INTRODUCTION. As shown in Figure 1a, the phenomena of lightningmesosphere-ionosphere

ABSTRACT Intense, transient quasi-electrostatic (QE) fields, which exist above the thunderclouds following a positive cloud-to-ground lightning discharge, can produce an upward travelling runaway electron (REL) beam. In the present work we calculate the REL density using the recently developed electrostatic heating (ESH) [Pasko et al., 1996] and QE models [Pasko et al., 1995]. The new two-dimensional REL-QE model expands the previously reported one-dimensional model [Bell et al., 1995], provides information on the lateral electron distribution in the beam and allows us to determine the ionospheric effects and the optical luminosities resulting from the simultaneous action of the QE fields on the ambient electrons and the runaway electrons. The REL beam radius changes with altitude, attaining its maximum value of -15 km at altitudes of - 90 km. The model includes the effects of the divergence of the beam on the electron number density. Optical emissions [Bell et al., 1995] and γ-ray emissions [Lehtinen et al., 1996] are calculated and compared to experimental observations. 1

INTRODUCTION As shown in Figure 1a, the phenomena of lightningmesosphere-ionosphere interactions include: Red Sprites Blue Jets Elves Terrestrial γ-ray bursts 1b: The mechanisms thought to cause them are shown in Figure heating by electromagnetic pulse (EMP, [Taranenko et al., 1993; Glukhov and Inan, 1996]), heating by quasi-electrostatic thundercloud field (QE, [Pasko et al., 1995]), runaway electrons driven by QE fields (REL, [Bell et al., 1995]). 2

The Sprite type of upward discharge is most commonly observed with sharp features and vertical structure, with response primarily in the red region of the spectrum, exhibiting short ( 1 ms) duration, and generally detached from the cloud-tops but extending to as high as 95 km altitude [Sentman et al., 1995]. Recently Pasko et al. [1995; hereafter referred to as I] proposed that Sprites are produced by the heating of mesospheric electrons by large quasi-electrostatic (QE) thundercloud fields. These QE fields appear following a positive CG lightning discharge, and their duration at any altitude is controlled by the local relaxation time of the E field [I]. This mechanism accounts for many of the observed aspects of these discharges but does not explain emissions below km which some Sprites exhibit. In the present work, we explore the possibility that high energy ( kev- MeV) runaway electrons (REL) may be involved in producing the low altitude portion of the optical output associated with Red Sprites (Figure 1). We examine this question within the context of the REL-QE model. It has been shown [Roussel-Dupre et al., 1994 and references therein] that secondary cosmic ray electrons of energy 1 MeV can become REL whenever the local E field exceeds the threshold field E t = 4πN mze 3 a mc 2, where N m is the number density of air molecules, Z 14.5, a 11, m and e are the 3

electron rest mass and charge, respectively, and c is the velocity of light. Beside optical emissions, REL may also be responsible for γ- ray emissions [Lehtinen et al., 1996] detected by the Burst and Transient Source Experiment (BATSE) detectors, located on the Compton Gamma Ray Observatory (CGRO), which was described by Fishman et al. [1994]. The experiment makes use of four γ-photon energy channels: - kev, -0 kev, 0-0 kev and > 0 kev. Since these bursts are associated with thunderstorm centers and in a few cases have been correlated with individual lightning flashes [Inan et al., 1996], they may be caused by the runaway electrons. 4

GENERAL REL-QE MODEL DESCRIPTION The initial thundercloud charge consists of ± Q at altitudes of km and 5 km respectively. The separated dipole charges are assumed to form over a large time (>0 s). The induced static charge distribution is then calculated using the electrostatic heating (ESH) model [Pasko et al., 1996]. Subsequently the positive half of the dipole charge is discharged to ground with a time constant τ s = 1ms. When the positive charge is discharged to ground, an E field appears at all altitudes well above the shielding space charge. Physically, the removal of +Q is equivalent to placing Q at the same point in space. The dynamics of the charge distribution is calculated using the quasi-electrostatic (QE) model [Pasko et al,, 1995], but taking into account the REL field and the change of conductivity associated with the slow secondaries produced by them. The effects of the geomagnetic field ( 5 5 T, assumed to be vertical in the model to preserve cylindrical symmetry) on the electron component of the conductivity of the atmosphere is included. 5

