Pilot system development in metre-scale laboratory discharge

In 2009 Cooray et al proposed a mechanism of x-ray generation in long laboratory sparks [20]. It was suggested that the x-ray bursts are caused by encounters of negative and positive streamer fronts. Although the negative discharge development process was simplified in this model, the main idea of streamer encounter as the emission source has recently been experimentally supported. With positive high-voltage pulse, x-rays indeed appear at the moment when positive corona from the HV electrode merge with negative corona from the grounded electrode [16]. Many encounters between individual streamers of opposite polarity occur at this moment. The development of negative discharges is more complex. We photographed the x-ray source region with ns-fast camera, and showed that positive streamers appear in the vicinity of the negative HV electrode [17]. The positive streamers originate from bipolar pilot systems and encounter nearby negative streamers. For more details and properties of the x-ray emission we refer to [10, 21, 22]; a statistical analysis is given in [23]. It is assumed that the x-rays are generated by high energy electrons in Bremsstrahlung process. The first attempt to measure such electrons directly has recently been published in [24].

Here it should be noted that a recent simulation of the x-ray emission questioned the streamer encounter mechanism [25]. While it was confirmed that such encounters dramatically increase the electric field between two streamer fronts to values much higher than the cold runaway breakdown, the field persists only for several picoseconds due to rapid rise of electron density. The ionization quickly collapses the field, giving no time for the electrons to accumulate high energy. It is possible, however, that the action of electric field on electron acceleration was underestimated in that work, as discussed in section 5.4. Nevertheless, in measurements the x-rays bursts and pilots coincide in space and time, so the precise role of the latter requires further investigation.

For the sake of consistency, we reproduce here the pilot development as reported in [10, 17], and discuss additional features. Figure 1 shows the pilot system development process. Every image was exposed for 50 ns and represents a single individual discharge at the moment of the most intense x-ray emission. The time delay between two consecutive discharges was at least 10 s. The depicted area is located below the HV electrode. The HV electrode tip is only visible in images (b) and (f) at the top in the middle. The electrical signals of the full discharge are shown in figure 2, the voltage U over the gap, the currents I_HV and I_GND through both electrodes and the x-ray signals. Two LaBr₃ scintillation x-ray detectors were placed next to each other. Both simultaneously registered a 400 keV signal. All images of figure 1 are taken with 50 ns shutter time that fell within the x-ray time window, or between $t=0.7-0.8~\mu$s. The pilot systems occur at an advanced stage of the discharge development. Returning to figure 1, we observe the negative streamers (1) that originate from the HV electrode (image (a)). Some of them leave isolated beads (2) behind during the propagation. We will call them 'streamer beads' or just beads; the reader should not confuse these with bead lightning [26]. In 2D images the beads appear at $2.5\pm 0.4$ cm intervals. Some beads become branching points of the negative streamer. Eventually the channels and heads of the branched streamers form the negative streamer corona.

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The streamer propagation velocity is measured by two different techniques: (i) by measuring the streamer trace length in an image with known exposure time and (ii) by measuring the displacement of a streamer head comparing two consecutive short exposure images taken by two cameras. The velocities are measured for many different streamers in different discharges; we further refer to [17] for details on negative streamer and corona development. We found that the negative streamer heads propagate at $\left(3.7\pm 0.2\right)\times$  10⁶  m s⁻¹, which is in agreement with previously reported in [12] for 2 m gaps. Since the projection into the camera plane always reduces distances, the actual distance and velocity is likely to be closer to the highest measured value than to the average. At the same moment, positive streamers start at the beads, either as single intense streak or as branches. The branches first move perpendicularly (image (c)) to the streamer channel driven by the streamer electric field, and then they start to propagate towards the HV electrode (images (b)–(f)). This gives the branches the appearance of a stack of the Greek letters $\Psi$. To substantiate the movement towards the electrode, we used two high-speed cameras triggered in sequence; the results have earlier in [17] and are reproduced here for consistency.

