Conditions for Inception of Relativistic Runaway Discharges in Air

Terrestrial gamma‐ray flashes are linked to growth of long bidirectional lightning leader system consisting of positive and stepping negative leaders. The spatial extent of streamer zones of a typical lightning leader with tip potential exceeding several tens of megavolts is on the order of 10–100 m. The photoelectric absorption of bremsstrahlung radiation generated by avalanching relativistic runaway electrons occurs efficiently on the same spatial scales. The intense multiplication of these electrons is triggered when the size of the negative leader streamer zone crosses a threshold of approximately 100 m (for sea‐level air pressure conditions) allowing self‐replication of these avalanches due to the upstream relativistic electron seeds generated by the photoelectric absorption. The model results also highlight importance of electrode effects in interpretation of X‐ray emissions from centimeter to meter long laboratory discharges, in particular, a similar feedback effect produced by generation of runaway electrons from the cathode material.

2 of 11 demonstrated that the relativistic feedback mechanism leads to amplification of the relativistic runaway avalanches allowing to explain observed features of TGFs, including their intensities and pulsing (Dwyer, 2003(Dwyer, , 2012Liu & Dwyer, 2013).
The existing evidence indicates that TGFs are closely associated with intra-cloud leader development and, in particular, with steps of negative lightning leaders (Heumesser et al., 2021;Kohn et al., 2020;Lu et al., 2010, and references cited therein). The processes leading to electric field enhancement around leader tip are therefore likely to play a role in TGF production. A streamer zone of a high potential (∼100 MV) lightning leader tip would not exceed several tens of meters (referring to values at sea-level air pressure), and it is reasonable to hypothesize that TGFs should originate from these relatively confined regions of space.
Photoionization of molecular oxygen due to extreme ultraviolet emissions from molecular nitrogen, N 2 , in the wavelength range 98-102.5 nm (Zheleznyak et al., 1982) is a fundamental process in positive corona discharges in air where electrons avalanche toward a sharp point that focuses electric field lines leading to significant field enhancements and electron multiplication due to electron impact ionization (Naidis, 1987). In a positive corona system the avalanche of electrons in bulk of discharge volume is initiated by specific distribution of photoionization far away from region of maximum electron density near the electrode where those photons are emitted (Naidis, 2005). There is a direct analogy between this photoionization feedback that occurs in the positive corona discharges in air and the relativistic feedback mechanism proposed by Dwyer (2003). The conditions for inception of relativistic runaway discharges in air, however, have not been defined using the same methodology as in the present literature on conventional positive corona discharges in air (Benilov et al., 2021;Naidis, 2005, and references therein). The definition of these conditions at small spatial scales ≤10-100 m (referring to dimensions of lightning leader streamer zones at sea-level pressure in air) represents a goal of this work.

Model Formulation
We note that the relativistic feedback problem is simpler than that of conventional positive corona in air in a sense that it does not contain non-similarity behavior introduced by the quenching of singlet states of molecular nitrogen responsible for photoionization in air (e.g., Naidis, 2005). In Pasko et al. (2022) we developed stationary and time dependent positive corona models in planar and spherical geometry and repeated previously published results on inception of positive corona discharges in air (Naidis, 2005). In this work we extend this modeling to the relativistic feedback problem. Following the convention in the corona literature we use a scaling factor δ = n/n 0 , where n is air number density and n 0 = 2.688 × 10 25 m −3 is a reference value corresponding to standard atmospheric conditions at sea-level in the Earth's atmosphere. We express results in terms of the reduced applied electric field E 0 /δ required for the inception of self-sustained relativistic runaway discharges in air versus the reduced physical dimension of space over which this electric field is applied, that is, dδ, where d denotes the gap size.
