The emission of energetic electrons from the complex streamer corona adjacent to leader stepping

We here employ a 2.5D cylindrical particle-in-cell Monte Carlo code with two spatial (r, z) and three velocity coordinates (v_r, v_z, v_θ) which has been used before (see e.g. [25, 50, 51]) and allows us to trace individual (super)electrons as well as to monitor the formation of bipolar streamers from a charge-neutral electron–ion patch

$$\begin{eqnarray}&&{n}_{e,i}(r,z,t=0)={n}_{e,0}\cdot \exp \left(-\left({r}^{2}+{\left(z-{z}_{0}\right)}^{2}\right)/{\lambda }_{0}^{2}\right)\end{eqnarray} \tag{ 1 }$$

centered in the middle of the simulation domain, i.e. z₀ = L_z/2, with a peak density of n_e,0 = 10²⁰ m⁻³ and a Gaussian length of λ₀ = 0.5 mm which is in agreement with initial conditions used in [27, 43, 52]. In nature, the initial density or shape might vary, see e.g. [53], but this is not crucial for the cases considered here; instead we need to ensure that a streamer incepts.

The size of the simulation domain, displayed in figure 2, is (L_r, L_z) = (6, 80 mm) (as in [27]) on a mesh with 150 × 1600 grid points. This grid is used to solve the Poisson equation

$$\begin{eqnarray}&&{\rm{\Delta }}\phi ={e}_{0}/{\epsilon }_{0}\cdot \left({n}_{i}-{n}_{e}\right)\end{eqnarray} \tag{ 2 }$$

for the electrostatic potential ϕ taking into account the effect of space charges. At the boundaries r = 0, L_r, we use the Neumann condition $\partial \phi /\partial r=0$, and at the boundaries z = 0, L_z, we use the Dirichlet conditions ϕ(r, 0) = 0 and ϕ(r, L_z) = E_amb · L_z where E_amb is the ambient electric field. We here consider two different ambient fields: E_amb = 50 kV cm⁻¹ ≈ 1.56E_k [27] and E_amb = 0.5E_k where we here and throughout the paper refer to E_k ≈ 3.2 MV m⁻¹ as the classical breakdown field in air at STP. E_amb = 50 kV cm⁻¹ is chosen as an upper limit for the electric field at the tip of a leader channel [54]. However, as we will discuss below, the complexity of the streamer corona will give rise to regions with a substantial amount of preionization which could potentially lower the local electric field; for this particular reason, we have chosen an ambient field of 0.5E_k. In the current simulation set-up, the applied electric fields are equivalent to voltages of 400 and 128 kV.

Zoom In Zoom Out Reset image size

We here trace individual (super)electrons interacting with ambient air. Unlike fluid models, tracing individual (super)electrons with a particle code allows us not only to obtain streamer properties such as the electron density or electric field distribution, but also to estimate the electron energy distribution. We include electron impact ionization, elastic and inelastic scattering as well as electron attachment and bremsstrahlung. Additionally, we apply a photoionization model where photons emitted from excited nitrogen ionize oxygen molecules locally and liberate additional electrons. More details of the applied Monte Carlo model are described in [25, 55].

Since electrons ionize molecular nitrogen and oxygen, the electron number grows exponentially leading to an electron avalanche and eventually a streamer. Due to limited computer memory, we use an adaptive particle scheme [25] conserving the charge distribution as well as the electron momentum such that every simulated electron is a superelectron representing $w\lesssim {10}^{5}$ physical electrons.

In Monte Carlo particle simulations, we include the collisions of electrons with ambient air where the nitrogen and oxygen molecules are put at random positions as an implicit background. The probability P_c of a collision of an air molecule with an electron with velocity v_e within the time interval Δt is ${P}_{c}=1-\exp \left(-{n}_{{\rm{air}}}{v}_{e}\sigma {\rm{\Delta }}t\right)$ where n_air is the number density of ambient air and σ the collision cross section.