Figure 3 shows two subsequent images made with 3 ns exposure time. The first image was placed on the red layer of an RGB picture, the second image was delayed by 10 ns with respect to the first one, and placed on the blue layer of the same picture. The arrows indicate the displacement from the red to the blue images. Clearly, many streamers move towards the cathode. The positive streamer velocity is difficult to measure correctly due to the limited extension in space and apparent dependence on other factors, such as the proximity of the HV electrode. We estimated velocity of the positive streamer head at $\left(1.5\pm 0.7\right)\times$  10⁶ m s⁻¹, or half the speed of the negative ones. As a result, positive streamers appear brighter in the images than negative — assuming equal intrinsic brightness. The entire structure—the negative corona, streamer beads with $\Psi$s, is called a 'pilot system' (4) in [1]. Pilots are encircled in images (c)–(e) by dashed ellipses.

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In our setup the last bead appears at $36.8\pm 6.3$ cm distance from the tip. We can use the streamer velocity data of figure 6 in [17] to go back to the moment that the negative streamer head was at this point (0.5 μs in figure 2). Then the applied voltage U was 500 kV, or the local electric field $E\approx 12$ kV cm⁻¹ assuming a 1/r-dependence for E/U. This field is close to the so-called 'stability field' E_s [27]. It also indicates that the streamer experiences a shortage in charge and electric field which hinders smooth propagation. The blockade may lead to beading. As a support for this suggestion we refer to [12] where it was demonstrated that a shorter voltage rise time, i.e. larger $\text{d}E/\text{d}t$, leads to smoother discharge development.

The upper parts of figures 1(c) and (d) show many $\Psi$s that are about to collide with negative streamers or the cathode. Such collisions provide the kick-off for the few electrons that become run-away in the electric field and produce x-ray by Bremsstrahlung a few nanosecond later [28]. We observed high frequency cathode current oscillations [10] simultaneously with the x-rays, and also attribute the oscillation to the collisions.

Figure 4 summarizes the pilot system development. It starts with a negative streamer that leaves one or more beads behind, that may grow into $\Psi$s, and that finally transforms into the bipolar structure. An attempt to combine this history with the streak photographs of [12] is hindered by the space-time convolution inherent to streak images. To this there is the addition of the larger gaps of 2 and 7 m versus ours of 1 up to 1.5 m and the longer voltage rise time of 6 μs versus 1.2 μs. Figure 6.1.5(b) in [12] shows at least three stationary beads at 10 times larger separation than ours that last for about 0.5 μs; the beads seem to appear 'out of the blue' in virgin air. Our images demonstrate that a precursor streamer initiates the beads.

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Though it is not apparent from the photographs presented here, previous observations on longer spark gaps show that the pilot system becomes a hot space leader if given sufficient time [12]. Figure 6.3.2 in [12] shows that the instantaneous velocity of the space stem ranges from $2\times {{10}^{4}}$ to $2\times {{10}^{5}}$ m s⁻¹; the higher velocity goes with the shorter voltage rise time. Such space stem behavior can be explained by subsequent launching of positive streamers starting from the first bead to the last. Naturally, this will appear as a moving space stem on streak photographs.

Two types of positive corona emanate from the beads. The first is a single positive streamer, for instance image(a) in figure 5; the second, our stack of $\Psi$s in image (b), has many streamers. The single positive streamer is significantly brighter and wider than all other streamers. The $\Psi$s are shown in detail in figure 5 image (b), and also in figures 1 images (c)–(f). The streamers follow the local electric field lines towards the HV electrode. There are no visible beads anymore, which indicates that they fade away quickly, in fact about 10 times quicker than those in figure 6.1.5(b) of [12].

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The faint speckle trace is visible in images (a) and (b) and indicated by small arrows. In image (a) it runs from the HV electrode tip down in the middle of the picture. In image (b) it comes from the left upper corner and goes down through the structure. It is a camera artifact. The camera's electronic shutter is switched off during the final breakdown, but some light can still leak through it and appear in the images as a trace. It helps to identify the streamer that grows into the final spark.

Figure 6 shows an example of the pilot reconnection. Both types of pilot systems, as described above, are clearly visible. A reconnection between positive streamers in STP ambient air was previously shown in [29] and possible mechanism proposed in [30]. It is shown here that the reconnection also occurs between negative streamers, in this case interconnecting two pilot systems. The characteristic curvature of the streamer path and termination on the edge of another streamer channel indicates that they merged.