The geometry of this one dimensional coordinate dependent model is depicted in Figure 1. The relativistic runaway electrons avalanche in discharge volume and emit the bremsstrahlung X-rays due to their interactions with air and anode material. The presence of anode and cathode electrodes is optional. The flux of electrons bombarding the anode is schematically shown as Γ ea . The X-rays are attenuated due to the photoelectric absorption, Compton scattering, and pair production. The secondary relativistic runaway electrons are produced due to photoelectric absorption of X-rays in the volume of air and from the cathode (both processes are schematically depicted by Γ es fluxes in Figure 1). In this process the photon energy is completely transformed into the electron energy, and the angular distribution of electrons is determined by the corresponding differential scattering cross section (e.g., Carron, 2007, p. 44). Under the self-sustained steady state conditions, the photoelectric feedback produces just enough secondary runaway electrons upstream of the avalanche to replicate constant Γ ea . The processes are analogical to the photoionization feedback in positive corona where each electron arriving at the anode creates on average just enough seed electrons in discharge volume through the photoionization to replicate itself (Naidis, 2005). Under these steady state conditions in the corona and in the relativistic feedback system one can introduce an effective surface anywhere in discharge volume (shown by vertical dashed line in ) and express the relationship between ion flux at that location Γ ic and Γ es as Γ es = γ se Γ ic , where γ se is the effective secondary electron emission coefficient due to volume photoionization in the positive corona system (Kaptzov, 1950, p. 610) or due to the photoelectric effect in the relativistic electron feedback system. The relation Γ es = γ se Γ ic provides a physically transparent connection to secondary electron emission from cathode surface due to the ion bombardment in conventional Townsend discharge theory. The ion flux Γ ic and the effective coefficient γ se , however, are generally not needed for solution of the corona (Naidis, 2005) or the relativistic feedback problems. Figure 2a depicts the friction force acting on electrons in air against the accelerating force from the externally applied field. The format of this figure is adopted from (Moss et al., 2006, please also refer to discussion and references therein). The force has units of eV/m and can be directly compared to the applied electric field E a to provide an intuitively simple insight into the expected motion of electrons at various energies ɛ. Figure 2a also provides a summary of various electric field thresholds that are important for description of gas discharge phenomena discussed in this paper: the thermal runaway threshold E c /δ ≃ 260 kV/cm (Gurevich, 1961), the conventional breakdown field E k /δ = 28.7 kV/cm defined by the equality of the ionization and dissociative attachment coefficients in air following models of these processes proposed by Morrow and Lowke (1997), and the minimum electric field required for development of relativistic runaway electron avalanche E t /δ = 2.76 kV/cm (see Coleman & Dwyer, 2006;Dwyer, 2003;Dwyer et al., 2012;Gurevich et al., 1992, and discussion therein). If electrons possess significant energy above ∼100 eV corresponding to the E c peak, they can continue to gain 4 of 11 energy (runaway) in relatively low applied fields E t < E a < E c . It had been discovered by Gurevich et al. (1992) that a fraction of secondary electrons produced by these runaways also can become runaways themselves leading to an avalanche multiplication of these electrons. The characteristic length quantifying this exponential growth is shown in Figure 2d (Dwyer et al., 2012, and references therein). Figure 2a also includes the electric fields in streamer zones of positive + cr ∕ = 4.4 kV/cm and negative − cr ∕ = 12.5 kV/cm lightning leaders. There are some variations around these numbers in the existing literature, with + cr typically quoted to be a factor of 2-3 lower than − cr . For consistency of numerical estimates we adopt the listed values, which were also used in previous publications (Celestin & Pasko, 2011;Moss et al., 2006). For an example of applied electric field E a = − cr shown in Figure 2a electrons with initial energy ɛ < ɛ run (E a ) ∼ 20 keV will decelerate, and those with ɛ > ɛ run (E a ) will gain more energy from electric field than they lose in collisions and will become runaway electrons.

