Streamer properties and associated x-rays in perturbed air

To investigate the motion and development of streamers, we trace electrons through air using a 2.5D cylindrically symmetric particle Monte Carlo code with two spatial coordinates (r, z) and with three velocity coordinates $({v}_{r},{v}_{\theta },{v}_{z})$. As in [21, 53], we initiate a Gaussian electron–ion patch with a peak density of ${n}_{e,0}={10}^{20}$ m⁻³ and a width of ${\ell }=0.2$ mm centered at ${z}_{0}=7\,\mathrm{mm}$. The size of the simulation domain is ${L}_{r}=1.25\,\mathrm{mm}$ in r direction and ${L}_{z}=14\,\mathrm{mm}$ in the z direction. The ambient field E_amb is 1.5 times the classical breakdown field in uniform air, ${E}_{k}\equiv 3.2$ MVm⁻¹, at standard temperature (300 K) and pressure (1 bar) and is pointing from the upper to the lower boundary; after every time step we solve the Poisson equation on a mesh with 150 grid points in r and with 1200 grid points in z direction. We have chosen ${E}_{{\rm{amb}}}=1.5{E}_{k}$ that is common in the literature and corresponds to a relatively weak field, allowing the initiation of a streamer from a single electron. This field strength and higher fields can be found in sprite streamers [30, 54] or laboratory discharges [18]. At the boundaries $(z=0,{L}_{z})$, we use the Dirichlet boundary conditions $\phi (r,0)=0$ and $\phi (r,{L}_{z})={E}_{{\rm{amb}}}\cdot {L}_{z}$, and at $(r=0,{Lr})$ we use the Neumann boundary conditions and fix the electric field strength to zero. Details about the collisions of electrons with air molecules and their numerical implementation can be found in [21, 55]. For the particle management, i.e. merging and splitting superelectrons, we use the same scheme as elaborated in [21, 55] and references therein.

One of the key ingredients governing the motion of streamers is the composition and the density of ambient gas. The mean free path Λ of electrons between two collisions with air molecules is ${\rm{\Lambda }}={(n{\sigma }_{t})}^{-1}$, where ${\sigma }_{t}$ is the total cross section of electrons scattering off air molecules and n is the air density. Equivalently, the motion and energy of electrons in a gas is governed by the reduced electric field E/n. Since the electron's mean free path and the reduced electric field depend on n, the spatial distribution of n is crucial to understand the motion of electrons in air.

To investigate the effect of perturbed air on the streamer dynamics, we simulate the motion of streamers in uniformly distributed air density ${n}_{0}\equiv 2.55\times {10}^{25}$ m⁻³ and in sinusoidal densities $n(r,z)$ depicted in figures 1(a) and (b):

with $a,b$ and c as tabulated in table 1. Additionally, we investigate the properties of streamers in air with the spatial distributions

with $\zeta =1\,\mathrm{cm}$ shown in figures 1(c) and (d). We have chosen sinusoidal functions since the air density is modulated as a sinusoidal wave after the appearance of a shock wave. With these density functions we simulate the effects of air density perturbations after shock waves and thermal expansion induced by streamer or spark discharges [32, 41, 42, 56] or in the vicinity of high-speed air flows creating large pressure gradients and subsequently large air gradients [49, 50, 52]. We refer to (1) and (3) with their gradients along the z-axis as perturbations parallel to the ambient electric field; likewise we refer to (2) and (4) as perturbations perpendicular to the ambient field. The functional shape is chosen such that there are different regions with different numbers of air molecules. We note here that, in reality, in long discharges not only one, but multiple streamers occur [58]. Although there are rarely models describing the interaction of several streamers amongst each other [59–61], it is thus not unlikely to assume that a couple of streamers perturb ambient air whilst others travel in such perturbed air region. Other than that, the actual shape of the chosen functions is not important for our conclusions.