RUNAWAY ELECTRONS (REL) The transport equation for REL can be written as: N R t +div( v R N R )= N R τ i + S o (z) where N R is the number density of the REL, v R c is their mean velocity, τ i is the ionization time constant, S o is the local source function for energetic cosmic ray secondary electrons. The REL velocity direction is calculated using the equation of motion of the electron, which includes action of electric and geomagnetic field and the dynamic friction force [Roussel- Dupré et al., 1994]. Figure 2 shows that REL move along magnetic field lines above km, where the collision frequency is small compared to the electron gyrofrequency, and along the electric field lines at heights km, where the collision frequency dominates. REL are produced by impact ionization of neutrals, and the resulting ions are left behind as positive space charge. The transport equation for these ions is: N is t = N R τ i + S o (z) where N is is the number density of the space charge ions. It can be shown that the effects of REL diffusion in velocity 6

space due to ionizing collisions and collisions with nuclei do not lead to significant widening of the beam in our model. The REL also produce low energy secondary electrons which do not become runaways but instead thermalize through collisions with the neutrals. 7

OPTICAL EMISSIONS The flux of low-energy secondary electrons will excite the neutral species through impact excitation, leading to optical emissions. The intensity of each optical line in Rayleighs is given by the expression [Chamberlain, 1978]: I k = 6 A k n k dz, where n k is the number density of excited particles in state k, A k is the radiation transition rate, and the integral is taken along the line of sight. The quantity n k is governed by the relation [Sipler and Biondi, 1972]: n k t = n k τ k + m n m A m + R k where τ k is the total lifetime of state k, and the sum over the terms n m A m represents increases in n k resulting from cascading from higher energy states, R k is the excitation rate of a given energy state k and is calculated on the basis of the number density and distribution over energies of low-energy secondary electrons. 8

γ-ray EMISSION (BREMSSTRAHLUNG) γ-ray production is characterized by the doubly differential cross-section for bremsstrahlung [Heitler, 1954] the crosssection of the bremsstrahlung process into a unit solid angle and a unit photon energy interval in the given direction Ω 0 and at the given photon energy k) can be written: 2 χ = Φ(Ω k Ω 0 ) χ 0 k, where Φ(Ω 0 ) is the angular distribution of emission. Φ(Ω 0 ) is forward-directed for relativistic electrons. For nitrogen and oxygen and for electron initial and final kinetic energies 1 MeV we can use the Born approximation to calculate χ/ k. The number of photons radiated by a unit volume of atmosphere per steradian per unit photon energy interval per second (specific emissivity) is: ɛ( r,k,ω 0 )= de 0 dω e f e ( r,e 0, Ω e )v 0 Φ(Ω 0 ) {2n N2 χ N k +2n O 2 χ O k }. In the 1-d model ([Lehtinen et al.]) it was shown that for the altitudes where most of the γ-rays are produced, the attenuation of radiation is small ( 1 %). 9

RESULTS AND COMPARISON WITH DATA Figure 4 shows REL density as a function of time. The slow decrease of their density after the discharge is due to slow relaxation of the electric field at the altitudes where the REL avalanche takes place. The calculations show that the REL cause optical emission of the same order as emission by the QE heating without taking REL into account. It occurs at lower altitudes ( km). Figures 5,6 present the intensity of the N 2 First Positive band (B 3 Π g A 3 Σ + u ). In the Figure 5, the earlier and wider part of the emissions at altitutes 90 km is also occurring without including REL into the model and is due to the QE heating. The later and narrower part, occurring at altitudes km, is caused by the change in atmosphere conductivity, which in turn is due to ionization by REL. For comparison, Figure 7 presents optical emissions for a smaller discharge (Q = 1C,τ s = 1ms), when the effects of REL are negligible. Figure 8 shows the predicted γ-ray fluxes at height 0 km as a function of horizontal range R s from the beam location. The results are consistent with 1-d model predictions [Lehtinen et al., 1996] and BATSE observations [Fishman at al., 1994]. Also, a significant fraction of the γ-ray bursts presented by Fishman et al. [1994] have the duration of 3-5 ms, which also agrees with present calculations.