Runaway Electrons in Air
The relativistic runaway electrons with number density n r and with normalized to unity energy distribution func- Figure S1a in Supporting Information S1) (where 0 = 7.3 × 10 6 eV) , and references therein) avalanche against the constant applied electric field with magnitude E a , and their avalanche multiplication is characterized by length shown in Figure 2d, r = 1 , and references therein), where t 0 = 2.76 × 10 5 V/m corresponds to the sea-level pressure E t value shown in Figure 2a. The deflection of electrons by nuclei of oxygen and nitrogen atoms in air leads to braking (bremsstrahlung) radiation characterized by the energy spectrum shown in Figure 2b. Due to the 1/ɛ γ dependence of the related cross section, the distribution is characterized by orders of magnitude higher number of photons at low energies 1-100 keV than around the 0 threshold. These photons are attenuated as they propagate in air due to photoelectric absorption, Compton scattering, and pair production. The corresponding attenuation lengths are shown in Figure 2c. For the spatial scales lδ ≤ 100 m of interest in present work the attenuation is dominated by photoelectric absorption and Compton scattering. At energies above ∼1 keV the electron emerging as a result of photoelectric absorption of a photon with energy ɛ γ carries essentially the same energy as photon energy (minus a small binding energy increment), that is, ɛ ≃ ɛ γ . The photoelectric absorption is the primary process that can effectively produce runaway electrons on spatial scales of interest in this work and in a given electric field E a as soon as the condition ɛ > ɛ run (E a ) is satisfied (see Figure 2a).
The relativistic runaway electrons are described by linear equation (n r ∝ S pe ): where the ionization coefficient α r = −1 r [m −1 ], c is speed of light in free space and the source term S pe describes production of runaway electrons due to the photoelectric effect. For a case when S pe = S 0 with S 0 being a constant background rate of production of runaway electrons, the equation has a one-dimensional (coordinate dependent) solution: r ( ) = 0 r ( r − 1 ) . For ≪ l r , r ( ) ≃ 0 and for ≫ l r , r ( ) = r 0 r , where r 0 = 0 r . This emphasizes that runaway electrons arriving at the end of model region after many multiplications (i.e., at the anode) are mostly defined by their seeds in a relatively narrow region of space with size l r . We also note that similarly to positive corona system (Naidis, 2005), the seed runaway electrons due to the photoelectric effect are mostly produced by the region of maximum density of runaway electrons at the end of the multiplication region. The solution of the photoelectric feedback problem requires finding a specific value of applied electric field E a = E 0 and related self-consistent distribution of S pe ( ) that provides full self-replication of runaway electrons. The constant background rate S 0 is generally not needed for the solution of the feedback problem, however, it can represent initial runaway seeds in the form of cosmic ray secondaries with typical values at lightning initiation altitudes being S 0 = 10 m −3 s −1 (McCarthy & Parks, 1992), or a production of runaway electrons in the regions populated by propagating streamers (Celestin & Pasko, 2011;Moss et al., 2006). This discussion also emphasizes that even under conditions when E a < E 0 , if S 0 > 0 there is a steady state solution and a significant exponential multiplication of runaway electrons.

Photoelectric Runaway Electrons Due To X-Rays
The X-ray generation by runaway electrons in air is characterized by doubly differential cross-section of the bremsstrahlung photon production 2 br Ω , where the angular dependence is the one for the emitted photon PASKO ET AL.
10.1029/2022GL102710 5 of 11 (Heitler, 1954, p. 245;Lehtinen, 2000, p. 45, and references therein). The X-ray emission frequency per unit energy per one runaway electron [1/eV/s] (Figure 2b) is: where (ɛ) = √ 1 − (1 + ∕( 2 )) −2 , m is electron rest mass, and br = ∫ Ω 2 Ω Ω . The total emission frequency of bremsstrahlung photons per one runaway electron [1/s] is = ∫ ( ) , where integration is performed over the energy interval 1 keV to 100 MeV shown in Figure 2b ( ∕ = 1.67 × 10 7 s −1 ). In the following we refer to these X-rays as air X-rays. As will be emphasized further the bremsstrahlung photon production scales quadratically with atomic number Z and in our representative calculations we will include tungsten (W) and aluminum (Al) as representative high and low Z materials (Z N = 7 and Z O = 8 in air, Z Al = 13, and Z W = 73).