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Numerical modeling of streamer development in air has currently reached a rather advanced stage. The streamers discharge has been modelled in full 3D space and based solely on microscopic physical mechanisms, e.g. [31]. However, obtaining numerically a fully developed branching fractal streamer pattern in this way is computationally difficult and still under development. To study the streamer structures with limited computational resources, one can introduce macroscopic physics, i.e. assumptions about the details of streamers which are not modelled microscopically [30]. The 'pilots' occur only at an advanced stage of a streamer discharge, by which we mean that the streamer corona have been fully developed and the individual streamers may have undergone possibly multiple branching. In order to understand the pilots, we are thus forced to follow the route of macroscopic (simplified) modeling. So that we can simplify the modeling, we make a rather crude assumption of spherical symmetry in the developed structured streamer discharge, with physical values being functions only of a single (radial) coordinate. The individual streamer branches are thus not considered but are treated collectively, and we represent the collective streamer effects in transverse (angular) direction in terms of the effective curvature, which is thus different from the curvature of individual streamer heads. The overall schematic representation of the model used is shown in figure 7.

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We shall model the development of a discharge in quasi-electrostatic approximation, i.e. neglecting the effects of electromagnetic waves. This can be done since the ratios of typical length and time scales, including the typical streamer velocities $1.5\times {{10}^{6}}-4\times {{10}^{6}}$ m s⁻¹ (given above in this paper), are much less than the speed of light. In principle, the relatively large  ∼1 m size of the electrode gap can give importance to electromagnetic effects of very fast phenomena happening at time scales $\lesssim$1 ns, which could occur, e.g. during the streamer collisions; this is a topic for our future research. The quasi-electrostatic equations may be represented as:

$$\begin{eqnarray}\left. \begin{array}{*{35}{l}} {{\nabla}^{2}}\phi & = & -\rho /{{\varepsilon}_{0}} \\ \quad \overset{\centerdot}{{\rho}}\, & = & -\nabla \centerdot \left(\sigma \mathbf{E}\right) \\ \quad \overset{\centerdot}{{{{N}_{\text{e}}}}}\, & = & \left(\nu _{\text{i}}^{\text{eff}}(E)-\nu _{\text{a}}^{\text{eff}}(E)\right){{N}_{\text{e}}}+{{S}_{\text{p}}} \end{array}\right\}.\end{eqnarray}$$

In these equations, $\mathbf{E}=-\nabla \phi$, ϕ, ρ are electric field, potential, and charge density, respectively, $\sigma =\text{e}{{\mu}_{\text{e}}}{{N}_{\text{e}}}$ is the electric conductivity, N_e is the electron density, $\nu _{\text{i}}^{\text{eff}}$ and $\nu _{\text{a}}^{\text{eff}}$ are effective ionization and attachment rates which describe propagation of streamers, which are different from the physical (microscopic) ionization and attachment rates (denoted here by ${{\nu}_{\text{i},\text{a}}}$), and S_p is the photoionization source.

This model includes a row of simplifying assumptions. First, the electron mobility ${{\mu}_{\text{e}}}$ is assumed constant because it varies very little in a large range of electric fields, e.g. from  ∼  0.05 m² s⁻¹ V⁻¹ at $E=0.3{{E}_{\text{b}}}$ to  ∼0.04 m² s⁻¹ V⁻¹ at $E=2{{E}_{\text{b}}}$, where ${{E}_{\text{b}}}\approx 3\times {{10}^{6}}$ V m⁻¹ is the electric breakdown field [32]. We may therefore justify our assumption of constant electron mobility by arguing that the discharge-related processes occurring at $E\lesssim {{E}_{\text{b}}}$ are much less important than at fields  ∼E_b. Second, we neglect ion conductivity due to the fact that ion mobility is at least by a factor of  ∼10² smaller [33]. Third, we neglect electron advection effects, since the velocity of electron drift is much smaller than the streamer velocity. This, in particular, leads to having no difference in propagation of positive and negative ionization fronts in our model, while the observed velocities differed by a factor of  ∼2 in the same background field. Fourth, we neglected electron diffusion D_e, valued at  ∼0.08–0.16 m² s⁻¹ in the range of electric fields of interest [34], because the characteristic Kolmogorov–Petrovskii–Piskunov velocity of the ionization $2\sqrt{{{\nu}_{\text{i}}}\left({{E}_{\text{b}}}\right){{D}_{\text{e}}}}\sim {{10}^{4}}$ m s⁻¹ [35, p 77] is also much smaller than the observed streamer velocities.

