DOI 10.1007/s11141-017-9765-3

Radiophysics and Quantum Electronics, Vol. 59, No. 12, May, 2017 (Russian Original Vol. 59, No. 12, December, 2016)

SELF-CONSISTENT MODEL OF A NIGHT SPRITE A. A. Evtushenko∗ and F. A. Kuterin

UDC 551.510.533

We propose a radially symmetric self-consistent model of a sprite at altitudes of 60 to 90 km in the region with a radius of 60 km. The perturbations of the densities of ions, electrons, and neutral particles, as well as the photon-emission intensities at the mesospheric altitudes are analyzed with respect to the sprite under night-time conditions. Because of the fast electric-field displacement in the upper part of the diffuse region of the sprite at altitudes of 78 to 81 km, radiation at the discharge axis stops earlier than that in the outer region, i.e., the toroidal structures of the electric field and the sprite radiation are observed. At altitudes of 83 to 87 km, the electrondensity decrease related to the increasing role of the dissociative attachment to molecular oxygen occurs, which significantly decreases conductivity at these altitudes.

1.

INTRODUCTION

Airglows at altitudes of up to 100 km, which correlate with the tropospheric thunderstorm activity and are conventionally called the Transient Luminous Events (TLEs), were discovered at the end of the 20th century. Immediately after the discovery of the high-altitude discharges, the issue of their influence on the atmospheric condition became topical. Direct measurements of the chemical-composition and conductivity perturbations in the region of the high-altitude discharge development are technically difficult and sometimes impossible. The mesosphere and the lower ionosphere are not readily accessible for studying by the groundbased methods since they are too high to be reached by aircraft and aerostats, while direct measurements can only be carried out by launching expensive rockets, which bring significant perturbation to the mesospheric condition during their motion. In this case, to ensure the time and space matching of the rocket launch and the high-altitude discharge is an insoluble problem. Therefore, the development of the detailed models of high-altitude discharges and the methods for solving the inverse applied problems, which allow us to obtain information on the mesospheric conductivity and composition perturbations, becomes of special importance. From the viewpoint of possible influence on the chemical composition of the atmosphere, sprites attract the main attention since they are the brightest TLA types (the brightness is up to 1.5 MR) and have a significant volume (up to 104 km3 ) and a high occurrence frequency in the terrestrial atmosphere (up to one event per minute) [1–3]. The first rather detailed single-point models were proposed in 2008 for the streamer part of the sprites under night-time conditions [4, 5]. Considerable ion and electron-density perturbations at altitudes of 70 km and below was predicted from the simulation results. The inconsistent allowance for the electric field is the main debating point when constructing these models, while calculations are performed for fields exceeding the breakdown ones by a factor of 3 to 5. In [6], it is shown that the sprites can be important sources of the NOx molecules in the tropics at an altitude of 70 km and locally perturb the nitrogen-oxide density (perturbation can reach 20%) above the regions of high lightning activity but fail to globally affect the NOx content. ∗

a [email protected]

Institute of Applied Physics of the Russian Academy of Sciences, Nizhny Novgorod, Russia. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 59, No. 12, pp. 1076–1086, December 2016. Original article submitted March 15, 2016; accepted August 8, 2016. c 2017 Springer Science+Business Media New York 962 0033-8443/17/5912-0962 