The number of runaway electrons produced per unit volume per second due to photoelectric effect in air due to air X-rays: | is the distance between point of source ′ and point of observation , k t = n(σ pe + σ c + σ pp ) is the total absorption coefficient including the photoelectric absorption, Compton scattering, and pair production in air, with respective cross sections σ pe , σ c , and σ pp , and the quantity ξ pe = 5 × 10 −3 denotes efficiency of production of runaway electrons accounting for the differential cross sections of the bremsstrahlung radiation and the production of electrons due to the photoelectric absorption (see Discussion section). The structure of Equation 3 accounts for volumetric emission source of photons, geometrical spreading and attenuation of photons as they propagate, and their absorption by target molecules, similarly to a formulation used for photoionization problems (e.g., Janalizadeh & Pasko, 2019, and references therein). The effects of bremsstrahlung induced photoionization on the propagation of positive streamers in different N 2 /O 2 mixtures have been discussed by Kohn et al. (2017). Figure 2c illustrates the characteristic absorption lengths of photoelectric absorption l pe = 1/(nσ pe ), Compton scattering l c = 1/(nσ c ), pair production l pp = 1/(nσ pp ), and total absorption l t = 1/k t . For comparison Figure 2c also shows the electron range l e over the same energy range. The photon cross sectional data are taken from http://www.nist.gov/pml/data/xcom. The electron range data are from https://physics.nist.gov/PhysRefData/Star/ Text/ESTAR.html. We solve Equation 3 by replacing it with an equivalent set of Helmholtz equations (Bourdon et al., 2007;Janalizadeh & Pasko, 2019) (with details of numerical implementation provided in Supplementary Information).
The photoelectric runaway electrons in air are also produced due to X-rays emitted by the anode upon bombardment by runaway electrons anode-X pe ( ⃗ ) . We consider aluminum (Al) and tungsten (W) as two representative electrode materials. The total number of runaway electrons produced per unit volume per unit time in air due to the photoelectric absorption is then pe . The runaway electrons are also emitted from the cathode due to the air and anode X-rays. Both Al and W materials have a factor of 10 3 higher mass densities as compared to air at sea-level pressure. The bremsstrahlung photon production from tungsten is enhanced by a factor of 10 2 due to Z 2 scaling of the related cross section. In comparison to air tungsten also has a factor of 10 4 higher cross section for photoelectric absorption. The calculations follow the integral approach defined in Equation 3 by replacing the X-ray source in air γ (ɛ γ ) with corresponding quantities defined for aluminum and tungsten Al,W ( ) , and the photoelectric absorption cross section in air σ pe (ɛ γ ) with corresponding Al and W cross sections

Results
The differential Equation 1 and integral Equation 3 are self-consistently solved for n r and S pe , respectively. The threshold electric field E 0 (E t < E 0 < E c ) required for inception of relativistic runaway discharges for different gap dimensions d is obtained by an iterative process. The solution is started by setting an arbitrary initial value of E 0 , initiating a primary relativistic runaway electron avalanche with initial density n rp (d) = 1 m −3 at the right boundary of the simulation domain at = d (Figure 1), and assuming S pe = 0 everywhere in the simulation domain. The 6 of 11 resultant primary relativistic electron distribution n rp produces photoelectric source of runaway electrons S pep that is used to find the next iteration n rs (with n rp (d) = 0). The density n rs produces S pes that gives next iteration n rss . The final E 0 is obtained by simple iterations (by increasing or decreasing E 0 ) until the volume integrated n rs and n rss agree. Specific spatial distributions are illustrated in Figure S3 in Supporting Information S1. Due to the linear dependence of n r on S pe and vice versa, the results do not depend on the specific value n rp (d) = 1 m −3 . The solutions identical to those shown in Figure 3 are also produced by assuming n rp (d) = 0 and initiating the iterations with an arbitrary constant value of S pe = S 0 . Figure 3 shows the resultant E 0 as a function of d, in reduced form.
