High-Energy Emissions Induced by Air Density Fluctuations of Discharges

1 Introduction

Terrestrial gamma-ray flashes (TGFs) are photon bursts with energies of up to 40 MeV originating from thunderstorms. They were first observed in 1994 from the Compton Gamma-ray Observatory satellite (Fishman et al., 1994) and later confirmed by other space-based observations (Briggs et al., 2010; Marisaldi et al., 2010; Smith et al., 2005). X-rays have also been observed at closer proximity to the source with sensors on the ground and on balloons (Dwyer, 2004; Mallick et al., 2012; Moore et al., 2001) and in high-voltage laboratory discharge experiments of long sparks (Babich et al., 2015; Dwyer et al., 2005; Kochkin et al., 2014, 2016). The X-ray and γ-ray are produced by high-energy, runaway electrons through the bremsstrahlung process; however, the acceleration process of these electrons is still under debate.

There are currently two theories explaining the origin of TGFs. One is that seed electrons from cosmic ray ionization of the atmosphere are born with energies in the runaway regime and are further accelerated by the ambient electric field in a cloud, forming a relativistic runaway electron avalanche (RREA; Babich et al., 2012; Dwyer, 2003; Gurevich et al., 1992; Gurevich & Karashtin, 2013; Wilson, 1925) including the feedback mechanism where high-energy electrons produce high-energy gamma rays through the bremsstrahlung process which subsequently produces secondary electrons and positrons through photoionization, Compton scattering, or pair production (Dwyer, 2003, 2007; Kutsyk et al., 2011; Skeltved et al., 2014). The other is that thermal (cold) electrons are accelerated into the runaway regime in the high, but very localized, field of streamer tips as well as by the enhanced electric fields in the vicinity of lightning leader tips (Babich et al., 2015; Celestin & Pasko, 2011; Chanrion & Neubert, 2008; Köhn et al., 2014; Köhn & Ebert, 2015; Köhn, Diniz, Harakeh, 2017) and subsequently turn into relativistic RREAs (Carlson et al., 2010, 2006; Köhn, Diniz, Harakeh, 2017; Moss et al., 2006). In the following we explore the streamer mechanism.

Lightning leaders propagate by means of a multitude of streamers. Streamers are ionization waves formed when the avalanche of thermal electrons creates space charge fields of magnitudes that approach the levels of the background field (Raizer, 1991).

Past models of electron acceleration by streamers suggest that runaway electrons are created in the high-field region of the streamer tips (Babich et al., 2015; Celestin & Pasko, 2011; Chanrion & Neubert, 2010) and that these are further accelerated in the larger-scale leader field to flux levels that can account for TGF observations (Briggs et al., 2011; Dwyer et al., 2008). However, the environment of the leader tip is very complex and to a large extent unknown, and there are currently no self-consistent models that account for the leader-streamer interaction and propagation of the leader, at least not on a plasma kinetic level. Early discussions of density perturbations in connection with discharges suggest that, before the formation of a hot conductive leader channel, streamers heat the air and induce a radial flow of neutral air molecules, which reduces the air density in the streamer path by up to 50% (Marode et al., 1979). Similar conclusions were reached for positive streamers in a point-plane electrode geometry in the more recent simulations of Eichwald et al. (1998), Eichwald et al. (2011), Kacem et al. (2013), Liu and Zhang (2014), and Ono and Oda (2004). Tholin and Bourdon simulated the hydrodynamic air expansion from a nanosecond-pulsed spark discharge in a point-to-point gap of 2.5-mm length and for a voltage pulse peaked at approximately 5 kV, hence in a maximum ambient field of approximately 20 kV/cm plate-electrodes equivalent (Tholin & Bourdon, 2013). Under such conditions, they found that spark discharges initiate pressure waves potentially decreasing the air density by a factor of 50%. By coupling the fluid equations of discharge dynamics and the hydrodynamic equations for the air flow, Agnihotri et al. observed that ambient air heats up to approximately 800 K within tens of nanoseconds within a mean ambient field of 17 kV/cm. This heating process and the induced pressure waves are effective enough to initiate electrical breakdown without the streamer mechanism with locally enhanced electric field tips (Agnihotri et al., 2017). Beyond air perturbations induced by shock waves and heating processes, civil transport aircraft, high-speed air vehicles, or the wind flow around (sharp) objects (Corda, 2017; Fleming et al., 2001; Gu & Lim, 2012; Gumbel, 2001; Lawson & Barakos, 2011) can initiate large pressure and thus air density gradients. The efficiency of the streamer discharge mechanism under more realistic conditions is therefore unclear (see, e.g., discussions in Dwyer et al., 2012).