In [7], the causes of the delay between the tropospheric discharge and the sprite are studied in detail on the basis of the previously developed chemical model [5] and the role of the dissociative attachment of electrons with the formation of the O− ion and the subsequent associative detachment is specially emphasized. The single-point model of the streamer part of the sprite is developed for the daytime conditions [8]. The sprite is shown to be initiated at lower altitudes compared with those under night conditions due to a much higher atmospheric conductivity. The atmospheric chemical perturbations at altitudes of 31, 42, and 54 km are analyzed. An ozone-density decrease after the sprite flare, which reaches 15% at an altitude of 54 km 15 min after the discharge, is predicted. The one-dimensional self-consistent model of the diffuse region of the sprite is developed in [9, 10]. It is shown that with allowance for a significant variation in the conductivity the electric field exceeds the breakdown value (by a factor of up to 1.5), which leads to the electron-avalanche formation and perturbation of the mesospheric chemical balance above 70 km. The dependence of the size of the diffuse region of the sprite on the lightning-discharge parameters in the troposphere is studied. Despite the achieved success during simulation of the effects of influence of the high-altitude discharges on the atmospheric chemical balance, only a few numerical models allowing for the radial nonuniformity during the discharge development have been worked out. A two-dimensional model for describing the mesospheric response to the perturbation of the slowly varying electric field, which is related to the discharge processes in the troposphere, is proposed in [11]. Because of the complexity and duration of the calculations, the chemical block of the model is significantly simplified and contains the generalized components of the positive, negative, and cluster ions, electrons, the O− ions, and several neutral components. The calculation results emphasize the importance of allowance for the processes of fast electron detachment from the O− ions to describe perturbation of the chemical mesospheric balance especially in the pre-breakdown fields. Experimental observations show that the sprites are far from always being ignited exactly above the cloud–Earth discharge and have horizontal displacement up to several kilometers. Some cases of the sprite ignition over the ring round the cloud–ground flare have been recorded. To study the radial effects in the high-altitude discharges, we upgraded the previously developed one-dimensional self-consistent model of the sprite under the night conditions [9, 10]. The model is considered in the axially symmetric formulation. The altitude and radial dynamics of perturbations of the electric field, chemical composition, conductivity, and optical radiation in the diffuse part of the sprite and above it are studied. 2.

FORMULATION OF THE PROBLEM

In this work, we present the radially symmetric model of the high-altitude discharge, which is the upgrade of the one-dimensional self-consistent plasma-chemical model of the sprite proposed in [9]. A system of 267 chemical reactions for 61 chemical components was used for describing the chemical condition of the mesosphere and optical emission at altitudes of 60 to 90 and 0 to 60 km in the radial direction. The main + + − and O− ions, and the neutral positive O+ 2 and NO ions, the cluster H (H2 O)n ions, the negative O 2 components of the nitrogen–oxygen mixture including the nitrogen and oxygen atoms in the excited states are allowed for. The chemical-reaction unit is mainly based on [12] and supplemented by the data from [4, 5, 13]. The complete list of the chemical reactions and components is given in [9]. The system of differential equations corresponding to the model includes the chemical-kinetics equations in the form   dNi = vk Ns , dt i k∈Ki

i = 1, . . . , 61,

(1)

s∈Jk

where i is the chemical-component index, Ki and Jki are the sets of the indices corresponding to the chemical reactions in which the ith chemical component participates, and vk is the rate of the kth chemical reaction. The initial densities of the chemical components are borrowed from the atmospheric climatic model (Whole Atmosphere Community Climate Model (WACCM)), which is a part of the generalized Community Earth System Model (CESM) of version 1.2 at the point with the coordinates 38◦ N, 0◦ E [14]. 963