The no electrode effects solution shown in Figure 3 gives the size of the region and the electric field value in that region that is required for inception of relativistic runaway discharges. As discussed in the model formulation section for any d and for corresponding E a < E 0 there is a stationary (if S 0 ≠ 0), or a damped solution, depending on external sources of runaway electrons. For E a > E 0 the system can be initiated by a single runaway electron and leads to very fast multiplication of electrons developing on time scales on the order of d/c. In the conventional positive corona system the E a > E 0 regime is accompanied by growth of electron density that is normally bounded by outflow of plasma through chamber walls containing the discharge, or by space charge effects modifying the applied field E a (Benilov et al., 2021). The gap sizes and electric fields in Figure 3 provide sufficient potential drops for the runaway distribution with 0 = 7.3 × 10 6 eV cutoff to be valid. The softer distributions are expected at reduced gap sizes and potential differences (<100 MV) (Celestin et al., 2015).
Under thundercloud conditions there are three principal pathways by which the relativistic feedback discharges can operate: (a) The local electric field staying below the E 0 threshold over physical space with dimension d, with no lightning leaders developed yet, but with streamer activity possibly present due to hydrometeors and their collisions (Jansky & Pasko, 2020, and references therein); (b) the increase of local electric field over the E 0 threshold over physical space with dimension d, and similarly to the previous scenario, with no leaders but possibly with streamers present; (c) A bi-directional IC leader discharge propagating and creating extensive regions of quasi-constant electric fields − cr and + cr in respective negative and positive streamer zones. The first scenario is expected to be of the most common occurrence in a typical thundercloud environment and may lead to quasi-stationary solutions and long duration X-ray emissions discussed in the model formulation section. These emissions should cease immediately following the lightning discharge removing the charge separation regions that created the initial high field enhancements (Eack & Beasley, 2015;McCarthy & Parks, 1992, and references therein). The second scenario would likely result in an abrupt growth of plasma conductivity and space charge effects similar to those modeled in (Dwyer, 2012;Liu & Dwyer, 2013), but on smaller scales of 10-100 m driven by the photoelectric feedback and leading to lightning leader initiation Stolzenburg et al., 2013). Depending on number of seed electrons (i.e., produced by streamers (Celestin & Pasko, 2011)) the elementary time scale of this process is defined by several round trips with speed of light over dδ ≃ 10-100 m, i.e., (0.066-0.66)/δ μs in good agreement with 10 μs narrow bipolar events (NBEs) (Tilles et al., 2020).