Here we take a step toward a more realistic scenario where the neutral density is considered inhomogeneous on spatial and temporal scales of streamers, as in the environment of a propagating leader tip. We imagine a multitude of streamers emitted from a leader tip, facilitating the leader propagation, and that filaments of neutral density depletion are created in the stepping process from heating of the neutral gas. We consider a scenario that can be accommodated in our model, whereby a streamer is propagating along the axis of a filament. In previous work, we have discussed air perturbations as well as their origin and focused on streamer properties in inhomogeneous air (Köhn et al., 2018). Conclusively, we found that the streamer velocities as well as the streamer morphology depend on the spatial distribution of ambient air. Depending on the perturbation level, electrons reach energies of up to several kiloelectron volts suggesting the production of X-rays in perturbed air. We here now continue and explore the possible effect of air perturbations on the emission of X-rays from streamer discharges.

Temporal and spatial scales of discharges, and the magnitude of the threshold electric field E, are inversely proportional to the neutral density, n, and perturbations to the neutral density could therefore potentially affect the streamer properties. The reduced electric field, E/n is often used when discussing discharge processes. In air at standard temperature and pressure, with a homogeneous density and electric field, the reduced breakdown field, E_k/n is ≈125 Td corresponding to E_k=3.2 MV/m at standard temperature and pressure where n = 2.55·10²⁵ m⁻³.

In the following we present proof-of-concept simulations that explore the impact of streamer-scale inhomogeneities on electron acceleration and bremsstrahlung radiation. Our model does not self-consistently account for air perturbations but is meant to identify the possible effects on streamers that may be induced by density perturbations, an area that until now is largely unexplored.

2 Methods

Following Babich et al. (2015), we hypothesize simple radial profiles of air and electron densities of channels formed by preceding streamers and compare with results obtained in uniform air without pre-ionization.

2.2 Air Density Perturbations

We model the effects of small-scale air density perturbations and ionization by a preceding streamer as suggested in Babich et al. (2015), Eichwald et al. (1996), Navarro-González et al. (2001), Hill and Robb (1968), Kacem et al. (2013), Liu and Zhang (2014), Marode et al. (1979), Villagrán-Muniz et al. (2003), and Plooster (1970). We choose sinusoidal air density perturbation in the radial direction with the minimum on the axis (r = 0) and the maximum at the outer boundary (r = L_r)

(1)

where n₀=2.55·10²⁵ m⁻³ is the background neutral density at sea level and ξ_{j = 0−4} is the perturbation amplitude of the j'th profile simulated. We consider the following levels of perturbations: ξ_j=0,0.25,0.5,0.75,1.0, which represent differences on the r axis of 0% to 100%. We note here that the ratio E_amb/E_k depends on r for j > 0. For example, for j = 3,E_amb(0,z)/E_k=6,E_amb(L_r/2,z)/E_k=1.5, and E_amb(L_r,z)/E_k=0.86. Hence, the effective electric field is strongly enhanced only in a small region around the symmetry axis. Various measurements have shown that electric fields in streamer discharges can reach field strengths of up to ≈10E_k (Kim et al., 2004; Pancheshnyi et al., 2000; Spyrou & Manassis, 1989) consistent with results of streamer simulations and analytic estimates (Chanrion & Neubert, 2008; Köhn et al., 2018; Liu & Pasko, 2004; Moss et al., 2006; Naidis, 2009; Qin & Pasko, 2014; Tholin & Bourdon, 2013). In the vicinity of lightning leader tips, calculations have shown that the enhanced electric field can exceed several times the breakdown field (Köhn, Diniz, Harakeh, 2017; Köhn & Ebert, 2015).