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We first look into the evolution of bidirectional streamers for heated or compressed air parallel to the ambient electric field; the air density profile of these scenarios is described by (1) or (3). The third column of figure 2 compares the electron density in uniformly distributed air (left half of each panel) with the electron density in perturbed air (1) (right half); the second column shows the corresponding electric fields. Panels (a)–(f) of figure 2 demonstrate that streamer inception at the position of maximal density elongation (${n}_{z,1}$ and ${n}_{z,2}$) does not alter the shape and the velocity of a streamer significantly since the number density of air molecules quickly reaches the density n₀ corresponding to uniform air. If the streamer is initiated in reduced air density (${n}_{z,1}$), the streamer fronts have advanced slightly further than in non-perturbed air; contrarily, if the streamer is initiated in higher air density (${n}_{z,2}$), the streamer fronts are slightly damped.

The effect on the shape and on the velocity is much more significant if the streamer inception takes place at the position of maximal and minimal slope of the air density (${n}_{z,3}$ and ${n}_{z,4}$), which leads to a rapid deviation from the air density n₀ of uniform air. Since the densities grow and fall to different sides, both fronts behave contrarily. In ${n}_{z,3}$ the negative front is faster than in n₀ whilst the positive front is slower, vice versa for ${n}_{z,4}$. Panels (i) and (l) demonstrate that ionization is pronounced ahead of the streamer in regions with high air density independent of the polarity of the streamer front; accordingly, for low density regions, the streamer front is thinner.

Table 2 shows the mean streamer velocities

where ${z}^{\pm }({t}_{\mathrm{0,1}})$ is the position of maximum electric field strength on the positive or negative streamer head after t₀ = 0.54 ns and t₁ = 2.59 ns, as well as the normalized mean velocities ${(v/{v}_{0})}^{\pm }$. The mean velocities in all cases are in the order of 10⁶ m s⁻¹. In all our simulations, negative fronts are faster than positive fronts, which is consistent with previous work [62, 64]. However, this is only the case for the very first time steps of the streamer evolution; at later stages, the positive front moves faster than the negative front [62]. Indeed, experiments performed by [65] and other modelling efforts [30] have shown that positive fronts move faster than negative ones because of their rather high peak field. However, since the motion of streamers in our simulations is governed by the air density profile, the following results can be extended very easily to faster positive fronts irrespective of the relative speed between the negative and positive front. In density ${n}_{z,1}$ both streamer fronts move approximately as fast as in constant air density n₀; in ${n}_{z,2}$ the negative streamer front is $\approx 20 \%$, the positive one $\approx 10 \%$ slower than in n₀. In ${n}_{z,3}$ and in ${n}_{z,4}$ the streamer fronts moving within the region of increasing air density move slower whereas the fronts within decreasing air density move faster compared to the motion of streamer fronts in constant air density. This is in agreement with observations made by Opaits et al [27] that (positive) streamers move faster in air densities small compared to n₀.

Figure 3(a) shows the electron density in ${n}_{z,3}$ (left half) and in ${n}_{z,4}$ (right half) after 2.59 ns; in the right half we mirrored the electron density plot at z = 7 mm compared to the density plot in the right half of figure 2(l). Hence, the spatial distribution of air molecules in both halves looks like ${n}_{z,3}$, and we can explicitly compare positive and negative streamers in the same air density profile. The negative streamer front moves faster than the positive one because of the different propagation mechanisms [62–64]; the negative streamer front moves through the motion of electrons against the electric field and the subsequent ionization in front of the streamer head facilitated by photoionization [17], whereas photoionization ahead of the streamer head is essential for the motion of the positive front. Given the mean velocities in table 2, the negative front moves approximately 40% faster than the positive one in the same density profile.