The electric-field perturbation at the mesospheric altitudes results from the tropospheric discharge processes. The development of the essentially powerful cloud–ground discharges (mainly of positive polarity) results in formation of a considerable uncompensated discharge in the cloud. The electric field E dynamics at the mesospheric altitudes is self-consistently simulated in accordance with the following differential equation (see, e.g., [15, 16]): dEext dE σE = + . (2) dt ε0 dt Here, σ is the electron conductivity, Eext is the absolute value of the external electric field of the uncompensated discharge from the cloud–ground lightning discharge, which is specified in the dipole approximation with allowance for the high conductivity of the Earth, E is the absolute value of the acting electric field at the mesospheric altitudes, and ε0 is the dielectric constant. In this case, the “acting” field is understood to be the field used for calculating the chemical-reaction rates during the high-altitude discharge. Equation (2) is approximate and describes the field in the mesospheric region in the quasi-static approximation thereby allowing for the external-field dynamics, which is stipulated by the current dynamics in the lightning channel, and the conductivity perturbations in the discharge region. In [15, 16], such a simplified approach is used for calculating the field of the M component of the lightning flash (with allowance for the realistic altitude conductivity profile) at the sprite-initiation altitudes and demonstrated high efficiency. It was established that it is exactly the quasi-static field expulsion from the region of high conductivity (related to both a sharp increase in conductivity with increasing altitude and the conductivity perturbations because of the increase in the electron temperature and density in the discharge region), which plays the key role in the discharge dynamics at the sprite-development times. In this work, it is assumed that Eq. (2) includes the absolute value of the electric field for calculating the chemical-reaction rates, which allowed one to take into account the correction related to the influence of the horizontal component of the field on the peripheral discharge dynamics. Of course, the systematic allowance for the self-consistent field dynamics would require solution of a much more complicated problem with allowance for the entire system of the Maxwell equations. The charged region is located at the altitude H = 10 km above the Earth surface and its radial size is L = 10 km. It is assumed that the uncompensated charge is distributed in a cloud according to the normal two-dimensional law with the characteristic standard deviation L/3. The maximum value of the current in the lightning channel amounts to 121.7 kA, which corresponds to the maximum dipole moment 790 C·km. Current in the lightning channel is specified by I(t) = I0 · (t/τ1 )2 · exp(−t/τ2 )/[1 + (t/τ1 )2 ]

(3)

for the parameters I0 = 150 kA, τ1 = 70 μs, and τ2 = 500 μs. The system of differential equations (1) and (2) of the chemical kinetics and electric field is independently solved at the nodes of the grid with the altitude and radius steps 100 m and 1 km, respectively. 3.

NUMERICAL-SIMULATION RESULTS

3.1.

Electric field

Figures 1 and 2 show the dynamics of the normalized electric field. The breakdown field 128 Td is reached at an altitude of 81 km 0.4 ms after the tropospheric-discharge onset. In what follows, the time when the breakdown field is reached at the discharge-initiation altitude is called the sprite onset. The electric field reaches a maximum of 185 Td at an altitude of 79 km 0.3 ms after the sprite onset. The upward propagation of the diffuse discharge occurs up to an altitude of 84 km with a velocity of 2 · 107 m/s for 150 μs. Downwards, the sprite reaches an altitude of 73 km. The downward-propagation velocity amounts to 2 · 107 m/s during the first 150 μs and decreases thereafter. The average downward velocity of the diffuse region of the discharge amounts to 4.7 · 106 m/s during the 1.7 ms propagation. On the discharge axis, an electron avalanche with the electron density increasing by over an order 964

Fig. 1. The normalized electric field θ at the discharge axis as a function of altitude.

Fig. 2. Radial dependence of the normalized electric field θ at the altitude z = 79 km.

of magnitude is formed at an altitude of 80 km in the lightning channel 1 ms after the current-flow onset, which leads to the conductivity increase and the electric-field expulsion from the discharge region. The field-expulsion effect is especially pronounced at altitudes of 79 to 81 km, which coincide with the altitudes of the maximum electron-density and conductivity perturbations. The fast electric-field expulsion leads to the formation of the toroidal structure of the discharge at the studied altitudes. In 1.2 ms, the electric field on the discharge axis is totally expelled, whereas the discharge persists at a radial distance of 20 km (see Fig. 2). With reference of the close connection between the electric field and the sprite emission in the first and second positive bands of molecular nitrogen, the sprite emission at altitudes of 79 to 81 km originates from the toroidal shape. This effect is observed less frequently with decreasing altitude since the maximum fields and, therefore, the conductivity perturbation decrease and the field persists in the discharge region for a longer time.

3.2.

