Global transients in ultraviolet and red-infrared ranges from data of Universitetsky-Tatiana-2 satellite



[1] Light detectors sensitive to wavelength ranges 240–400 nm and beyond 610 nm (which we refer to, for simplicity, as the UV and Red bands) on board Universitetsky-Tatiana-2 satellite have detected transient flashes in the atmosphere of duration 1–128 ms. Measured ratio of the number of Red photons to the number of UV photons indicates that source of transient radiation is at high atmosphere altitude (>50 km). Distribution of events with various photon numbers Qa in the atmosphere found to be different for “luminous” events Qa = 1023 – 1026 (with exponent of differential distribution –2.2) and for “faint” events Qa = 1021 – 1023 (with exponent − 0.97). Luminous event parameters (atmosphere altitude, energy released to radiation, and temporal profiles) are similar to observed elsewhere parameters of transient luminous events (TLE) of elves, sprites, halo, and gigantic blue jets types. Global map of luminous events demonstrates concentration to equatorial zones (latitudes 30°N to 30°S) above continents. Faint events (with number of photons Qa = 1020 – 5⋅ 1021) are distributed more uniformly over latitudes and longitudes. Phenomenon of series of transients registered every minute along satellite orbit (from 3 to 16 transients in one series) was observed. Most TLE-type events belonged to series. Single transients are in average fainter than serial ones. Some transients belonging to series occurs far away of thunderstorm regions. Origin of faint single transients is not clear; several hypothetical models of their production are discussed.

1 Space Detectors of the Atmospheric Ultraviolet and Red Radiation

[2] Moscow State University satellite “Universitetsky-Tatiana-2”(later- on Tatiana-2 satellite) was launched on 20 September 2009 to solar synchronous polar orbit with the height of 820–850 km and inclination of 98.8° [Sadovnichy et al., 2011]. The main goal of the scientific program of this satellite as well as of the first satellite of the same class “Universitetsky-Tatiana” [Sadovnichy et al., 2007] is a study of space near the Earth by measuring charge particle flux at the orbit and radiation from the atmosphere. The data discussed in this paper were collected from 20 October 2009 to 17 January 2010.

[3] Radiation from the atmosphere was measured by detectors in two ranges of wavelength: 240–400 nm (UV band) and >610 nm (Red band). Both detectors were photomultiplier tubes of Hamamatsu type R1463 with multi-alcali cathode and borosilicate window covered by different filters: UVS1 (240–400 nm) and KS11 (>610 nm). It is worthwhile to note that the quantum efficiency of PM tube cathode is uniform in UV band (QE ~20%) and is rapidly decreasing in Red range (QE = 6% at 600 nm and QE = 1% at 800 nm). Field of view of detectors is oriented to nadir and is determined by the collimator. For UV detector, collimator was done as 80 holes of diameter 0.8 mm in the black 2.2 mm thick cover; for Red detector, the number of holes was twice less. From the orbit height of 850 km, those detectors observe the atmosphere area with effective diameter of 300 km (effective area 7 ⋅ 104 km2).

[4] In both detectors, conversion of analog signals to digits was done every microsecond by one ADC with the help of the multiplexer. Much larger wave form sample (1 ms) was used in recording the event temporal profile of 128 ms by summing digital signals measured every microsecond. Data on 128 ms trace were recorded in operative memory, and every millisecond new trace data were compared with the previous one. If the maximal 1 ms signal in the new trace was larger than that in the previous one, the new trace was kept as a reference trace for comparison with the next trace. In the end of every minute, the last trace with the largest 1 ms signal was rerecorded in the main memory. The main memory data were transmitted to the mission center when satellite flied over the Moscow region.

[5] The gain of both PM tubes was controlled by the atmospheric UV glow so that the average ADC signal in UV was kept constant (ADC code N = 16) with the relaxation time of 0.25 s. The PM tubes gain was recorded as code M of digital-analog converter controlling the PMT voltage. In 128 ms trace, the gain was constant.

[6] In parallel to detectors of the atmosphere radiation in UV and Red bands, detector of charged particles at the orbit was operated. Charge particle detector was plastic scintillator plate of area 350 cm2 with R1463 PM tube detecting scintillation light gathered from the plate by a light guide. The main aim of this charge particle detector was a search for charge particle flux “flashes” in correlation with the atmospheric radiation flashes. Electronics of the charged particle detector was similar to radiation detectors. For more detailed description of the detectors electronics, see the work of Sadovnichy et al. [2011], Sadovnichy et al. [2007]; Garipov et al. [2006].