SUMMARY AND DISCUSSION The 2-d model predicts the radius of the REL beam of the order 15 km, (see Figure 3) which is determined mainly by the structure of the post-discharge quasi-electrostatic field. The divergence of the REL beam, as can be seen from Figure 2, can play role only at altitudes < km. The radial diffusion of electrons due to scattering is found to be negligible. As shown in Figure 6a, the optical emissions due to REL tend to peak in the 55- km altitude range, and substantial intensity can be found at altitudes as low as 45 km. Thus the predictions of the REL model fit the observations reasonably well. In comparison with other mechanisms, we note that the predicted optical emissions for QE heating [I] and EMP heating [Taranenko et al., 1993; Glukhov and Inan, 1996] peak at -95 km altitude and have no significant intensity below km. We have calculated the bremsstrahlung γ-ray fluxes that would be associated with a beam of 1 MeV electrons, accelerated in an avalanche process by quasi-static thundercloud fields. The calculated γ-ray fluxes are comparable to those measured by the BATSE detectors on the Compton Gamma-Ray observatory. Major features of the BATSE observations can be explained by the runaway MeV electron model. 11

REFERENCES Bell, T. F., V. P. Pasko, and U. S. Inan, Runaway electrons as a source of red sprites in the mesosphere, Geophys. Res. Lett., 22, 2127, 1995. Chamberlain, J. W., Physics of the aurora and airglow, Academic Press, New York, 1961. Fishman G. J., P. N. Bhat, R. Malozzi, J. M. Horack, T. Koshut, C. Kouveliotou, G. N. Pendleton, C. A. Meegan, R. B. Wilson, W. S. Paciesas, S. J. Goodman, H. J. Christian, Discovery of intense gamma-ray flashes of atmospheric origin, Science, 264, 1313, 1994. Glukhov, V. S. and U. S. Inan, Particle simulation of the time-dependent interaction with the ionosphere of rapidly varying lightning EMP, Geophys. Res. Lett., 23, 2193, 1996. Heitler, W., The Quantum theory of radiation, 3rd ed., Clarendon, Oxford, 1954. Inan, U. S., S. C. Reising, G. J. Fishman and J. M. Horack, On the association of terrestrial gamma-ray bursts with lightning and implication for sprites, Geophys. Res. Lett., 23, 17, 1996. Lehtinen, N. G., M. Walt, U. S. Inan, T. F. Bell and V. P. Pasko, γ-ray emission produced by a relativistic beam of runaway electrons accelerated by quasi-electrostatic thundercloud fields, Geophys. Res. Lett., 23, 2645, 1996. Pasko, V. P., U. S. Inan, Y. N. Taranenko and T. F. Bell, Heating, ionization and upward discharges in the mesosphere due to intense 12

quasi-electrostatic thundercloud fields, Geophys. Res. Lett., 22, 365, 1995. Pasko, V. P., U. S. Inan and T. F. Bell, Ionospheric Effects due to Electrostatic Thundercloud Fields, to be submitted to JATP, 1996. Roussel-Dupré, R. A., A. V. Gurevich, T. Tunnel and G. M. Milikh, Kinetic theory of runaway breakdown, Phys. Rev., 49, 2257, 1994. Roussel-Dupré, R. A. and A.V. Gurevich, On runaway breakdown and upward propagating discharges, J. Geophys. Res., 1, 2297, 1996. Sentman, D. D., E. M. Wescott, D. L. Osborne, D. L. Hampton, M. J. Heavner, Preliminary results from the Sprites94 campaign: Red Sprites, Geophys. Res. Lett., 22, 15, 1995. Sipler, D. P., and M. A. Biondi, Measurements of O( 1 D) quenching rates in the F region, J. Geophys. Res., 77, 62, 1972. Taranenko, Y. N., U. S. Inan and T. F. Bell, The interaction with the lower ionosphere of electromagnetic pulses form lightning: excitation of optical emissions, Geophys. Res. Lett.,, 2675, 1993. Taranenko, Y., and R. Roussel-Dupré, High altitude discharges and gamma-ray flashes: a manifestation of runaway air breakdown, Geophys. Res. Lett., 23, 571, 1996. 13

Fig. 1a. Lightning-mesosphere interaction phenomena (not to scale). Elves ~90 km γ-rays Sprites Blue Jet + + + - - ++ +CG -- - Cameras

Fig. 1b. Mechanisms of lightning-mesosphere interactions (not to scale). Ionization EMP heating QE heating ~90 km γ-rays Runaway electrons EMP and QE Fields + + + - - ++ +CG -- - Cameras

Fig. 2. Electric field lines (black) and REL velocity lines (red): (a) vertical geomagnetic field (our model); (b) geomagnetic field tilted at 45ßto the vertical. (maximum REL density) Q=225 C; t=1.2 ms Q=225 C; t=1.2 ms 0 - - 0 (a) 0 - - 0 (b)