Conductivity

One of the main applied issues, which are encountered during the high-altitude discharge simulation, is related to the study of the conductivity-perturbation dynamics significantly affecting the propagation conditions of the very low frequency (VLF) radio waves. On the one hand, the conductivity variation can influence radio communication, while, on the other hand, it can be a tool for studying the sprites themselves. Correct simulation of the conductivity dynamics requires self-consistent calculation of the electric field and conductivity, which has been accomplished in this work. A significant conductivity variation is observed at altitudes of 75 to 90 km. Two regions with qualitatively Fig. 3. Conductivity σ as a function of time on different conductivity dynamics can be singled out in the the discharge axis at altitudes of 75 km (curve 1), discharge region. At altitudes of 75 to 82 km, the con- 79 km (curve 2), 83 km (curve 3), and 85 km ductivity is considerably decreased at the initial discharge (curve 4). stage because of an increase in the electron temperature and collision frequency (see Fig. 3). A sharp increase in the charged-particle densities is observed during the sprite development and the avalanche-ionization formation, which leads to the proportional conductivity increase. Actually, the conductivity at altitudes of 77 to 81 km reaches 5–6 μS/m during the discharge and several seconds thereafter, which is comparable with the conductivity at an altitude of 86 km under the unperturbed conditions. At an altitude of 75 km, 965

Fig. 4. The altitude-time dependence of the electron density Ne at the discharge axis.

Fig. 5.The radial-time dependence of the electron density Ne at the altitude z = 83 km.

the conductivity-perturbation relaxation occurs in several seconds. At the same time, the characteristic relaxation time amounts to several hundreds of seconds at altitudes of 80 to 83 km. The altitudes in the range 83–86 km exhibit specific electron dynamics, i.e., the electron density, as well as the conductivity drop during the discharge because of the increasing role of the attachment reactions and recovery to the pre-sprite level occurs during several seconds. At an altitude of 78 km, the conductivity-perturbation radius reaches 35 km.

3.3.

Electrons

A significant perturbation increase in the electron density is observed at altitudes of 74.5 to 86.5 km in the region with a radial size of up to 30 km (see Figs. 4 and 5). The diffuse region of the sprite can be divided into two parts with qualitatively different electron-density dynamics. The electron density increases in the lower part of the discharge from 74.5 to 82.5 km and decreases at altitudes of 82.5 to 86.5 km. In the unperturbed night-time mesosphere, the neutral-particle ionization is the main source of electrons, while sink is related to recombination with positive ions. Along with these processes, one can observe formation of the O− 2 ions upon fast attachment of the electrons to molecular oxygen and dissociative attachment to O3 . The produced O− 2 ion interacts with atomic oxygen, which leads to the formation of O3 and electrons during the associative detachment. In the above-shown chain, the chemical-reaction rates are high and O− 2 is not accumulated at the studied altitudes. Strong electric-field perturbation during the discharge leads to the redistribution of the electron sources and sinks. During the discharge, the molecular-oxygen and nitrogen ionization reactions whose rates are the nonlinear functions of the electric-field intensity, are the main sources during the discharge. As for other charged particles, an avalanche increase in the electron density occurs with a delay with respect to the tropospheric discharge onset, i.e., a sharp density increase at an altitude of 81.5 km is observed only 0.75 ms later. For 1.4 ms, the electron-density perturbation wave propagates downwards and reaches an altitude of 75 km. The vertical propagation velocity of the electron-density perturbation amounts to 4.65 · 109 m/s. At the beginning of the sprite, the main role is played by the reaction with molecular oxygen, which is related to a higher rate of this reaction when the breakdown value is slightly exceeded by the field. As the discharge develops, the nitrogen-ionization role increases and reaches 70%, which is determined by the nitrogen percentage in air, such that the nitrogen- and oxygen-ionization rates become almost equal. The main sink for the electrons is related to their attachment to molecular oxygen with O− formation according to the reaction O2 + e → O− + O. Maximum electron-density perturbation at an altitude of 78 km amounts to 198 cm−3 , which by a factor of 275 exceeds the equilibrium value 0.72 cm−3 . The electron-density relaxation time increases with increasing altitude from several seconds to 1000 s at altitudes of 75 and 82 km, respectively. The 966