The electric breakdown occurs above E_b, where the ionization prevails over attachment. However, in a 1D situation the propagation of negative streamers occurs above the negative streamer sustainment field ${{E}_{-\text{s}}}$, which is lower than E_b. Thus, we choose the functional dependence of the effective 1D rates $\nu _{\text{i},\text{a}}^{\text{eff}}(E)$ in such a way that the ionization occurs at $E>{{E}_{-\text{s}}}\approx 1.25\times {{10}^{6}}$ V m⁻¹ [36]. We approximate these rates with power functions:

$$\begin{eqnarray}&&\nu _{\text{i}}^{\text{eff}}(E)={{\left(E/{{E}_{-\text{s}}}\right)}^{\alpha}}/{{\tau}_{\text{i}}},\quad \quad \nu _{\text{a}}^{\text{eff}}(E)={{\left(E/{{E}_{-\text{s}}}\right)}^{\beta}}/{{\tau}_{\text{i}}}\end{eqnarray} \tag{ 1 }$$

where ${{\tau}_{\text{i}}}\approx 7$ ns is the typical ionization time.

We note that the empirical data for the physical values of ${{\nu}_{\text{i},\text{a}}}$ [34] may also be fitted with power-law functions (1), but we of course must use E_b instead of ${{E}_{-\text{s}}}$ in these formulas. In the case of the physical values ${{\nu}_{\text{i},\text{a}}}$, the best-fitting coefficients are $\alpha =5.5$ and $\beta =1.1$.

Another source of ionization growth is S_p, the photoionization, which is one of the mechanisms responsible for ionization front propagation (the other mechanisms being the neglected electron advection and electron diffusion). The photoionization is a non-local source

$$\begin{eqnarray}&&{{S}_{\text{p}}}\left(\mathbf{r}\right)={\int}^{}{{S}_{\text{i}}}\left({{\mathbf{r}}^{\prime}}\right)F\left(\mathbf{r}-{{\mathbf{r}}^{\prime}}\right)\,{{\text{d}}^{3}}{{\mathbf{r}}^{\prime}}\end{eqnarray}$$

where ${{S}_{\text{i}}}=\nu _{\text{i}}^{\text{eff}}(E){{N}_{\text{e}}}$ is the 'local' ionization rate. We model it as the 'exponential profile' model [37]:

$$\begin{eqnarray}&&F(r)=\frac{C}{{{ \Lambda }^{2}}}\frac{{{e}^{-r/ \Lambda }}}{4\pi r}\end{eqnarray}$$

where $\Lambda$ and $C\ll 1$ may be understood as the characteristic 'length' and the 'strength' of photoionization, respectively. This model is chosen for computational efficiency, because we can find S_p as the solution of Helmholtz equation:

$$\begin{eqnarray}&&\left(1-{{ \Lambda }^{2}}{{\nabla}^{2}}\right){{S}_{\text{p}}}=C{{S}_{\text{i}}}\left(\mathbf{r}\right).\end{eqnarray}$$

The electrode gap is modelled as 1-dimensional interval $x\in \left[0,L\right]$, where the emitting electrode (the cathode in the case of the negative discharge study considered here) is at x  =  0, while the grounded electrode is at x  =  L  =  1 m. The system is assumed to be symmetric in the transverse direction. Voltage $\phi (0)=V(t)$ is applied at x  =  0, while x  =  L is held at $\phi (L)=0$. The discharge is started by a small initial ionization at t  =  0, x  =  0.

The curvature of ionization front $\varkappa$ is included through the expression for divergence of vectors $\parallel x$:

$$\begin{eqnarray}&&\nabla \centerdot \equiv {{\partial}_{x}}\,\,+\,\varkappa.\end{eqnarray}$$

For example, a spherically-symmetric system may be modelled by taking $\varkappa =2/r$. For the results presented here, we take a constant value $\varkappa =\text{const}>0$ for simplicity, which is equivalent to the transverse area of the discharge growing exponentially with distance. This may be justified by the streamers branching repeatedly with a fixed interval, and the transverse area being proportional to the number of streamers.