2 Transient Events Characteristics

[7] The first results of transient event observation by detectors of Tatiana-2 satellite were published elsewhere [Sadovnichy et al., 2011; Vedenkin et al., 2011]. In this paper, we present more detailed data with larger statistics.

[8] Both UV and Red detectors (and charge particle detector as well) were triggered by a signal from the UV detector. Every minute, the UV detector selects the trigger event with maximal 1 ms signal in analysis of 468 measured traces. If in this period there was no transient atmospheric phenomenon, detectors were triggered by maximal signal expected from statistical fluctuations of the average photon flux in the detector field of view. For standard signal distribution, every minute, a signal of about 5σ is expected as the largest one (σ is the standard deviation from the average). Signals much larger than 5σ are rare (for example, one signal more than 10σ is expected in 105 minutes). In experiment, the signal of ≥ 10σ occurs every 10 minutes in average and, over some regions of the Earth, every 1 minute. It shows that triggering system selects signals above statistical fluctuations of the atmosphere glow. Origin of selected signals is discussed later on.

[9] Calibration of the detector signal in the number of photoelectrons in the PM tube cathode was done before flight in measurements of single photoelectrons [Garipov et al., 2006]. Hamamatsu R1463 tube selected for the detector has good resolution for single photoelectrons. For conversion from measured number of photoelectrons to number of photons coming to PM tube cathode values of quantum efficiency δ presented by Hamamatsu) were used(〈δ〉 = 20% for UV detector and 〈δ〉 = 2% – for Red detector).

[10] Every transient event was presented by the number of photoelectrons q measured every 1 ms in trace of 128 ms. Number of photoelectrons was converted to number of photons Q arriving in the detector field of view as Q = q/δ. There is alternative possibility that photoelectrons in the PM tube cathode are produced by photons generated in the glass of PM tube by charge particles of cosmic rays or of radiation belts. In the “Universitetsky-Tatiana” mission, this possibility was tried in use of PM tube closed to outer light [Garipov et al., 2006]. In closed tube, some small signals (10% of signal in the tube open to the atmosphere glow) were detected only in the vicinity of South Atlantic Anomaly (SAA) where trapped electron flux is thousand times higher than out of it. The data on charge particle flux in SAA were confirmed in measurement by charge particle detector in Tatiana-2 experiment. Charge particle signal in radiation detectors when it is out of SAA is less than 1% of the atmosphere signal.

[11] Temporal profiles of observed events are presented in Figure 1. They could be differentiated as follows: (a) short single pulse, (b) repeating short pulses, and (c) structural longer pulse.

Figure 1.

Examples of msec transient temporal profiles. Y-axis is ADC code of 1 ms signal. (a) Short (1-5 ms) single pulse (secondary pulses add less than 50% to the integral photon number). (b) Multiple short pulses. (c) Longer pulse. Upper traces are for UV band, bottom traces are for red band.

[12] In assumption that detected photons are isotropically radiated in the atmosphere, full number of radiated photons in 128 ms Qa was calculated as

display math(1)

where R is the distance from the transient event to the detector and S is the summary area of collimator entrance holes. For UV detector, S = 0.4 cm2; for Red detector, it is S = 0.2 cm2. For UV detector at distance R ∼ 800 km, the ratio Qa/Q = 2.2 ⋅ 1017.

[13] For comparison with other results on atmospheric transients, the energy released in UV or Red bands in the atmosphere was calculated as

display math(2)

where ε is the average photon energy (for UV band, ε = 3.5 eV; for Red band, ε = 1.75 eV).

[14] In period October 2009 to January 2010, Tatiana-2 detectors orbited the Earth 797 times in 320 hours of operation time in shadowed “night” part of the orbit. At “day” part, no transient events were found as background of scattered solar radiation was very high and gain of detectors PMTs was 5 orders of magnitude lower than at “night” part. At local nights in off-line analysis of the experimental data, 2628 transient events were selected with average rate of 0.13 min-1 or 10− 4hr− 1km− 2. In this analysis, additional selection criterion was applied: the maximal UV signal measured in 1 ms time sample has to be in ADC codes N >80. Selected transient events found to be distributed in wide range of photon numbers: Q ∼ 5 ⋅ 102 − 5 ⋅ 108 (Qa ∼ 1020 − 1026). The observed range of photon number is much wider than the available range of ADC codes in one time sample (1–1024) due to the following: (a) summing data from all 128 time samples in one trace and (b) measurement of signals at variable PMT's gain (variable intensity of atmosphere glow).