For the third scenario, the branches of lightning leaders significantly modify the externally applied electric field as they focus electric field lines on their tips and therefore effectively screen (i.e., reduce) electric field in other regions of space. In the negative leader stepping process (during the negative corona flashes) the field of highly conducting negative leader tip is temporarily unshielded and can reach values significantly exceeding the conventional threshold E k (E k is included for the reference in Figure 3). For low potential leaders with a streamer zone size − st ≤ 10 m, the field near the negative tip can reach 1.5E k (Bazelyan & Raizer, 2000, p. 68). The analysis presented in (Celestin & Pasko, 2011, Figure 7, and discussion therein) indicates that with an increase of the leader potential and for − st ∼ δ100 m this field moves closer to E k . In both cases, however, this high field E ≥ E k only exists several centimeters from the leader tip and can not be sustained over dδ = 20-40 m required for the open air inception of the relativistic runaway discharge in accordance with Figure 3. For a leader tip potential U l the sizes of negative and positive streamer zones can be evaluated as − st = l∕(2 − cr ) and + st = l∕ ( 2 + cr ) , respectively (see Celestin & Pasko, 2011;Moss et al., 2006, and discussion therein). For a leader with U l = 10 MV the streamer zone has spatial extent − st = 4 m. For a high potential leader with U l = 300 MV that leads to TGFs (Mallios et al., 2013) − st = 120 m. At a representative 11 km TGF altitude δ ≃ 0.275 and − st = 436 m. The speed of light round trip over 436 m is ∼3 μs indicating that ∼50 μs energetic in-cloud pulses (EIPs) require several round trips before high-fluence TGFs are produced (Tilles et al., 2020). The smooth EIP sferics may represent the manifestation of relatively large volume of growing conductivity and current as opposed to the very high frequency (VHF) noisy streamers expected to coexist in the same physical space (Tilles et al., 2020). The 10-50 μs are broadly consistent with times required for runaway electrons producing streamers to traverse the above mentioned spatial scales. The quantitative modeling of related processes indicates that slow low-frequency (LF) pulses are likely generated directly by the TGF sources (Berge et al., 2022). It can be seen from Figure 3 that as soon as the streamer zone size reaches the threshold dδ ≃ 100 m for negative leader tip and dδ ≃ 1,000 m for positive leader tip the inception conditions for the relativistic runaway discharge will be satisfied. Since both positive and negative streamer zone sizes scale with field as 1/E and the required gap size for the relativistic feedback drops faster than 1/E 2 (see Figure 3), for any given leader with potential U l the conditions for relativistic runaway discharge will always be first satisfied in the negative streamer zone. We associate the situation when the negative streamer zone reaches this threshold with a condition when terrestrial gamma ray flash is produced by stepping lightning leaders. The occurrence of TGFs in association with positive leaders is not excluded but is rare (Hare et al., 2016).
Although representing a bi-directional leader as a long unbranched conductor is a reasonable approximation that provides insight into the leader fields observed experimentally (Pasko, 2014), the actual leaders exhibit significant branching with VHF dark positive leaders developing extensive networks before development of negative leader branches (Boggs et al., 2022;Mallios et al., 2013). The potential drop in each branch decreases after the branching (Celestin & Pasko, 2011). For branched leaders the length of the streamer zone would not necessarily satisfy the relativistic runaway condition during each step (i.e., the leader may continue stepping propagation without producing TGFs (Cummer et al., 2015)). The amplification of runaway avalanches even under conditions when E a < E 0 may lead to observable X-ray emissions from leader streamer zones (Moore et al., 2001), up to fluences approaching TGFs (Celestin et al., 2015).

Discussion
Figure 3 provides illustration of effects produced by model electrodes made of aluminum or tungsten (Figure 1). For simplicity for cases considered it is assumed that the same material is used for both cathode and anode. As noted in the model formulation section, the principal effects produced by electrodes include X-ray production from the anode upon bombardment by runaway electrons, and the production of relativistic runaway electrons from the cathode due to the photoelectric effect. Figure 3 indicates that aluminum (being a low atomic number material) does not significantly modify the open air solutions. The tungsten solutions significantly modify the E 0 8 of 11 threshold by lowering it to values slightly above and below the E k value for small gap sizes dδ ∼ 10 m. Consistent with discussion in the model formulation section, the seed runaway electrons released from the cathode play the dominant role in comparison with effects of X-rays emitted from the anode. Although these results are obtained for highly simplified planar geometry, they indicate a potential importance of the electrode material and configuration for interpretation of X-ray production from laboratory sparks (see discussion by da Silva et al. (2017), Parkevich et al. (2022), Stankevich and Kalinin (1967), and extensive list of references therein). The buildup of runaway electrons in the discharge volume may not be easily observed due to bright streamer, pilot and leader activity. However, related conductivity changes may be responsible for the observed correlation of X-ray fluxes and the cathode current pulses (Kochkin et al., 2015). It is remarkable that even in a case when no air volume X-rays and photoelectrons are produced (in the model these processes can be simply switched off), the relativistic electron runaway feedback with low E 0 threshold exists solely due to the production of relativistic electrons from the cathode due to X-rays emitted by the anode (please refer to the no air volume X-rays or photoelectrons solutions shown in Figure 3). In the present modeling we exclude effects related to Compton back scattering of X-rays from the cathode and also production of the characteristic (K α ) X-rays from the cathode and anode. The K absorption edge for tungsten, for example, is ∼58 keV. These photons emerging from the cathode in the same direction as the runaway avalanche can provide an efficient additional channel for seed runaway electrons through either photoelectric absorption, Compton scattering or pair production (we note that at ∼5 MeV pair production in tungsten is comparable to Compton scattering).