In the process of the nonlinear development of ionization growth, the ionization developed into multiple persistent peaks (seen in figure 8) with complex dynamics (i.e. moving in both directions) which may be interpreted as various luminous features of streamers. In particular, the first peak in an ionization wave may be interpreted as the streamer head, while the consequent peaks, which move in the same direction or become stationary, may be interpreted as beads. Under some conditions (specified in the next paragraph) we observed a particular class of such peaks, which exhibited the following stages of evolution: (1) an ionization peak appears at the most advanced point of the ionization wave; (2) the ionization is extinguished in the part of the gap between this peak and the emitting electrode; (3) as the voltage at the emitting electrode increases further, a reverse ionization wave separates from the peak and moves towards the emitting electrode, while a direct ionization waves moves towards it, until they collide; (4) after even further voltage increase, the ionization peak continued its movement toward the grounded electrode. In view of the observations reported above, these ionization peaks may be interpreted as the pilots which exhibit similar behavior, namely that they launch ionization waves in both directions: positive streamers towards the emitting electrode (cathode) and the negative streamers towards the grounded electrode (anode). Figure 8 presents a snapshot of the process described above (at stage 3). Note that we used a slower-growing shape of ionization function ($\alpha =3$ instead of 5.5). The higher values of power coefficient α create steeper ionization edges. The simulations with other values of α were also performed (e.g. $\alpha =1$, $\beta =0$) and they also produce similar results (i.e. formation of the 'reverse streamers' and the HV electrode current pulsations, see below). The chosen lower value of α in a 1D case creates a smoother front, and thus simulates the uncertainty in the position of the individual streamer heads (they may be distributed around some average position in x); unfortunately, the exact value of α which is best suited for this is not known to us at the current stage of research. We also chose the value of the photoionization length $\Lambda$ so that the simulation produces approximately the observed streamer velocities, while the value of C is about the same as can be obtained from experimental data [38].

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As we see, the variations in the ionization rate functional dependencies on E did not change qualitatively the result of having 'islands' of ionization and reverse ionization waves (stages 2–3 described above). By varying other parameters we preliminary conclude that this stages only appear when (1) the voltage has a stage when it gradually increases with time and (2) the photoionization effect is rather large. Although the propagation of streamers does need a large photoionization, and the 'islands' of stage 2 appeared even at small values of C, the reverse ionization waves (stage 3) only appeared when C exceeded a certain threshold.

Beside pilots and reverse streamers, another interesting outcome of the presented 1D model were the quasi-periodic current pulses at the high-voltage electrode (cathode), shown in figure 9, which reproduce, at least qualitatively, the experimentally observed pulses shown in figure 2. The current was calculated taking into account both conductivity and displacement currents:

$$\begin{eqnarray}&&I={{A}_{0}}\left({{\varepsilon}_{0}}\overset{\centerdot}{{E}}\,+\sigma E\right){{|}_{x=0}}\end{eqnarray}$$

where the area of the cathode A₀  =  0.03 m² is chosen so that the current is of the same order as those in figure 2.

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In the past, the pilots and stepping mechanism were studied numerically in long negative sparks (leaders) by 1D modeling [1, 39]. However, these models treated separately the stages of the discharge which lead to the development of the pilots and stepping and therefore also take into account transition from the streamer corona to a leader discharge. In contrast to these models, we demonstrated that the leader development is not necessary for the pilot formation, and the similar stepping stages appear automatically from the nonlinear nature of the system without artificially subdividing the process. The colliding individual streamers were considered by [20] in the context of x-ray production by high electric field, but the streamer configuration was taken as an input on the basis of previous results and only E field was calculated. The modeling of a streamer collision, taking into account the microscopic physical processes, was also performed by [25]. We emphasize that, in contrast to the presented model, the last two works describe collision of two pre-existing streamers and do not deal with the development of the global streamer discharge system.