[15] Transient differential and integral distributions on photon number Qa in UV and Red bands are presented in Figures 2 and 3. Maximum in differential photon number distribution in UV band indicates the experimental event threshold: Q ∼ 1.5 ⋅ 103 (Qa ∼ 3 ⋅ 1020; Euv ∼ 170 J). This threshold is much lower than luminosity (released energy) observed in known transient luminous events (TLEs): starting from tens KJ [Chen et al., 2008]. In the present experiment, Qa distribution changes at values Qa ≃ 1023 = Euv ≃ 50 KJ – close to TLE threshold. At " low luminosity" range of 1021<Qa<2 ⋅ 1023, differential Qa distribution could be approximated by power law with exponent − 0.97 ± 0.04 (solid line in Figure 2). For Qa>2 ⋅ 1023, the power law exponent changed to − 2.20 ± 0.13. It is interesting to note that the observed power law even at TLE range is “hard” so that full energy released to transient UV radiation is concentrated in the largest events with energy releases more than 1 MJ – rare events in the present experiment.

Figure 2.

Differential distribution of transients on number of photons Qa. Circles are numbers of photons in UV band. Triangles are numbers of photons in the red band. Lines present power law approximation of the experimental data.

Figure 3.

Integral distribution of transients on number of photons Qa. Circles and triangles as in Figure 1.

[16] Event temporal profiles have a general tendency to correlate with transient photon number: events with Qa<5 ⋅ 1021 are mostly short (Figure 1a), events with large photon numbers Qa>1024 are mostly long with duration of tens- to- hundred ms (Figure 1c). In many intermediate events, short pulses are repeated several times in trace of 128 ms (Figure 3b). Profiles of short single and multipulses were measured before in Tatiana-1 experiment with smaller wave sample 16 μ s at trace duration of 4 ms. The rise time of the pulses were about 0.2–0.3 ms and duration of ~1 ms [Garipov et al., 2011]. They were interpreted as fluorescence of streamer discharges in the atmosphere [Milikh and Shneider, 2008]. In TLE models, overall luminosity was explained as radiation from many streamers [Valdivia et al., 1997; Raizer et al., 1998].

[17] An interesting parameter of transient events is ratio P of photon number in the Red band to photon number in the UV band. P-ratio was measured for every 1 ms interval of the transient event and summed up to trace period of 128 ms. A difficult point in such measurement is the saturation of ADC in measurement of UV signal profile (Red signals measured by ADC are ten times lower than the UV due to lower quantum efficiency of PMT's cathode). To get the true P-ratio, only UV events with ADC codes N in limited range were considered: 80<N<1024.

[18] For pulses with duration <5 ms, P-ratio distribution is presented in Figure 4 as clear histogram. This distribution has maximum at P = 3.6 with half maximum width ΔP = ± 1.5. P-ratio for photons summed over 128 ms trace is presented in Figure 4 by a dashed histogram. This distribution has maximum at P = 1.5 with half maximum width ΔP = ± 1. Statistics of short pulses are larger than statistics of 128 ms traces due to events with several short pulses in one trace.

Figure 4.

P- ratio distribution. Open histogram for photons in short (<5 ms) pulses. Dashed histogram for photons summed over 128 ms trace.

[19] P-ratio as a measure of flash radiating spectrum gives important information on its origin. It was shown in ISUAL experiment [Kuo et al., 2005] and confirmed in model calculations [Milikh et al., 1998; Gordillo-Vazguez et al., 2011] that lines of molecular nitrogen in near UV (337, 356, 390 nm) are significant part of TLE radiation in contrast to lightning spectrum enriched in radiation of excited atomic nitrogen and oxygen in visible and Red-IR bands [Orville and Henderson, 1984]. In Figure 5, available from above-mentioned works spectra from red sprite (bottom panel) and from lightning (upper panel) are presented. Evidently the ratio of photon number radiated in visible-IR band to photon number radiated in UV band could be used for distinguishing origin of observed flashes. In lightning observation by space detector LIS [University of Washington, 2011], events were selected as flashes in narrow OI band around 777 nm. In the present work, experimentally measured P-ratio P ∼ 1 − 5 is close to ratio observed in TLE.