Previous experiments and simulations [44, 47, 49, 56, 57] suggest that shock waves and thermal expansion by leaders and also by the small-scale discharge modes, the streamers, are capable of perturbing the vicinity of their location up to 80% of the ambient air level [47]. Within several tens of ns, the complex streamer corona simultaneously consists of a multitude of streamers [58] amongst which some reach large currents. Therefore, they develop ionization-heating instability such that they are capable of disturbing the air density experienced by other streamers. For n_air, we therefore choose the ansatz

$$\begin{eqnarray}&&{n}_{{air}}(r)={n}_{0}\left(1-\xi \cos \left(r\cdot \pi /{L}_{r}\right)\right),\end{eqnarray} \tag{ 3 }$$

depicted in figure 3, with a global minimum on the symmetry axis (r = 0) and a global maximum on the outer boundary (r = L_r). The sinusoidal form has been computed by Marode et al [44] and is here meant to capture the minimum air density in the proximity of the channel axis driving air molecules to the exterior boundary. Otherwise, the actual form of n_air is not crucial and we here limit ourselves to ξ = 0, 0.25, 0.5 since Marode et al calculated perturbation levels of 50% as an upper limit in the vicinity of a streamer. Note that the time t_D of air molecules to diffuse back to uniform density is in the order of ${t}_{D}\simeq {L}_{r}^{2}/{D}_{{\rm{air}}}\approx$ 1.8 s with D_air ≈ 2 × 10⁻⁵ m² s⁻¹ [59] which is much larger than the simulation time of the order of several nanoseconds allowing us to assume a stationary distribution of air molecules.

Zoom In Zoom Out Reset image size

As discussed by Babich et al [27, 42], streamers leave behind residual ionization affecting the motion of successive streamer and leader channels. The reminiscent density n_pre of the previous streamer channel is modeled by

$$\begin{eqnarray}&&{n}_{{\rm{pre}}}={n}_{{\rm{pre}},0}\cdot \exp (-{r}^{2}/{\lambda }_{{\rm{pre}}}^{2}),\end{eqnarray} \tag{ 4 }$$

where ${n}_{{\rm{pre}},0}={10}^{10}\mbox{--}{10}^{13}$ m⁻³ determines the peak density and ${\lambda }_{{\rm{pre}}}=0.5{\lambda }_{0},1.0{\lambda }_{0},1.5{\lambda }_{0}$ the width of the preionized channel. This approach is advocated, firstly because each streamer discharge has its own characteristic minimal radius depending on the streamer velocity and the ambient gas density [60], secondly because the charge and the width of the preionized channel diffuse with time [42]. In addition, Sadighi et al [61] have shown that such levels of preionization prevent streamers from branching.

After a preceding discharge, the time to readjust the electric field is in the order of some ns μs [51] which is significantly smaller than the diffusion time t_D. Hence, the screening of the electric field is negligible in the current set-up which justifies to run simulations in E_amb = 1.56E_k given a fixed potential difference.

Figure 2 shows the initial electron–ion patch together with the preionized channel (λ_pre = λ₀ and n_pre,0 = 10¹² m⁻³). It illustrates how the initial electron–ion patch is embedded in the preionized channel. Note that the channel is extended radially as much as the electron–ion patch which is not visible because the colorbar is limited down at n_e = 10¹¹ m⁻³.

In this section we will give a brief overview how to understand and how to calculate the production rate of runaway electrons; we therefore adopt the discussion of equations (10) and (11) in [27]. The runaway rate ν_run(E) at STP as a function of the electric field strength, i.e. the number of runaway electrons per unit time, used here is computed first for nitrogen through Monte Carlo simulations [62]; these Monte Carlo results are then fitted by an exponential function [27] to obtain

$$\begin{eqnarray}\begin{array}{rcl}{\nu }_{{\rm{run}}}(E) & = & 3.5\times {10}^{-24}\ {{\rm{s}}}^{-1}\\ & \times & \exp (-{(2.166\times {10}^{-7}{{\rm{m}}{\rm{V}}}^{-1}\cdot E)}^{2}\\ & & +3.77\times {10}^{-6}\ {{\rm{m}}{\rm{V}}}^{-1}\cdot E).\end{array}\end{eqnarray} \tag{ 5 }$$

Although this rate is valid for nitrogen only, we here use it for air since the cross sections for the ionization and excitation of nitrogen and oxygen are almost identical except for the very small range below 20 eV and since the percentage of oxygen in air is ≈20% only.