The Compton scattering and pair production in air are included in the present modeling as photon attenuation processes but not as processes leading to generation of feedback runaway electrons. As indicated by data shown in Figure 2c the Compton scattering interaction has a characteristic distance l c δ of approximately 100 m over the 10-100 keV range of energies, which are of primary interest in this work. As demonstrated above, in order for the feedback process to be effective, the seed runaway electrons should be produced in the relatively narrow region with characteristic length l r upstream from the avalanche. The values of l r δ shown in Figure 2b indicate that for the fields − cr and + cr of primary interest in this work, the 100 m Compton range l c δ appears to significantly exceed l r δ (i.e., l r δ ∼ 7 m at E a = − cr ). Furthermore, the kinematics of Compton scattering only produce electrons in the forward direction with respect to the incoming photon, and the backward moving bremsstrahlung photons produced by the relativistic runaway avalanche would preferentially produce electrons that would move against the primary runaway avalanche, decelerate, and would need to experience additional scattering interactions to become a part of the runaway population. The backward moving bremsstrahlung photons can Compton backscatter (θ = π) and depending on their incoming energy ɛ γ with respect to the electron rest energy mc 2 would possess significant energy ′ ≤ 2 ∕2 = 255 keV ( ′ = mc 2 /(2 + mc 2 /ɛ γ )). These photons with energy ′ can contribute to the runaway population by both photoelectric absorption and Compton scattering. However, due to the original Compton scattering event having low probability and long range l c δ ≥ 100 m (i.e., l c δ > l r δ), we consider these effects as not dominant producers of feedback runaway electrons. On the same physical grounds we exclude pair production, and for the considered reduced dδ scales of interest, that is, 10-100 m, retain only the photoelectric absorption as the dominant relativistic feedback factor.
Having considered l c δ and l pp δ in Figure 2c, the Compton scattering and pair production are important feedback factors for dδ values ∼1-10 km (Dwyer, 2012;Liu & Dwyer, 2013). Our model values of E 0 /δ at these scales are generally higher than those shown by Dwyer (2003), Figure 3 and Skeltved et al. (2014), Figure 6, and this could be attributed to the contribution of Compton scattering and pair production on these scales. In the dδ range of 10-100 m the open air onset values obtained in this work are a factor of 2 higher than those reported by Dwyer (2003), Figure 3 and Skeltved et al. (2014), Figure 6, and this can be attributed to these authors employing the simulation domain with 10 km vertical extent filled with air at sea-level pressure leading to reduction of range scales and therefore overestimation of frequencies of all physical interactions by up to a factor of 3. Additionally, for the 10 km, the speed of light round trip time is 66/δ μs leading to modeled TGF pulse durations on the order of 1 ms (Dwyer, 2012;Liu & Dwyer, 2013) significantly exceeding the observed ∼10 μs and ∼50 μs durations of NBEs and EIPs, respectively (Tilles et al., 2020), and slow LF pulses (Berge et al., 2022).