Figure 5.

Spectrum of lightning radiation (upper panel) [Orville and Henderson, 1984]. Spectrum of sprite radiation (bottom panel) [Milikh et al., 1998].

[20] Measured P-value helps to evaluate the altitude of light source as P-ratio depends on the atmosphere density (altitude). Expected P-ratio in the atmospheric electric discharges was calculated for the sum of 1PN2 and 2PN2 excitation states by the following formula:

display math(3)

where q is the rate of corresponding transitions depending on excitation cross-sections and electron energy distribution. The characteristic value of ratio inline image is ≈ 10. Radiation life time of excitation levels 1PN2and 2PN2 are τr(1PN2)≈ 8 ⋅ 10− 6 s and τr(2PN2)≈ 9 ⋅ 10− 8 s. νd is the rate of collisions in which the excitation energy is lost without radiation, νd = σ ⋅ VT ⋅ nm(H) (σ is collision cross-section, assumed as kinetic ∼ 10− 15cm2), VT = (4/3) × [8kT(H)/πm]1/2 is molecular velocity relative to average temperature velocity, H is altitude in the atmosphere, and m is nitrogen molecular mass. Molecular density was assumed as nm(H) = 1.8 ⋅ 1015 ⋅ exp[−(H − 70)/H0)] where H0 = 7 km is the parameter of the exponential atmosphere. Taking into account comparatively weak temperature dependence of velocity V, it was assumed that atmosphere temperature T(H) linearly decreases with altitude from 237 K at H = 50 km to 173 K at H = 80 km (neglecting more complicated temperature regime around mesopause). Calculated P-ratio as a function of the atmosphere altitude is presented in Figure 6. Comparison of the experimental P-ratio distribution presented in Figure 4 with the calculated ratios indicates the range of altitudes H in the atmosphere corresponding to the experimental P values are H ≃ 50 km for 128 ms events and H ≃ 80 km for 1–5 ms events.

Figure 6.

P-ratio as a function of altitude in the atmosphere (calculation).

3 Transient Events Global Distribution

[21] Position of every transient event was determined in geographical coordinates by Universal Time (UT) of the event. In 642 of all 797 orbits available for analysis, there was at least one transient event. Global distribution of all transient events is presented in Figure 7. Night period of observation has distinct borders: latitude 60°N in the Northern and latitude 30°S in the Southern Hemisphere. The distribution has an evident component concentrated above continents in equatorial regions of America, Africa, and Indo-China.

Figure 7.

Global distribution of all transient events. Points are coordinates of transient events.

[22] Trying to separate two transient components: 1. known transient luminous events and 2. newly observed “faint” transients, comparison of global distributions of events with various photon number Qa was made. It was found that event concentration to equatorial region above continents become weaker for events with less photon number Qa. Global distribution of transients having " small" photon numbers Qa<5 ⋅ 1021 (Figure 8) was compared with distribution of transients with large photon numbers Qa>5 ⋅ 1021 (Figure 9). Evidently faint transients show less concentration to equator than luminous events. There are regions which transients avoid—deserts of Sahara and Australia, parts of Siberia.

Figure 8.

Global map of transients with Qa<5 ⋅ 1021.

Figure 9.

Global map of transients with Qa ≥ 5 ⋅ 1021.

[23] It was found that many transients were detected above cloudless regions. To differentiate cloud and cloudless regions, the data from the NASA overview [2011] were used where IR cloud map is available every 3 hours. Taking into account that position of large cloud region do not change much in 3 hours, the “cloud” and “cloudless” regions were marked, and number of transient events was compared in both areas. Among analyzed 720 events, 236 (33% ± 6%) was found to be in cloudless regions. Distribution on transient photon number in cloudless regions is rich in faint events (transients with Qa<5 ⋅ 1021 are 22% ± 7% in contrast to 4% ± 0.5% in cloud regions).