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Panels (b) and (c) show the electron density (left half) and the mirrored electron density (right half) in air densities ${n}_{z,3}$ (b) and ${n}_{z,4}$ (c) giving us the opportunity to immediately compare the different elongations of the negative and the positive front. It shows that in ${n}_{z,3}$ the negative front moves almost twice as fast as the positive one; in ${n}_{z,4}$ both streamer fronts move equally fast. Since the air density decreases with increasing z for ${n}_{z,3}$, i.e. against the electric field, the reduced electric field increases in the direction of electron drift and the electron energy is enhanced. Since the air density increases downwards, the reduced electric field in that region decreases and the electron motion is damped. In ${n}_{z,4}$ the opposite takes place: the air density increases against the electric field, i.e. in the direction of electron drift, and decreases at the positive tip, i.e. along the electric field. Hence, electron motion at the negative front is damped whereas the electron energy in the vicinity of the positive front is favored, such that both fronts move equally fast. Controlling the air density profile parallel to the ambient electric field, is thus a mean to align the mean velocities of negative and positive streamers.

Figure 4 compares the electron density of a streamer initiated at z = 7 mm in uniform air density (left half of each panel) with the electron density of a streamer in air density (3). Equation (3) describes the spatial distribution after a shock wave where the majority of air molecules is shifted to the upper boundary. After 1.05 ns the motion of both streamer fronts is suppressed; at the negative tip the air density is increased, hence the reduced electric field E/n is decreased and the electron motion is damped. At the positive front, the air density is decreased much more than for ${n}_{z,4}$. Although the reduced electric field is thus increased and electrons gain more energy than in uniform air, the ionization coefficient is decreased for very small air densities and so is the probability of photoionization.

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The motion of the negative front in air density (3) is damped during the whole simulation. Whilst the mean velocity of the negative streamer between 1.05 ns (a) and 2.24 ns (d) in uniformly distributed air is $1.09\times {10}^{6}$ m s^–1, the average velocity is $0.66\times {10}^{6}$ m s^–1 in perturbed air density, hence a factor of approximately 1.6 slower.

The propagation of the positive streamer front is, however, not damped during the whole simulation. Figure 5 zooms in on the electron density into the region $z=0.5\mbox{--}2.5\,\mathrm{mm}$ between 1.34 ns and 1.71 ns; here the air density varies from $0.1{n}_{0}$ at 0.5 mm to $0.42{n}_{0}$ at 2.5 mm. We observe that a small electron inhomogeneity is produced by UV photoionization at approximately 0.5 mm (panel (a)). It first turns into an electron avalanche (panel (b)) and then further into a negative streamer (panels (c) and (d)). Panels (e) and (f) show the electric field after the same time steps as in panels (c) and (d). They show the typical field pattern of a streamer with enhanced field strengths at the channel tips and vanishing field in the body. Here we also note that there are several local electron patches below 2.5 mm, supporting the initiation of a negative streamer. Since we use a 2.5 dimensional Monte Carlo code, the patches illustrated in figure 5 actually have a cylindrical shape. We note that fully three-dimensional simulations would be desirable to confirm our observation of the spontaneous inception and motion of a negative streamer. However, although we cannot exclude that this is a numerical artifact, our microphysical interpretation of the observed phenomena, i.e. the production of local electron patches through photoionization and the further multiplication of the electron number, is independent of the dimensionality of the used Monte Carlo code. Subsequently, the positive front of the original streamer and the newly formed negative streamer encounter and accelerate each other [53, 66] such that the positive front finally moves faster than in uniformly distributed air. However, we here strongly emphasize that the production of such an electron inhomogeneity and the formation of a negative streamer is a rare event. Fortunately, we were able to capture such an event, but we do not expect this pattern to occur in all simulations with the same air density profile and ambient electric field.