Fig. 6. Radial-temporal dependence of the O− 2 ion density at the altitude z = 80 km

Fig. 7. Radial-temporal dependence of the O− ion density at the altitude z = 78 km.

electron-density perturbation radius increases from 10 to 40 km at altitudes of 75 and 79 km, respectively. The electron-density decrease at altitudes of 82.5 to 86.5 km with a pronounced radial dependence (see Figs. 4 and 5), which is related to reaching the almost breakdown values by the field at these altitudes, is a specific feature of the discharge. The effectiveness of the reaction O2 + e → O− + O considerably increases and the electron avalanche is not formed, which is also observed during the displacement from the discharge axis. The electron density at an altitude of 83 km and a radius of 15–30 km decreases two times compared with that in the unperturbed state (see Fig. 5). Once the discharge is over, the electron density is recovered to the unperturbed values in 1s mainly because of the electron detachment from negative ions.

3.4.

Ions

The O− 2 accumulation occurs during the electron avalanche development in the discharge and the subsequent density relaxation of other negative ions. A significant density perturbation of O− 2 is observed at altitudes of 74.5 to 84 km with a radius of up to 42 km (see Fig. 6). Relaxation of the O− 2 density perturbation has a strong altitude dependence. At altitudes of 74.5 to 76.5 km, the perturbation relaxation time amounts to several tens of seconds. At altitudes of 76.5 to 81.3 km, one can observe a sharp relaxationtime jump to almost 600 s. Above 81.5 km, the perturbation is small and the relaxation time does not exceed 1 s. The O− 2 ion density perturbation at an altitude of 81 km starts 1 ms after the current flow onset in the tropospheric lightning channel. The O− 2 density perturbation propagates downwards and upwards for 1.4 and 0.5 ms, respectively. The radial perturbation propagates to 15 km from the discharge axis for at most 1 ms. Maximum O− 2 density perturbation is substantially displaced to the bottom of the diffuse part of the sprite, observed at altitudes of 76 to 78 km, and amounts to 115 cm−3 . The lifetime of the maximum O− 2 ion densities is equal to 7 s at an altitude of 76 km, such that the perturbation radius reaches 20 km. At altitudes between 76 and 78 km, we observe the density-perturbation peak after the discharge end from the 1st to the 9th second, which is related to the ion conversion of other negative ions. The radial perturbation size does not exceed 10 km. The O− 2 density perturbation has strong radial dependence. At an altitude of 75 km, the perturbation radius does not exceed 10 km. With increasing altitude, the perturbation radius increases and amounts to 20 and 35 km at altitudes of 76 and 78 km, respectively, and exceeds 40 km at altitudes of 80 to 82 km. Above 83 km, the ion-density perturbation is almost absent. The reaction of the dissipative electron attachment to molecular oxygen, which is also the main electron sink, is the main source of the O− ions. The relaxation time of the density perturbation of these ions is small and they almost immediately disappear after the field-perturbation end. The maximum density is reached at an altitude of 78 km and amounts to 30 cm−3 . The radial density perturbation reaches 30 km 967