In most existing theoretical treatments (see chapter 12.3 in [40]) the streamer channel is understood as a linear structure along which the ionization (and therefore the electric field and current) vary monotonously. For example, the ionization is highest at the head of the streamer channel and gradually decreases in the backward direction. However, the high-speed videos of sprites [41] reveal luminous 'beads' which move along each individual streamer channel. We may therefore propose a hypothesis that a streamer channel is not monotonous but has peaks of enhanced conductivity and/or field which manifest themselves as 'beads'. The presented simulation results supports this hypothesis. Namely, there are multi-peak structures in figure 8 even when the peaks are moving in the same direction (so that they are not parts of disconnected streamers). However, another interpretation of multi-peak structures in the simulation results is also possible. Since we modelled a whole group of streamers, the peaks may be just heads of separate monotonous streamer channels or groups thereof. The beads and periodic structures in ionization waves were attributed to the attachment instability by [42], who presented a linear-wave analysis of this phenomenon in their appendix A. The mechanisms included in the two models are slightly different (e.g. electron advection in [42] versus the effective curvature in the present work), so this similarity requires further investigation.

Pilots strikingly resemble high-altitude discharges known as sprites [41, 43]. Peculiar features such as glowing beads, streamer branching on beads, counter-propagating streamers originated from the beads, difference in brightness of the top and bottom parts and finally reconnection, characterise both phenomena. For visual comparison we refer to figure 16 in [17]. Two known types of sprites, carrot and column, can be directly compared to two types of pilots, as shown in section 3.3. The similarity is clear, despite the fact that in the laboratory pilots are pointed towards the sharp electrode tip, while sprites in nature originate from a dispersed charge region and appear vertically in 2D images.

The last concern of scientific community regarding different polarity of pilots and sprites has recently been eliminated. It has been known since 1999 that sprites are not uniquely associated with positive cloud-to-ground (+CG) lightnings, but can also be triggered by negative  −CG flashes [44]. However, the community continued to be skeptical in accepting sprite polarity asymmetry and existence of 'negative sprites' (private conversations at AGU/EGU meetings). Five more evidences of sprites following a negative  −CG discharge were published in 2016 [45]. With simple considerations put forward in the next section, we cannot yet explain the existence of pilots with opposite polarity. In this case, new experimental studies of positive laboratory discharges with different voltage rise times are highly desirable.

As it was mentioned in the Introduction, the pilots may transform into space leaders [1]. We may also speculate how pilots determine the polarity asymmetry between the modes of propagation of negative and positive leaders. Let us consider the conditions for formation of a system of forward and reverse streamers moving towards each other, such as the one that appeared during stage 3 of the simulation described in section 4.2, and consider the differences for two different leader polarities.

In a negative leader discharge, the electrostatic field in front of the leader converges towards its tip. The negative forward streamers, being closer to the electrode, experience a higher field than the positive reverse streamers that are initiated from a position further away from the leader tip and are on their way to encounter the negative forward streamers. This is consistent with the fact that negative streamers need a higher field to support their propagation than the positive streamers. Thus, both forward and reverse streamers may exist at the same time.

On the other hand, in a positive leader discharge, the forward streamers are positive while the reverse streamers (if they appeared) would be negative. We see that the lower field, which the reverse negative streamers would experience, cannot support their propagation. Thus, we do not expect formation of reverse streamers in the positive-leader case, the ionization gap (if it appeared) would be filled only by forward positive streamers.

The role of the decreasing field in the differences between positive and negative leader propagation was also discussed by [1], where they suggested that in the negative leader corona the forward-moving electrons attach in the lower field, which interrupts the current and leads to stepping.

Another implication of experiments and modelling is that the positive and negative streamers, which travel towards each other, will collide at a certain moment of the discharge. This may lead to a increased electric field in the gap between them, which subsequently leads to generation of high-energy electrons [24]. It has been proposed as the possible mechanism of x-ray production in laboratory spark discharges [20].

This mechanism has been modelled by [25] who found that the number of x-ray photons produced may not be sufficient to explain observations. However, we think that the electric field in [25] has been underestimated. The boundary conditions with fixed potential, represent a perfectly conducting boundary and lead to image charges, whose field reduces the field in the modelling domain. Thus, the actual field may be higher and also occupy a bigger volume. On the other hand, a higher field between the two colliding streamers would also lead to an increased streamer velocity, which would reduce the time of the existence of the region with high field. Since there are two counteracting effects, it is not clear without repeating the full simulation whether corrected boundary conditions would lead to increase of x-ray production. Here, we may note that the observations confirmed the coincidence of occurrence of x-rays in laboratory sparks with colliding positive and negative streamers [24], even if the exact mechanism of electron acceleration is still open for discussion.