In an electric field of 6E_k, the ionization length 1/α_ion(E), where α_ion(E) is the Townsend coefficient (Chanrion & Neubert, 2008), is 3 μm, while the ionization length amounts to approximately 111 μm in a field of 0.86E_k. Thus, on the boundaries r = 0 and r = L_r, the ionization length is small enough to allow the formation of streamers yet significantly different such that electrons experience the effect of nonuniform air. We here note that the case j = 4 is an extreme case which we use to conclude our parameter study extrapolating perturbations of 80% and above. The functional shape of the density is meant to capture the scale, the density minima, and the radial gradients of the perturbations. Other than that, the function chosen is not important for our conclusions.

2.3 Pre-ionization

In order to address the impact of pre-ionization, we simulate conditions without pre-ionization and with pre-ionization n_e,0=10¹² m⁻³ as used by Babich et al. (2015) and also used in Nijdam et al. (2011). Other simulations (Köhn et al., 2018; Köhn, Chanrion, & Neubert, 2017a) indicate that the electron density in the streamer channel left behind is in the order of 10¹⁶–10²⁰ m⁻³.

After a discharge, the time of field screening inside the previous channel is determined by the times for electron attachment and ion-ion recombination. The time of electron attachment in air is

(2)

with the oxygen density n_O2=0.2 n, air density n, and with k_att=2·10⁻³⁰ cm⁶/s (Kossyi et al., 1992). After electron attachment, the time of ion-ion recombination is calculated through

(3)

with k_rec,1=10⁻⁷ cm³/s and k_rec,2=2·10⁻²⁵ cm⁶/s (Pancheshnyi et al., 2005). For n = n₀, it is t_att=20 ns and t_rec≈7.8 fs; for n = 0.1n₀, it is t_att=2 μs and t_rec≈0.7 ps. As we will discuss in section 9, the lifetime of air perturbations is in the order of 50 ms and thus significantly larger than the time to readjust the electric field.

The electric field will diffuse within time (Banks et al., 1990; Neubert et al., 1996) where ν_en is the collision frequency of electrons and ω_pe the plasma frequency. If we approximate the maximum collision frequency at standard temperature and pressure ν_en≈8.45·10¹² s⁻¹ and noting that the plasma frequency ω_pe ranges from 5.65·10⁷ s⁻¹ to 5.65·10¹¹ s⁻¹ for pre-ionization levels of 10¹²–10²⁰ m⁻³, the electric field diffusion time becomes τ_diff=26.52 ps to 2.65 ms. Thus, levels of pre-ionization  m⁻³τ_diff>100 ns are longer than the time scale for air perturbations (Marode et al., 1979) or the time scale for streamer simulations at sea level (Köhn et al., 2018; Köhn, Chanrion, & Neubert, 2017a). For the field to reestablish itself within a previous streamer body, we must place an additional assumption that either the time between the old streamer and the new streamer is longer than these time constants or that a new potential wave is propagating into the streamer body from the leader tip as discussed by Babich et al. (2015) and Bazelyan and Raizer (2000).

3 Results

3.1 Temporal Evolution of the Electron Density

The electron densities, n_e, of the streamers without pre-ionization are shown in Figures 1a–1h. To ease comparison with the unperturbed case, which is usually considered in simulations of streamers, the left halves are for unperturbed air, j = 0, and the right halves for j = 1 to 4. We have chosen to show the results at three times, determined by the maximum time that can be accommodated in the simulation domain. For the two smaller perturbation levels the simulations can run longer because the streamers develop and propagate slower (t = 1.03 ns), and for the two higher perturbation levels we must stop the simulations earlier because the streamers develop rapidly in the low-density regions (t = 0.37 ns for ξ_j=0.75 and t = 0.12 ns for ξ_j=1.0). In the supporting information we have added the temporal evolution of this comparison. In order to compare the streamer evolution at the same time step, panels (e–h) show the electron density for all cases after 0.12 ns.