Figure 4 shows that for fields of up to 10E_k ν_run varies between approximately 10⁻²⁴ and 10⁸ s⁻¹. For electric fields above 6E_k, it is ν_run(E = 6E_k) ≈ 2.94 s⁻¹, ν_run(E = 7E_k) ≈ 990.66 s⁻¹ and ν_run(E = 8E_k) ≈ 1.28 × 10⁵ s⁻¹; hence, the local electric field has a significant effect on the production rate of runaway electrons. The rate ν_run allows us to estimate the number k_RE of runaway electrons per unit length. For a negative front moving with velocity v_neg, the temporal variation of the number N_RE of runaway electrons obeys the differential equation dN_RE(t) = k_RE(t)v_negdt which is equivalent to

$$\begin{eqnarray}&&{k}_{{RE}}(t)=\displaystyle \frac{{\dot{N}}_{{RE}}(t)}{{v}_{{neg}}(t)},\end{eqnarray} \tag{ 6 }$$

where the time depending number N_RE of runaway electrons is [27]

$$\begin{eqnarray}&&{N}_{{RE}}(t)={\int }_{\bar{t}=0}^{t}{\int }_{{\bar{V}}_{{\rm{sim}}}}{\nu }_{{\rm{run}}}\left(E\left(\bar{r},\bar{z},\bar{t}\right)\right)\cdot {n}_{e}\left(\bar{r},\bar{z},\bar{t}\right){\rm{d}}\bar{V}{\rm{d}}\bar{t}\end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray}&&=2\pi {\int }_{\bar{t}=0}^{t}{\int }_{\bar{r}=0}^{{L}_{r}}{\int }_{\bar{z}=0}^{{L}_{z}}{\nu }_{{\rm{run}}}\left(E\left(\bar{r},\bar{z},\bar{t}\right)\right)\cdot {n}_{e}\left(\bar{r},\bar{z},\bar{t}\right)\bar{r}{\rm{d}}\bar{r}{\rm{d}}\bar{z}{\rm{d}}\bar{t},\end{eqnarray} \tag{ 8 }$$

where the volume integral over the simulation domain takes into account the variation of the electron density and the electric field in space and time. Note that the electron density is calculated from Monte Carlo simulations whilst ${\nu }_{{run}}$ is the analytic fit (5), and that any explicit dependence on r and z is integrated out in (8). Yet, N_RE depends on time t; so does the evolution of the streamer fronts. Therefore, N_RE depends on the streamer length implicitly. However, because of this implicit dependency, we use the time derivative of N_RE and the streamer velocity instead of the spatial derivative of N_RE.

Zoom In Zoom Out Reset image size

Babich et al [27] have already solved the fluid equations of negative streamers in preionized air without the effect of air perturbations focusing on the production of runaway electrons in a field of E_amb = 50 kV cm⁻¹. Figure 5 compares the on-axis electron density (a) and the on-axis electric field (b) of the negative streamer front for n_pre,0 = 10¹² m⁻³ computed by Babich et al and computed by MC particle simulations showing a very good agreement in the streamer channel. There is a slight deviation after 2 ns which is compensated again after 3 ns. In all considered cases, however, our results show fluctuations which do not occur in the previous results. Yet, this is not surprising since a particle code normally shows more fluctuations than a fluid code, see. e.g. [63]. At the tips, the electric field peaks as smoothly as for the fluid code. Beyond the streamer channel the electron density is larger than the electron density calculated by the fluid equations. In contrast to the fluid model, the particle Monte Carlo code allows us tracing the space-time energy distribution of each individual electron and, as a consequence, computing the spatial electron distribution more accurately. The plateau-like behavior of the electron density after 2 and 3 ns is an evidence of the polarization self-acceleration of the electron swarm moving ahead of the streamer body, as observed in experiments with nanosecond discharges at STP conditions [17, 64, 65]. Such electrons, ahead of the streamer and therefore in the channel, are continuously accelerated in the strong moving field at the streamer tip. They overtake the main bulk of electrons and are therefore disconnected from the main streamer body [66, 67]. The applied particle code is capable of capturing this effect unlike the fluid approach. Therefore, we reason that our results are more accurate than the ones obtained by the fluid model.