and propagates with a delay of about 1.5 ms after the sprite onset (see Fig. 7). −3 and reached at an altitude of 78 km, while the The maximum O+ 2 -ion density is equal to 255 cm −3 altitude region with perturbations exceeding 230 cm is in the range 77–79 km. The lower perturbation boundary is sufficiently sharp and is located at an altitude of 73 km. The upper boundary of the O+ 2 ion perturbation at an altitude of 82 km is diffuse and slight density perturbations are observed to almost 90 km. The radial size of the O+ 2 ion density perturbation strongly increases with increasing altitude, i.e., the radius amounts to 105 km and exceeds 30 km at altitudes of 75 and 78 km, respectively. One can also observe a delay during the radial propagation of the perturbation, i.e., at an altitude of 78 km and for a radius of 3 km, the delay is 0.7 ms. The density relaxation of this ion strongly depends on the altitude. The longest relaxation is observed at an altitude of 82 km, at which return to the pre-sprite values takes 100 s, whereas only several seconds are required at an altitude of 7 km. The N+ 2 ion density under the unperturbed conditions of the night-time mesosphere is small. A pronounced N+ 2 density perturbation is observed only during the discharge and quickly decays after its completion. The N+ 2 density perturbation is observed at altitudes of 73 to 82 km with a radial size of up to 40 km. The maximum value 105 cm−3 is reached at an altitude of 78 km.

3.5.

Radiation and nitrogen in excited state

The main radiation of the high-altitude discharge is for the first and second positive nitrogen bands. The second-band radiation is related to the transition N2 (C) → N2 (B) + hν(2 PN2 ). Since the radiation lifetime N2 (C) is short, the N2 (C) density perturbation does not exceed 1 cm−3 despite a significant increase in the N2 (C)-formation rate according to the reaction N2 + e → N2 (C) + e in the electric field. The main radiation in the second nitrogen band is observed at altitudes of 75.0 to 81.5 km, such that the radial size of the radiation region reaches 30 km with the maximum at the sprite axis at an altitude of 78 km where the volume velocity of the photon emission amounts to 1.35 · 107 cm−3 ·s−1 . The radiation lifetime of the N2 (B) state is much longer than that for N2 (C). The transition from N2 (C) with radiation in the second band and nitrogen excitation during collisions with electrons in the electric field are the main sources of N2 (B) and radiation in the first positive nitrogen band is the main sink. The altitude range of N2 (B) perturbation is from 75 to 81.5 km, while the maximum perturbation 405 cm−3 is reached at an altitude of 78 km. The N2 (B) density dynamics is entirely determined by the field and its relaxation. The N2 (B) state has the longest lifetime among all the excited states of molecular nitrogen taken into account. The transition from N2 (B) with radiation in the first positive band, i.e., Fig. 8. Radial-temporal dependence of the volume N (B) → N (A) + hν(1 PN ) and nitrogen excitation dur2 2 2 rate of the photon emission in the first positive ing the collisions with electrons in the electric field are the band of molecular nitrogen at the altitude z = main sources of N (A). The N (A) density perturbation 2 2 78 km. range is at altitudes of 74.5 to 82.5 km and the maximum density perturbation reaches 1.73 · 104 cm−3 . The maximum radiation intensity in the first positive band is located on the sprite axis at an altitude of 78 km and amounts to 6.4 · 107 cm−3 ·s−1 . The intensity ratio is well correlated with the data of the experimental observations of the sprites (see Fig. 8). With increasing altitude, the perturbation-relaxation time increases from 2 to 5 ms at 74.5 and 82.5 km, respectively. The radial perturbation size for N2 (B) and N2 (A) reaches 35 km. At an altitude of 78 km, the N2 (B)-density perturbation starts at the time moment 1 ms at the discharge axis and propagates in the radial direction for 0.8 ms. In this case, the entire perturbation at the 968

discharge axis has enough time to relax for 0.3 ms. Therefore, the perturbation toroidal shape is observed for the state N2 (B) unlike the state N2 (A), i.e., perturbation is absent on the discharge axis, while it is present at a radial distance of 15 km and greater. Such a dynamics of the N2 (B) density is related to the electric-field dynamics and its fast expulsion from the discharge region with a high conductivity perturbation. The radial dependence of radiation in the first and second nitrogen bands is the most pronounced at an altitude of 78 km. Intense radiation on the sprite axis starts 0.9 ms after the cloud–ground discharge onset. By the time 1.2 ms, the field radially propagates to a distance of 22 km. In what follows, radiation on the sprite axis at this altitude stops because of the field expulsion from the discharge region. The radiation-wave propagation continues up to 30 km and farther. Therefore, when the sprite is sufficiently developed, only its outer shell glows. The atomic-oxygen radiation at the wavelengths 557 and 630 nm and the atmospheric lines of molecular oxygen were also taken into account during the simulation, but the corresponding radiation intensity is by an order of magnitude smaller than that in the second positive band of molecular nitrogen. 4.