The ambient electric field is pointing downward such that positive streamers propagate downward and negative streamers upward. Panels (a–h) show that for small perturbations, below 50%, the positive and negative streamer fronts both develop and propagate almost with the same pattern as in unperturbed air, although faster with increasing ξ_j. For high levels of perturbations, the negative streamer develops and propagates faster than the positive and it becomes difficult to identify a positive front. The differences in their properties come from the underlying mechanisms of their propagation. Negative streamers are primarily driven by electron impact ionization as they propagate in the same direction as electrons are accelerated (against the field), whereas positive streamers primarily propagate by means of photo ionization of O₂ from excitation of N₂ by electrons accelerated ahead of the streamer and against its direction of propagation (e.g., Zheleznyak et al., 1982). For the high values of ξ_j, the reduced density of air molecules reduces the production of photoelectrons important for the positive streamers and increases the mean-free path of electron ionization, allowing electrons to move longer distances and achieve higher energies between collisions in the negative streamers.

The changes in streamer formation and acceleration in perturbed air are illustrated by considering the mean velocities of streamers. Because the streamers continue to accelerate at the end of the simulations and we stop the simulations at different times, it is not meaningful to calculate and compare the velocities themselves. Rather, we determine the mean streamer velocities in perturbed air normalized to the corresponding unperturbed velocities. Since we stop the simulations for perturbation levels of 75% and 100% within 1 ns, it is hard to identify clear positive fronts and hence we are not able to determine normalized velocities of positive streamers in these cases. The mean velocities are calculated from the position of the fronts at t = 0.01 ns and t_end. The results are shown in Table 1. We find that normalized velocities of both polarities increase with increased level of perturbation reaching ≃70 for the negative polarity at ξ = 1. This is consistent with experiments (Briels et al., 2008), theoretical considerations (Ebert et al., 2010), and numerical simulations (Liu & Pasko, 2004; Pancheshnyi et al., 2005) showing that positive and negative streamers move faster for higher so-called reduced fields, that is, higher electric fields for fixed air density or for reduced air density in a constant electric field.

Table 1. The Mean Velocities, , and Normalized Mean Velocities Ξ^± :=(v_j/v₀)^± of Negative and Positive Streamer Fronts at Different Levels of Perturbation (No Front Is Identified for Positive Streamers for ξ_4,5=0.75,1.0) and the Maximum Electron and Photon Energies ε_e,ε_γ as Well as the Photon Number N_γ Without and With Pre-ionization (PI)

		tend					εe	εγ				εe,PI	εγ,PI
j	ξj	(ns)	(mm/ns)	(mm/ns)	Ξ+	Ξ−	(keV)	(keV)	Nγ	(mm/ns)	(mm/ns)	(keV)	(keV)	Nγ,PI
1	0.25	1.03	0.84	1.39	2.9	1.9	0.2	—	—	0.40	1.00	0.1	—	—
2	0.50	1.03	3.83	4.96	13.2	6.8	0.2	0.1	3	3.28	3.32	0.2	—	—
3	0.75	0.37	—	9.50	—	12.7	3.0	3.0	777	—	5.97	1.0	0.4	9
4	1.00	0.12	—	59.85	—	70.4	100	30	21,816	—	59.68	50	15	5,247

• Note. v₀ is the velocity of the negative and positive streamer fronts in uniform air.

Figures 1i–1l illustrate the importance of pre-ionization on the temporal evolution of the electron density for the same perturbation levels and time steps as in panels (a)–(d). The left half of each panel shows the electron density with a pre-ionization level of 10¹² m⁻³ and the right half without pre-ionization, hence as the right halves of panels (a)–(d). Additionally, Table 1 compares the velocities of the streamer fronts without and with pre-ionization. In all considered cases, the streamer fronts move slower in the presence of pre-ionization (Nijdam et al., 2011) for the same ξ_j but still faster than streamers in uniform air. However, the effect of streamers being faster than in uniform air, increasing with ξ_j, is still prevalent in the presence of pre-ionization.