[24] One may consider events with photon number Qa<5 ⋅ 1021 as “instrumental noise.” There are several reasons for these events not being noise. They are certainly not statistical fluctuations of photon number of atmosphere glow (see section 2). They are not PMT outbursts as most of the events with photon number Qa<5 ⋅ 1021 were detected simultaneously in UV and Red bands. “Small” signals are not signals from flux of charged particles generating Cherenkov radiation and fluorescence in the PMT glass. In this case, flux of particles would be simultaneously registered by large area charge particle detector—part of the Tatiana-2 instrument. Data of charge particle detector were analyzed, and no signals simultaneous to UV–Red signals were found during the whole operation period [Vedenkin et al., 2011].

[25] Existence of “faint” or “dim” transients different from TLE was also observed in ISUAL measurements by far-ultraviolet detector [Chang et al., 2010]. Possible atmospheric origin of faint transients is discussed in the next sections.

[26] Interpretation of luminous transients in the present experiment, their separation from lightning is based on comparison of their characteristics with previous data on transient luminous events (TLEs). Transient luminous events were separated from lightning in analysis of image observations, giving data on transient development in upper atmosphere. Transient events in the present work were observed as specific temporal profiles in time scale of 1–128 ms. They are similar to profiles observed in TLE in the same energy released range (>10 KJ). Global maps of various TLE, measured in ISUAL experiment events [Chen et al., 2008], were compared with event maps presented in Figure 9. In ISUAL experiment, rate of “elves” generally prevail on other TLE types (sprites, halo, and gigantic blue jets). Elves are concentrated in equatorial zone (30°N to 30°S) and occurring not only above continents but at lesser rate above the ocean (specifically in North Atlantic, North Pacific regions). In our experiment, events with Qa>5 ⋅ 1021 have similar global distribution. Most of those events are short 1–5 ms pulses (type “a” and “b” in Figure 1). They could be considered as elves-type events in our experiment. Events with the largest photon numbers Qa ∼ 1024 − 1026 (Euv ∼ 1 − 10MJ) have longer profiles (type “c” in Figure 1) and are more likely sprites.

4 Phenomenon of Series of Transients

[27] In Tatiana-2 satellite experiment, a new feature of transient phenomenon was observed: sequences of transients registered every minute (Figure 10). A sequence of Ns transients with Ns>2 was called “series” as such sequences are statistically improbable (for average transient rate ~0.1 per minute). Transient event not accompanied by other transients in the next 2 minutes was called “single.” More than 50% (1519) of all transients (2628) were detected in series which shows importance of this feature in phenomenology of transients detected in the present experiment. Analysis of “single” and “serial” transient global distribution showed a large difference: single transient distribution does not show correlation with the equatorial region (Figure 11) while serial transients strongly correlate to equatorial region above continents (Figure 12). Notable is excess of single transients at high geomagnetic latitudes in the Canada region.

Figure 10.

One day (14 orbits) of transient observation. Black points are every minute coordinates of the satellite (at local night in the latitude range from 30°S to 60°N) and triangles are coordinates of detected transients.

Figure 11.

Global distribution of “single” transients.

Figure 12.

Global distribution of “serial” transients.

[28] Most of the single transients (90% ± 5%) are short pulses of types “a” and “b” presented above in Figure 1. Percentage of short pulses in serial transients is much less: 60% ± 6%. There is also a difference between single and serial transients in their distribution over UV photon number Qa (Figure 13). Maximum of single transient distribution is shifted to smaller Qa, and rate of single transients at Qa = 1021 − 1023 is lower than the rate of serial transients. This feature correlates with that mentioned in section 2 about the relation between transient duration and photon number: short transients have less photon numbers than longer ones. Ratio of photon number in the Red band to photon number in the UV band (P-ratio) for single and serial transients also differs. In Figure 14, P-ratio distribution measured in traces of 128 ms is presented for single transients by black histogram and for serial ones by shaded histogram. Average P value for single transients P = 0.5 ± 0.2 is much less than the average one for serial transients P = 2.5 ± 0.3. In assumption that excitation of molecular nitrogen is a source of observed transient radiation, this experimental trend indicates much lower height in the atmosphere of single transients (Figure 6).

Figure 13.

Differential distribution of single (triangles) and serial (circles) transients over UV photon number.

Figure 14.

Distribution of single (black histogram) and serial (shaded histogram) transients over P-ratio.