The number of relativistic runaway electrons created by the photoelectric absorption is controlled by two principal factors: (a) the differential cross sections of the bremsstrahlung photons and photoelectrons ( Figure S2 in Supporting Information S1); (b) the applied field E a dependent extent of energy interval (from ɛ run (E a ) to ∞) in which electron can gain energy and runaway (Figure 2a). For the first factor, the bremsstrahlung photons that facilitate feedback are launched backwards with respect to the avalanching runaway electrons. In order to PASKO ET AL.
10.1029/2022GL102710 9 of 11 contribute to the avalanche the runaway electron produced by these photons should be launched in the direction of the avalanche (i.e., in the range of angles π/2 < θ < π, where θ is measured from the photon direction). As evident from Figure S2a in Supporting Information S1, an isotropic approximation for the bremsstrahlung photons is a good approximation at low energies ≤100 keV, with the backscattering reduced at higher energies. The backscattering variation for the photoelectric electrons illustrated in Figure S2b in Supporting Information S1 is even sharper, indicating, in particular, over an order of magnitude reduction at θ = 3π/4 (135°) at 1 MeV with respect to the 10 keV value. We note that the lower runaway bound energies ɛ run (E a ) for the fields − cr and + cr of primary interest in this work reside in the 10-100 keV range (see Figure 2a) and the discussion above clearly indicates that both cross sections are highly anisotropic at these energies. To account for the reduction in the production runaway electrons due to these anisotropies we introduced in the model formulation an efficiency parameter ξ pe = 5 × 10 −3 . The model results appeared to be relatively insensitive to this parameter. In particular, at dδ = 100 m the increase to ξ pe = 5 × 10 −2 or reduction to ξ pe = 5 × 10 −4 led only to ∼13% variations in E 0 values shown in Figure 3, not affecting any principal conclusions of this work. At dδ = 1,000 m the same changes resulted in ∼4% variations. The second factor (the energy interval available for electron to runaway) practically involves the energies from ɛ run (E a ) to the 0 cutoff defining sharp drop off of γ (ɛ γ ) at high energies (Figure 2b). The effects related to this factor are accurately included in present modeling and depending on specific range R and air pressure p can exhibit up to eight orders of magnitude variations accounted by the propagator functions ( Figure S1b in Supporting Information S1).

Conclusions
The photoelectric absorption plays a dominant role in feedback amplification of the relativistic runaway electron avalanches on spatial scales ∼10-100 m in sea-level pressure air in the Earth's atmosphere. Using the same physical methodology as for conventional positive corona discharges in air, the conditions of inception of relativistic runaway discharges in air in terms of gap size, applied electric field and air pressure have been formulated. The results are relevant to understanding of physical conditions required for seeding of high conductivity regions in virgin air leading to lightning leader initiation, and understanding of intense fluxes of gamma rays produced by stepping leaders. The obtained model results indicate, in particular, that TGFs observed in association with stepping of negative lightning leaders can be explained by the spatial growth of the leader streamer zone, and a significant and source independent multiplication of relativistic runaway electrons occurs when a threshold of approximately 100 m (for standard air pressure conditions at sea-level) is reached. Below this threshold, the dominant seeding mechanism of runaway electrons is the thermal runaway electron production by streamers. A similar feedback amplification produced by the photoelectric absorption and generation of runaway electrons from the cathode material may be relevant to explanation of observed X-ray emissions from centimeter to meter long sparks under laboratory conditions.

Data Availability Statement
The original data presented in this paper as figures may be downloaded from the link https://doi.org/10.26208/5SGE-R820. by the Institut Universitaire de France (IUF). J Jansky gratefully acknowledges financial support from the grant VAROPS (DZRO FVT 3) granted by the Ministry of Defense of the Czech Republic. This study was conducted during a sabbatical visit of V P Pasko to the Plasma Physics Laboratory (LPP), Ecole Polytechnique, Palaiseau, France, and to the Laboratory of Physics and Chemistry of the Environment and Space (LPC2E), University of Orleans, France in the Fall of 2022. Hospitality of colleagues and staff at LPP and LPC2E is gratefully acknowledged.