Zoom In Zoom Out Reset image size

We here commence our study of the streamer evolution in preionized air only, similar to the set-up discussed in [27].

Figure 6 compares the electron density and the electric field in non-ionized air with the electron density and the electric field in preionized air with ${n}_{{\rm{pre}},0}={10}^{12}$ m⁻³. After 1.62 ns the streamer length and the electric field are equally large irrespective of n_pre,0. After 2.84 ns, however, the streamer in preionized air overtakes the streamer in non-ionized air and the field at the streamer tips is slightly more enhanced. In addition the density of the preionized channel has reached values of above 10¹⁵ m⁻³ distributed uniformly beyond the streamer tips which is not the case in the absence of preionization. Finally, after 3.44 ns, the streamer channel in preionized air has completely grown into the preionized channel (within the simulation domain) and has proceeded approximately twice as much as the streamer in non-ionized air.

Figures 7(a) and (b) show the streamer velocities of the negative and positive fronts in non-perturbed air. It shows that initially streamers move comparably fast for different levels of preionization. After a few ns, however, streamers in the highest level of preionization, here n_pre,0 = 10¹³ m⁻³, accelerate more effectively than streamers in non-ionized air, followed by streamers in air with descending order of preionization.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

The different acceleration of streamers in non-ionized and in preionized air is a result of the space-charge induced electric field. In the early stages of the streamer development there is no significant contribution of the preionized air channel. However, after several time steps depending on ${n}_{{\rm{pre}},0}$, the streamers grow into the preionized channel and hence the electric field at the tips energizes the channel electrons in the vicinity of the streamer head. These channel electrons subsequently gain enough energy to ionize molecular nitrogen and oxygen and create additional space charge in the proximity of the streamer head; thus, the electric field and the velocities of streamers in highly ionized air exceed the field and velocities of streamers in less ionized air.

Let us now turn to the production properties of runaway electrons from streamers in preionized air as they might occur after the reconnection of the space stem with a stepping leader. Note that the runaway threshold energy depends on the electric field [13]: for an electric field of 1.56E_k sea-level equivalent the runaway threshold energy amounts to 5 keV; for a field of 0.5E_k, it amounts to approx. 21 keV. These are, however, the runaway thresholds in a homogeneous electric field; in concrete streamer simulations, the field at the tip actually grows to field strengths larger than 0.5E_k or 1.56E_k and therefore, the runaway threshold energy becomes smaller than 5 or 21 keV.

Table 1 shows that for E_amb = 1.56E_k ${\max }_{t}\left({k}_{{RE}}\right)$, the maximum of k_RE(t) over time, is smallest in non-ionized air and increases until n_pre,0 = 10¹¹ m⁻³ since the maximum electric field at the streamer tips increases with the level of preionization until n_pre,0 = 10¹¹ m⁻³. This increase results from the more enhanced electric field at the streamer tips for larger preionization. In turn, above n_pre,0 = 10¹¹ m⁻³, the high electron density in the vicinity of the streamer front screens the electric field at the tips and thus limits the maximum field strength and the maximum number of runaway electrons. Table 1 also compares our results with the runaway rate k_RE,Babich by Babich et al [27] which is defined in a different manner:

$$\begin{eqnarray}&&{k}_{{RE},{\rm{Babich}}}=\displaystyle \frac{{N}_{{RE}}({t}_{{z}_{f}=3{\rm{cm}}})-{N}_{{RE}}({t}_{{z}_{f}=2.9{\rm{cm}}})}{0.1\ {\rm{cm}}}.\end{eqnarray} \tag{ 9 }$$