[29] Global distribution of serial transients (Figure 12) is similar to global distribution of luminous transients (Figure 9). They show concentration to the equatorial parts of continents. The global map of lightning rate observed in LIS experiment (Figure 15 for November 2009) was compared with the map of serial transient rate (Figure 16). Note that transient data are available only at local nights, i.e., at latitudes from 60°N to 30°S). It is also important to note that measured transient rate is only the lower limit value as the trigger system did not allow to measure rate >1 event/min. In the most active areas (South America, Africa, and Indonesia-Australia), the transient rate ≃ 10− 3 events/km2 is as high as lightning (>10%). In some parts of the ocean (North Atlantic, North Pacific, and Indian oceans) both lightning and transient rates are higher than those for other ocean regions. These transient rates are even higher than the lightning ones. Some transients were detected in cloudless areas where lightning are not expected. In Figure 17, an example of transient series in region with known cloud cover and lightning distribution is presented. The satellite goes from South to North, and the first three transients were detected in the thunderstorm region. The next three transients are in the cloudless region where lightning was not detected. In the presented example, distance between lightning active zone and detected transients in a cloudless region is about 103 km. Transients in series with greater distances from thunderstorm regions (∼ 5 ⋅ 103 km) were also detected; see an example of series in South America (Figure 10).

Figure 15.

Global lightning rate.

Figure 16.

Global transient rate (km2 hour)-1.

Figure 17.

Example of serial transient events (red points) detected in the region with known cloud cover (white regions) and lightning (blue points).

5 Discussion and Conclusion

[30] Tatiana-2 detectors allowed us to select transients in UV and Red bands with duration of 1–128 ms all over the globe in wide range of photon number Qa, radiated in the atmosphere. An important feature of the present experiment is observation in nadir direction and triggering by flashes in near UV (wavelengths 240–400 nm). Such triggering suppressed the detection of lightning occurring in detector FOV. Observed P-ratio of Red photon number to UV photon number in the range of 1<P<10 indicated a high altitude (50–80 km) of observed transients in the atmosphere.

[31] Distribution of transients over photon numbers Qa shows two ranges with a changing exponent of the power law approximating differential distribution: “-1” for 1021<Qa<1023 and “-2” for Qa>1023. This change of exponents is observed at photon numbers far over the detector threshold Qa = 1020 and indicates a physical threshold in the production of “luminous” transients.

[32] Several kinds of correlations underline a difference between luminous and faint transients (with “large” and “small” photon numbers): faint transients are shorter in time and more uniformly distributed in the global map, and they are frequently observed in cloudless regions; luminous transients are concentrated near the equator and above continents, and they have a tendency to be observed in series, occurring every minute on satellite route. We suggest that most luminous events are TLE with some combination of elves, sprites, and gigantic blue jets.

[33] The ratio of transient to lightning rate is ≥10% in thunderstorm areas near the equator above continents.

[34] Faint (small in photon numbers Qa<5 ⋅ 1021) transients are a new phenomenon. Most of them do not belong to transient series and do not show correlation with thunderstorm areas. Experimental data on transients in wide range of luminosity did not meet final interpretation. Several hypothetical models could be considered. a) All transients (luminous and faint) are the result of electromagnetic pulse (EMP) between clouds and lower ionosphere generated by lightning. Luminous events are produced directly above the lightning (they are TLE events: elves, sprites, etc). Faint events are produced at some distance from lightning, sometimes hundreds of kilometers away from it. Excitation of the atmosphere at large distance from EMP was long ago observed in experiments with pulses from powerful radio stations [Gurevich, 2007]. In this scenario, EMP spreading to large distances between clouds and ionosphere may be a “trigger” for transient event generation (at least for " faint" transients). b) Faint transients are light of lightnings scattered at semi-transparent clouds in the upper atmosphere at large distance from lightning. c) Faint transients are rather small streamer discharges in lower ionosphere caused by turbulences in charge distribution. d) Some of faint transients are activated by bursts of charge particles precipitating from near Earth space to the atmosphere. In this assumption, experimentally observed excess of faint transient rate in Canada (Figure 8) is explained.

[35] More data on faint transients is needed, particularly on their development in the atmosphere. The data expected from Tracking Ultraviolet Setup (TUS) having large light collector on board of Lomonosov satellite [Panasyuk et al., 2012] will be of great interest.


[36] Authors are grateful to A.V. Gurevich for helpful discussions. Support from Creative Research Initiatives of MEST/NRF, from RFFI grants 09-02-12162-ofi_m, 10-05-01045a, and 12-05-31025-mol_a are gratefully acknowledged. Reviewers' remarks are appreciated with gratitude.