Fig. 3. REL density at 0.9ms and 1.2ms after the onset of the discharge of Q=225 C: 90 Q=225 C; t=0.9005ms: Nr, 1/m^3 (a) 2D plots > 5 4-3 (m ) 90 (maximum) Q=225 C; t=1.2ms: Nr, 1/m^3 3 2-0 < 1 0-0

(b) on the axis (r=0) Q=225 C, t=0.9005ms Q=225 C, t=1.2ms 0 - -5 0 5 Nr, m^(-3) 0 - -5 0 5 Nr, m^(-3)

Fig 4. Time dependence of REL density 5 225 C: Maximum REL density -2 225 C: Total REL charge (abs. value) -4 Nr, m^(-3) 0 Nr, m^(-3) -6-8 - -5 0 0.5 1 1.5 2 2.5 3 t, ms -12 0 0.5 1 1.5 2 2.5 3 t, ms

Fig. 5. 2D plots of the intensity of First Positive N band (in Rayleighs): 2 (a) effect of ionization by REL included into QE emissions; (b) REL emission only (no QE); (c) no REL (QE only).

t=0.5 ms after the onset (a) (b) (c) Q=225 C; t=0.03ms: 1st pos N_2 (R) Q=225 C; t=0.03ms: 1st pos N_2 (R) Q=225 C; t=0.03ms: 1st pos N_2 (R) 8 90 90 90 > - 0-0 - 0 < 7 6 5 4 3 2 (R)

t=1.2 ms after the onset (a) (b) (c) 90 Q=225 C; t=1.2ms: 1st pos N_2 (R) Q=225 C; t=1.2ms: 1st pos N_2 (R) Q=225 C; t=1.2ms: 1st pos N_2 (R) 8 90 90 > (R) 7 6 5 4-0 - 0-0 < 3 2

t=2 ms after the onset (a) (b) (c) 90 Q=225 C; t=2ms: 1st pos N_2 (R) Q=225 C; t=2ms: 1st pos N_2 (R) Q=225 C; t=2ms: 1st pos N_2 (R) 8 90 90 > (R) 7 6 5 4-0 - 0-0 < 3 2

t=2.9 ms after the onset (a) (b) (c) 90 Q=225 C; t=2.9ms: 1st pos N_2 (R) Q=225 C; t=2.9ms: 1st pos N_2 (R) Q=225 C; t=2.9ms: 1st pos N_2 (R) 8 90 90 > (R) 7 6 5 4-0 - 0-0 < 3 2

Fig. 6. Intensity of First Positive N band (in Rayleighs): (a) on the axis at different moments in time; (b) maximum intensity. 2 0 Q=225 C 9 Q=225: Maximum intensity of 1st pos N_2 t=2.9ms t=2.0ms t=1.2ms t=0.5ms (b) (a) I, R 8 QE heating dominates REL dominate 0-5 0 5 1st pos N_2, R 7 0 0.5 1 1.5 2 2.5 3 time, ms

Fig. 7. Plots of the intensity of First Positive N band (in Rayleighs) 2 for a smaller discharge (Q=1 C) Q=1 C; t=0.05ms: 1st pos N_2 (R) 90 > 8 (R) Q=1 C; t=1.2ms: 1st pos N_2 (R) 90 7 6 5 4-0 < 3 2-0

90 Q=1 C; t=2ms: 1st pos N_2 (R) > 8 7 (R) 8 Q=1: Maximum intensity of 1st pos N_2 7-0 < 6 5 4 3 2 I, R 6 5 4 3 2 0 0.5 1 1.5 2 2.5 3 time, ms

Fig. 8. REL γ-ray emission fluxes at the satellite height, integrated over BATSE energy ranges: (a) dependence on horizontal range; (b) flux exactly above the source. flux, photons/(sec-m 2 ) 7 6 5 (maximum) Q=225 C; t=1.2 ms measurements for > 0 kev [Fishman et al, 1994] 4 - kev -0 kev 0-0 kev (a) > 0 kev 3 0 0 0 0 R s, km 0 0 0 flux, photons/(sec-m 2 ) 2.5 2 1.5 1 0.5 3 x 4 Q=225 C, photons > 0 kev (b) 0 0 0.5 1 1.5 2 2.5 3 time, ms