COMPARISON WITH EXPERIMENTAL DATA

To check the performed calculations on the basis of the proposed axially-symmetric plasma-chemical model of the sprite, the dependence of the radial size of the diffuse-discharge region on the maximum current (maximum dipole moment) in the lightning channel has been plotted. According to the numerical experiments, the maximum field on the discharge axis is reached at an altitude of 78.3 km for which the numerical calculations of the maximum and minimum sizes of the diffuse region of the sprite were performed. The minimum boundaries of the sprite are determined from the field value 128 Td, which is critical for the discharge development. Another curve corresponding to the field value 88 Td, which is characteristic of the halo was plotted in Fig. 9 for estimating the maximum possible size of the glow region. The maximum current in the tropospheric lightning channel is laid off on the horizontal axis and the second horizontal axis corresponds to the maximum dipole moment (this is done for convenient comparison with the experimental data). Comparing the model calculations and the experimental data is always rather difficult. The main difficulty is related to the fact that the data, which simultaneously contains information on the optical image, the value of the maximum current force, or the maximum dipole moment, are scarce. Figure 9 shows six experimental points taken from [17] (points 1, 3, 4, and 6) and [18] (points 2 and 5), which are used as basis for constructing the approximation curve. For larger dipole moments, the numerical model yields overestimated results for the perturbation radius, which is in particular related to the necessity of correcting Eq. (2) for large deviations from the discharge axis. For small dipole moments, the model yields a too small glow radius or its absence, which can be due to the complicated field dynamics and various discharge nonuniformities, which are not allowed for in our formulation of the problem. For the medium values of the dipole moment, which are the most typical of the night-time sprites, the model yields the values that are in satisfactory agreement with the experimental data. Important information on the discharge dynamics, characteristic values of the electron temperature and the electric field can be obtained from the spectrometric studies of the high-altitudes discharges, which are carried out from both the ground and the satellites. Figure 10 shows the ratio of the radiation intensities in the first and the second positive bands of molecular nitrogen at various altitudes and the total ratio for the entire studied discharge volume. The maximum value of the intensity ratio in the bands at the specified altitudes is located in the range 0.23–0.25 and equals 0.21 for the entire discharge region, which only slightly exceeds the experimental-observation data [19]. The satellite-borne observations [21] yield an overestimated value of the ratio of these intensities at an altitude of 78 km, which differs from the calculated results and can be due to illumination from the initiating cloud–ground flash or the technical issues of the radiation recording.

969

Fig. 9. Radial size of the diffuse discharge region as a function of the maximum current Imax and the maximum dipole moment Pmax . Curve I corresponds to θ = 88 Td, curve II, to θ = 128 Td, and the dashed curve approximates the experimental data. 5.

Fig. 10. The ratio of the radiation intensities RBR in the second and the first positive bands of molecular nitrogen at altitudes of 72, 76, 78, and 80 km (curves 1–4, respectively) and for the entire discharge region (thick curve).

