an arbitrary scale, we would like to convert it to real units with a single scaling factor that would be applied everywhere. The scaling factor is calculated as the ratio of the data to the pseudopower averaged over all observations (N): X X DEMETER ðlat; lonþ Pseudopower ðlat; lonþ lat lon scaling factor ¼ N [25] In determining the scaling factor, we include only observations where LGWs dominate. Low frequencies where hiss or EIW effects dominates are excluded, as are Figure 9. Spatial plots of the scaled pseudopower climatology along with the averaged DEMETER data. All plots represent 5 khz. The columns are model daytime, DEMETER daytime, model nighttime, and DEMETER nighttime. The rows are January, April, July, and October. 4478

9 Figure 10. Maps showing average contribution of VLF magnetic field spectral density from LEO to equatorial magnetic fields, (a) 1100 UT in January, (b) 1100 UT in July, (c) 2300 UT in January, and (d) 2300 UT in July. frequencies above the LHR where quasi-electrostatic waves may dominate. High latitude observations where auroral phenomena occur are also excluded. The longitudinal approximation invoked in the Helliwell calculation breaks down for small dip angles, and so we exclude invariant latitudes below 20. Within the set of data limited as described above, we calculate a scaling factor only from the fraction of data representing relative rank by spectral density. The latter helps to exclude outliers, to minimize the contribution of the instrumental cutoff, and to emphasize locations where there are many 0+ whistlers. Figure 8a shows the log scaling factors calculated in this way for each month of the year at selected frequencies, day and night. Figure 8b shows the associated correlation coefficients between the observations and the climatology. [26] Figure 8a reveals a systematic temporal variation in the scaling factor, with a large peak in Northern summer and a smaller peak in Southern summer. This is due at least in part to the overly simplified transionospheric attenuation treatment. It is well established that the ionosphere has a distinct seasonal cycle that we are ignoring here. Roughly, the same monthly trend is evident day or night. It is also clear that the daytime and nighttime values fall within different distributions, and the scaling factor is nearly an order of magnitude higher during the day (average 4.5 daytime and 3.7 nighttime). That is, given only a single scaling factor based on nighttime data, the model would underestimate the spectral densities by a systematic factor during the day. This also very likely relates to our simple transionospheric transfer function, and is consistent with the analysis of Tao et al. [2010] who compare the Helliwell curves to a transfer function derived from more recent rocket measurements and found that they overestimate absorption more during the day than during the night (Figure 9 in that work). [27] The correlation coefficients shown in Figure 8b reveal a very similar temporal variation. This suggests another possible contributing factor to the monthly trend: there is increased lightning activity at high latitudes during summer, while more lightning is equatorial near the equinox (driven primarily by the solar zenith angle differences between the seasons). Lightning at high latitudes is converted to LGW more efficiently (due to higher magnetic dip angle, smaller electron column densities, and the larger prevalence of ducts), which provides a larger signal above the background and mitigates some of the stochastic component in the signal. Overall, the correlation coefficients shown in Figure 8b reveal values consistent with a significant relationship between the model and the data, while still not accounting for all of the variance. [28] Given the apparent systematic nature of the monthly trends in both the scaling factor and correlation coefficient across frequencies and local time, we apply a different scaling factor for each month, day, and night. Figure 9 gives an overview of the scaled model values compared to the DEMETER averages for 5 khz, day and night, January, April, July, and October. The equatorial region is clearly in 4479