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Figure 6 shows the electron energy distributions in non-perturbed (solid) and in perturbed air ${n}_{z,5}$ (dashed) after 1.83 ns. The solid line shows a typical streamer-like energy distribution with energies of at most 100 eV [55]. In contrast, the dashed line shows an energy distribution with energies of up to 3 keV. As stated by Cooray et al [67], the collision of two encountering streamers could eventually lead to fields sufficiently high to produce electrons with energies of several hundreds of keV. In contrast, recent work [53, 66] has shown that the collision of two streamers alone cannot produce a sufficient number of runaway electrons. However, here we observe the production of a significant number of electrons above 500 eV between two colliding streamers in perturbed air which could eventually lead to the production of x-rays with energies of a few keV.

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For perturbations perpendicular to the ambient electric field, for example induced by shock waves, e.g. induced by bypassing lightning strikes or neighboring streamers [68], the spatial distribution of air molecules can be approximated by equation (2). Figure 7 shows the electron density and the electric field after 2.59 ns. The first row shows the electron density in the case that the air density is 5% higher than n₀ at r = 0 and decreases for increasing r; the second row shows the electron density in case the air density is 5% smaller than n₀ at r = 0 and increasing for increasing r. Since the streamer develops close to r = 0, its evolution is damped in the first case and driven in the second case. This is the same effect as we have observed in figure 2 for variations in the z direction. Table 2 shows that, in the first case, both streamer fronts are slower by approximately 40% whereas they are faster by a factor of up to 66% in the second case. The third and the fourth rows show the electron density and electric field if the air density equals n₀ in the vicinity of streamer inception and then starts increasing (third row) or decreasing (fourth row) for increasing r. In these cases, the streamer fronts are equally fast as streamers in uniform air, see table 2. In the third row we observe a slight streamer quenching and the emergence of another electron avalanche in the low air density region, which is comparable to the emergence of the electron avalanche in figure 4(b). The fourth row shows that, if the air density first decreases, the streamer follows the air density resulting in turning away from the symmetry axis. Additionally, the reduced electric field is enhanced due to the reduced air density. Since we use a Monte Carlo code with cylindrical symmetry, any deflection from the symmetry axis resembles branching. This resembles the effect of a magnetic field influencing the streamer motion. In strong magnetic fields between 1 T and 10 T, streamers at standard temperature and pressure have been observed to bend and branch compared to streamers in vanishing magnetic fields [69, 70]. This is the case for magnetic fields sufficiently strong such that the gyration frequency of electrons exceeds the collision frequency of electrons scattering off air molecules (see e.g. [31, 71] for discussions of the gyration and collision frequency). Additionally, we see a widening of the electron density in the r direction.

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Figure 8 compares the electron density in uniform air (left half) and in air density (4) (right half) after different time steps. The shape of the air density (4) describes the spatial distribution of air molecules after a shock wave where the majority of the air molecules is located at the exterior of the simulation domain. Spherical and cylindrical shock waves associated with lightning leader propagation create large over-pressures such that the air density in its vicinity is reduced by up to 100% [42, 43]. Figure 8 shows that in the latter case the electron density in the direction of the negative front grows much faster than in air density n₀. Since for the second case the air density is smaller in the vicinity of the symmetry axis, the the drift speed of electrons upwards is larger and the ionization of air molecules ahead of the initial electron patch are facilitated.

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The positive front is less pronounced than the negative front since the decreased air density close to the symmetry axis reduces the probability of photoionization events.

Figure 9 compares the energy distributions of electrons in uniform and in perturbed air and shows the energy distribution of Bremsstrahlung photons in perturbed air after 0.17 ns. Whereas in non-perturbed air electrons have reached energies of up to only 100 eV, electrons gain energies of up to 20 keV in perturbed air. As a consequence, high-energy electrons produce Bremsstrahlung photons with energies of up to 20 keV.

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Figure 10 shows the position of Bremsstrahlung production; every single point in that figure represents one Bremsstrahlung producing event. Since the majority of air molecules is moved away from the symmetry axis, no Bremsstrahlung photons are produced in its vicinity. Instead, electrons are first accelerated in the low air density region with its high reduced electric field, subsequently reach the high density region and finally create Bremsstrahlung photons by scattering at air molecules.

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