CONCLUSIONS

We have developed a radially-symmetric plasma-chemical self-consistent model of the electric-field perturbation influence on the mesospheric composition, which was used for describing the sprite influence on the mesospheric composition. It has been shown that the conditions for initiating the high-altitude discharge (reaching the breakdown field) are initially satisfied at an altitude of 81 km 0.7 ms after the current-flow onset in the tropospheric lightning channel. In what follows, the breakdown field propagates upwards up to an altitude of 83 km and downwards to 74 km. It has been established that the mesospheric conductivity at the sprite-initiation stage decreases by almost two orders of magnitude because of the increase in the frequency of electron collisions with neutral particles with increasing temperature, which contributes to the discharge development. As the electron avalanche is formed, conductivity is significantly increased, which leads to the field expulsion and discharge end. It has been shown that a decrease in the electron density, which is related to an increase in the role of dissociative attachment to molecular oxygen, is observed. This phenomenon substantially reduces the conductivity at these altitudes. Taking the radial dependence during the discharge simulation into account allows one to reveal the effect of the electric-field rejection and termination of radiation from the discharge axis, while radiation at a distance of 15 to 20 km from the axis continues. Therefore, we have shown the toroidal shape of the field and the radiation regions in the upper part of the discharge. We thank E. A. Mareev for discussion of the manuscript of the article and S. O. Dement’eva for useful remarks. This work was supported by the Russian Science Foundation (project No. 16–17–00132). REFERENCES

1. C. L. Kuo, A. B. Chen, J. K. Chou, et al., J. Phys. O, 41, No. 23, 234014 (2008). 2. H. C. Stenbaek-Nielsen, D. R. Moudry, E. M. Wescott, et al., Geophys. Res. Lett., 3829 (2000).

27, No. 23,

3. H. C. Stenbaek-Nielsen and M. G. McHarg, J. Phys. O, 41, No. 23, 234009 (2008). 4. D. D. Sentman, H. C. Stenbaek-Nielsen, M. G. McHarg, and J. S. Morrill, J. Geophys. Res., 113, No. D11, D11112 (2008). 970

5. F. J. Gordillo-Vazquez, J. Phys. D. Appl. Phys., 41, No. 23, 234016 (2008). 6. E. Arnone, A. K. Smith, C.-F. Enell, et al., J. Geophys. Res. Atmospheres, 119, No. 11, 6958 (2014). 7. A. Luque and F. J. Gordillo-Vazquez, Nature Geosci., 5, No. 1, 22 (2012). 8. H. Winkler and J. Notholt, Atmos. Chem. Phys., 14, No. 7, 3545 (2014). 9. A. A. Evtushenko and F. A. Kuterin, Radiophys. Quantum Electron., 56, No. 11–12, 853 (2014). 10. A. A. Evtushenko, F. A. Kuterin, and E. A. Mareev, J. Atmos. Sol.-Terr. Phys., 102, 298 (2013). 11. N. Liu, J. Geophys. Res., 117, No. A3, A03308 (2012). 12. I. A. Kossyi, A. Yu. Kostinsky, A. A. Matveev, and V. P. Silakov, Trudy IOFAN, 47, 37 (1994). 13. A. Mitra, Influence of the Solar Flares on the Earth’s Ionosphere [Russian translation], Mir, Moscow (1974). 14. http://www.cesm.ucar.edu/models/cesm1.2/cam/ . 15. S. A. Yashunin, E. A. Mareev, and V. A. Rakov, J. Geophys. Res., 112, No. D10, D10109 (2007). 16. E. A. Mareev and S. A. Yashunin, Izv. Atmos. Oceanic Phys., 46, No. 1, 69 (2010). 17. E. A. Gerken, J. Geophys. Res., 107, No. A11, 1344 (2002). 18. E. M. Wescott, H. C. Stenbaek-Nielsen, D. D. Sentman, et al., J. Geophys. Res., 106, No. A6, 10467 (2001). 19. T. Kanmae, H. C. Stenbaek-Nielsen, M. G. McHarg, and R. K. Haaland, Geophys. Res. Lett., 37, No. 13, L13808 (2010). 20. T. Adachi, H. Fukunishi, Y. Takahashi, et al., Geophys. Res. Lett., 33, No. 17, L17803 (2006).

971