10 Figure 11. VLF equatorial magnetic field spectral density in pt 2 /Hz plotted as a function of frequency for a range of L shells with the colors representing different months of the year going from red in January to purple in December as indicated. The thick black curve represents our annual average. The thick blue curve in each plot represents the values used in the study of Abel and Thorne [1998a], while the red curves represent the results of Meredith et al. [2007]. The thick solid red curve is a digitized representation of their data, and the thick dashed red curve is their fitted model. (a) L = 1.5, (b) L = 2, (c) L = 2.5, and (d) L = 3. need of a better treatment of transionospheric attenuation and/ or sferic to LGW transformation, especially at night. This fact is consistent with the breakdown of the longitudinal approximation there. For our purposes, the current treatment is sufficient because such low magnetic latitudes correspond to field lines with their apex values in the ionosphere rather than within the plasmasphere. The overall agreement is good. 8. Calculating Magnetic Fields at the Equatorial Plasmasphere [29] The above treatment has provided a climatology of whistler wave electric field intensity from lightning at LEO. In the absence of energy loss terms, a conserved quantity for the propagating electromagnetic fields should be the energy flux, often called the Poynting vector, while the corresponding electric and magnetic field intensities will depend on the local value of the refractive index, including its dependence on the angle of the wave vector with respect to the external magnetic field. Our approach to calculating the wave intensities inside the plasmasphere is to convert our electric field intensities at LEO into a time averaged Poynting flux. Then, assuming that the power is confined to flux tubes, transport it to a final location and convert to magnetic field intensities. For the Poynting flux conversions, we follow the treatment from Stix [1992]. In addition, we assume that the waves have radial k-vectors, such that DEMETER is a good measure of (one component of) the magnitude of the electric field intensity for these waves, and the wave normal angle is equivalent to 90 minus the geomagnetic inclination angle. We again take geomagnetic field values from the IGRF and plasma properties at DEMETER altitudes from the IRI. The plasmaspheric density model is identical to that described in Starks et al. [2008]. [30] Figure 10 shows the results of the calculation for some representative instants. The plot represents the contribution to the equatorial magnetic field along the field line running through that location at 660 km for the month and hour indicated. We have now switched to universal rather than local time in order to show what is integrated over when calculating the average wave intensities at a given instant in time. [31] The results of this calculation can be compared to previous estimates of the average VLF power from lightning in the magnetosphere. Figure 11 presents our spectral density estimates as a function of frequency and month at a range of McIlwain L shells (calculated from the IGRF) along with estimates from Abel and Thorne [1998b] and Meredith et al. [2007], while Figure 12a presents the total intensity as a 4480

11 Figure 12. (a) Annual average VLF magnetic field intensities in pt 2 plotted as a function of L shell. The black curve represents the results of this study. The blue curve represents the values used in the study of Abel and Thorne [1998a], while the red curve represents the results of Meredith et al. [2007]. (b) VLF magnetic field intensities in pt 2 plotted as a function of L shell. The colors represent different months of the year going from red in January to purple in December as indicated while the black curve represents the annual average. function of L shell. The previous spectral density estimates lie mostly within our monthly variation. Our average magnetic field intensities are significantly higher at low L shells, and somewhat lower at high L shells. This may be due in part to the sparse sampling of the CRRES mission at low L shells (due to its orbit) upon which the Meredith results are based, and a growing contribution from magnetospheric chorus at the higher L shells as discussed in Meredith et al. [2007]. It should be remembered that the total magnetic intensity is highly dependent on the bandwidth and that each of the representations in Figure 12a utilizes different bandwidths. Finally, we present the total intensity as a function of L shell for each month of the year in Figure 12b. We see that there is a significant variation in intensity as a function of L shell throughout the year as the major lightning centers shift with season. 9. Conclusion [32] Starting from OTD/LIS lightning data representing , we have constructed a climatology of VLF spectral density from lightning at LEO. Key assumptions include as follows: (1) a linear relationship between optical flash rate and VLF power flux; (2) VLF attenuation of whistler energy goes as the inverse distance in the EIW (includes EIW attenuation and sferic-whistler coupling); (3) the latitudinal component of transionospheric attenuation can be reasonably represented by the Helliwell curves; (4) 0+ whistler energy travels along field lines and shows up in the conjugate hemisphere without attenuation; and (5) the 5 years of DEMETER survey data reasonably represent one component of spectral density in LEO. Values are available for each month of the year and every 2 h of local time on a 1 1 spatial grid. The so constructed climatology shows complicated variation as a function of time and space. [33] This climatology represents a baseline treatment and there are various improvements that could be implemented. Most importantly, an improved model of average VLF attenuation should be employed, which would capture seasonal variation. It would also be relevant to include the EIW cutoff and proton whistler dispersion, which are so evident in the data. It should be kept in mind that the observations to which this model is tied were mostly taken during the occurrence of a deep solar minimum. Solar minimum conditions affect the state of the ionospheric plasma, the phase space density of energetic electrons throughout the inner magnetosphere, and the location of the plasmapause, all of which should modify the observed spectral power densities. [34] The purpose of this investigation was to give estimates of the VLF power in the plasmasphere at a 2-h cadence for each month of the year. In order to characterize the sources and sinks of energetic electrons in the radiation belts, the spectral power densities derived here must be used as input to a calculation of pitch angle diffusion coefficients, which could then be used to estimate particle lifetimes. Given that diffusion coefficients scale roughly with magnetic field intensities, the climatologies reported here should have a significant impact on calculated particle lifetimes, especially in the inner radiation belt. [35] Acknowledgments. The authors would like to thank R. A. Quinn, G. P. Ginet, J. M. Albert, and the Stanford VLF group, notably R. K. Said and N. G. Lehtinen, for useful discussions. This work was based in part on observations with the electric field experiment ICE embarked on the DEMETER satellite launched by the Centre National d Etudes Spatiales. The authors thank J. J. Berthelier, the PI of this instrument, for the use of the data. 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