3.2 Occurrence of High-energy Electrons and X-rays

The energy distributions of electrons at the end of the simulations are shown in Figure 2. The distributions in unperturbed air are shown as reference (solid curves) together with the distributions in perturbed air without pre-ionization (dashed lines) and with pre-ionization (circles). The distributions of bremsstrahlung photons in perturbed air are shown without pre-ionization (crosses) and with pre-ionization (squares). We see that, as expected, the number of electrons and the maximum electron energy increases with the perturbation level. The maximum electron energies for ξ_{2 − 4}, reached at the end of the simulations, are approximately 200 eV (200 eV with pre-ionization), 3 keV (1 keV), and 100 keV (50 keV) after 1.03, 0.37, and 0.12 ns. In comparison, the maximum electron energies in uniform air are ≈100 eV, which is in agreement with earlier results of electron energies of streamer discharges in perturbed air (Köhn et al., 2018). The higher energies are caused by the higher-reduced electric field close to the axis, which allows for stronger electron acceleration. For perturbations of 75% and 100%, the generation rate of runaway electrons above 1 keV is approximately 3.8·10¹² s⁻¹ and 3.4·10¹⁷ s⁻¹, respectively. Schaal et al. (2012) performed ground-based observations of high-energy emission from natural and rocket-triggered lightning and subsequently estimated the generation rate of energetic electrons producing X-rays. They found rates of approximately 10¹² s⁻¹ - 10¹⁷ s⁻¹ which agrees very well with our simulation results.

In the cases of pre-ionization, the electron number densities are reduced for ξ_{3 − 4} because the pre-ionization space charge tends to reduce the field; however, the reduction is modest and of the order of a factor of 2. The reduction in the maximum electron energy is more significantly caused by a lower acceleration of electrons. In such a configuration, the generation rate of electrons above 1 keV is 0.4·10¹² s⁻¹ for ξ₃=0.75 and 0.1·10¹⁷ s⁻¹ for ξ₄=1.0 which is smaller than without pre-ionization but still in the range determined by Schaal et al. (2012).

We also observe the occurrence of photons at the higher perturbation levels, ξ_{2 − 4}. Their production is a direct consequence of the existence of energetic electrons. After the acceleration of electrons in the low-density region, the local electric field at the streamer head moves some electrons to higher-density regions where they create X-rays through the bremsstrahlung process. The energy and number density of photons reflect the properties of the electrons. The lower density of energetic electrons for n₃ suppresses the photon distribution to just a few photons. The maximum photon energy is 3 keV without and 400 eV with pre-ionization. For n₄, on the other hand, the difference in the photon spectra is modest; the maximum photon energy in this case is 30 keV without and 15 keV with pre-ionization. This difference is caused by the sensitivity of the spectra to the time the simulations are stopped, affecting the case of n₃ more strongly because the high-energy tail of electrons has not had time to fully develop in this case. The results suggest, therefore, that the amplitude of the bremsstrahlung photon spectra for pre-ionization is modified similar to that of electrons, that is, by a factor of ∼2. The maximum energy of electrons, ε_e, and photons, ε_γ, as well as the photon number N_γ with and without pre-ionization are shown in Table 1.

4 Discussion and Outlook

Discussing the influence of air perturbations on the production of X-ray and γ-ray from electric discharges, we have to distinguish the effects of streamers from that of leaders. In laboratory experiments of long discharges (Kochkin et al., 2012, 2014), multiple streamers propagate in close proximity to each other and experiments have shown that X-rays occur within tens of nanoseconds when multiple streamers are concentrated in a small volume around the electrode. The observed currents at the grounded electrode and the high-voltage electrode during the production of X-rays in these experiments are in the order of 100 A . The work of Marode et al. (1979) shows that streamers with peak currents of several hundreds of milliampere perturb air by a factor of 50% in a radius of several micrometers within tens of nanoseconds. Hence, the effect of density perturbations from bypassing streamers is not negligible and can influence the properties of neighboring streamers. The lifetime of the perturbations in the streamer corona is in the order of  ms with a diffusion coefficient D_air≃2·10⁻⁵ m²/s (Cussler, 1997). This is large enough for following streamers to encounter inhomogeneities created by preceding streamers. For instance, it is observed both in lightning of the atmosphere and for sprites in the mesosphere that streamers may follow paths created by earlier streamers (Nijdam et al., 2014). In addition, spherical and cylindrical shock waves associated with lightning leader propagation create large overpressures such that the air density in its vicinity is reduced up to 100% (Liu & Zhang, 2014; Plooster, 1970). There is no doubt, therefore, that the environment around leader tips is complex, highly inhomogeneous, and dynamic. We have shown for the rather simple case of perturbations from a single streamer filament perturbation that such environment is likely to enhance electron acceleration and bremsstrahlung radiation from streamers relative to a homogeneous air.

Our simulations suggest that inhomogeneities in the background air density, resulting from streamers ahead of lightning leaders, from shock waves associated with lightning leaders or from neighboring streamers simultaneously propagating close to each other, may have a profound effect on electron acceleration and bremsstrahlung radiations in streamers. There are two simultaneous effects at play which are not present in uniform air. One is that electrons can gain high energies in the center regions of streamers where the air density is reduced, and the other is that the air density is high at the edge of the streamers allowing high electric fields to be established in the ionization wave. Thus, our results suggest the thermal acceleration mechanism may play a role as a source of energetic radiation as observed from lightning and laboratory sparks.

Table 1 shows the number of photons-produced ranges between 3 and about 22,000 within 0.12 to 1.03 ns depending on the perturbation level and the existence of pre-ionization. For comparison, the total number of photons in a TGF is estimated from satellite observations to be in the range of ≈10¹¹–10¹⁸ photons with energies between several kiloelectron volts to tens of million electron volts, with the lower photon number limit determined by the instrument sensitivities (Gjesteland et al., 2010; Østgaard et al., 2015). If the streamer zone of a lightning leader tip consists of approximately 10⁶ streamers as assumed by Celestin and Pasko (2011), the average production of one streamer is ≈10⁵–10¹² photons above 1 keV. Both the photon energies and the number of photons in TGFs are then much higher than obtained in our simulations. We point out, though, that our simulations are stopped very early because of the limitations on the simulation domain size. Subsequently, we miss the initiation of a relativistic RREA and thus a further amplification of the number of high-energy electrons. We can then only conclude that density perturbations enhance the number of relativistic electrons and significantly increase the photon flux relative to the unperturbed case.

In laboratory experiments, on the other hand, the estimated number of photons produced in a discharge was 10³–10⁴ (Kochkin et al., 2016; Nguyen et al., 2008) and the estimated number of photons being produced by one single streamer 2·10⁻³ (Nguyen et al., 2010).

Hence, the average number of photons produced by single streamer discharges ranges from 10⁵–10¹² photons for TGFs to photon for X-rays measured in laboratory discharges. For perturbation levels of 75% or 100% we observe 777 photons (9 with pre-ionization) or 21,816 (5,247), respectively, which lies in-between this number range.

Laboratory discharges likely have lower photon energies and photon numbers in a discharge relative to naturally occurring TGFs because of the limited size of laboratory experiments and the energy of a discharge. Because of computational limitations, the simulations presented in this manuscript are performed in a small spatial domain. In this way, the laboratory setting is closer to our simulation scenario. In order to improve our understanding of the role of density perturbations in facilitating the production of TGFs, it would be desirable to run further simulations in a larger domain for several milliseconds instead of nanoseconds as well as to treat air perturbations self-consistently accompanying the electron motion, the streamer development, and the emission of X-rays. However, as computational costs are currently too high, we conclude then that radiation enhanced by density perturbations is a likely candidate accounting for radiation in high-voltage experiments giving us a hint about the production of TGFs in perturbed air, and we point out that future, more realistic, simulations of streamer discharges and the associated emission of X-rays potentially need to involve the gas dynamics of ambient air.