This definition includes the number of runaway electrons produced between the time steps when the front has reached z_f = 2.9 and 3.0 cm, to the end of their simulations. Since previous simulations and our simulations behave slightly differently, we compare k_RE,Babich with ${\max }_{t}\left({k}_{{RE}}\right)$ calculated from our simulations. k_RE,Babich reaches its maximum for n_pre,0 = 10¹² m⁻³ and amounts to 2 × 10⁹ m⁻¹ whereas ${\max }_{t}\left({k}_{{RE}}\right)$ in our simulations reaches its maximum for n_pre,0 = 10¹¹ m⁻³ and amounts to ≈2.12 × 10⁹ m⁻¹. Thus, we find a good agreement between these two maxima within one order of magnitude of n_pre,0. Additionally, we here confirm the findings by Babich et al [27] that the number of runaway electrons is enhanced if streamers move in preionized air. We also observe both with Monte Carlo or fluid simulations that the runaway rate per unit length decreases for smaller and larger preionization densities n_pre,0. Since, we observe the same tendency for the two different models, we reason that the one order of magnitude difference is due to the different definitions of (6) and (9): (6) gives the production rate at each moment of time during the simulation, whereas (9) determines the production rate only at the final stage of the streamer development.

Table 1.  The maximum number of produced runaway electrons per unit length, ${\max }_{t}\left({k}_{{RE}}\right)$, the maximum electron energy $\,{\max }_{t}\,{E}_{{\rm{kin}}}(t)$, and the maximum electric field max_t E as a function of n_pre,0. For comparison, we also show k_RE,Babich (9).

npre,0 (m−3)	$\,{\max }_{t}\,{k}_{{RE}}(t)$ (m−1)	kRE,Babich (m−1)	$\,{\max }_{t}\,{E}_{{\rm{kin}}}(t)$ (eV)	$\,{\max }_{t}\,E$ (MV m−1)
0	2.92 × 103	—	252	22.43
1010	1.88 × 108	2 × 104	358	31.05
1011	2.12 × 109	5 × 107	16 883	33.08
5 × 1011	1.68 × 106	—	13 579	23.04
1012	8.06 × 106	2 × 109	10 160	23.03
1013	7.48 × 106	9 × 108	1367	22.98

Table 1 also shows that similarly to the maximum runaway rate, the maximum electron energy increases until n_pre,0 ≈ 10¹¹ m⁻³ and decreases for higher levels of preionization. Whereas the maximum electron energy in non-ionized air is approx. 250 eV, the maximum electron energy for n_pre,0 = 10¹¹ m⁻³ amounts to approximately 17 keV and decreases to approximately 1 keV for n_pre,0 = 10¹³ m⁻³ which is sufficiently high for electrons to overcome friction and initiate a relativistic runaway electron avalanche [22].

After we have discussed the sole effect of preionization, we now focus on the effect of air perturbations coinciding with the preionization of air as it might occur in the complex streamer corona in the proximity of lightning leaders.

As we have shown in [50], streamers move faster in perturbed air with $\xi \leqslant 0.5;$ similarly we have discussed in section 3.2 that streamers move faster in preionized air with n_pre,0 = 10¹¹–10¹⁵ m⁻³. As a result from the combination of both effects, we have observed in our simulations that streamers move fastest in perturbed and preionized air with $\xi \leqslant 0.5$ and n_pre,0 = 10¹¹–10¹⁵.

Figure 8(a) shows ${\max }_{t}\left({k}_{{RE}}\right)$, as a function of n_pre,0 for different levels of air perturbation. It shows that in non-ionized air the number of runaway electrons increases with the level of air perturbation. As for the streamer velocities, the increase of the number of runaway electrons results from the enhanced reduced electric field in the vicinity of the symmetry axis (see [51]). If the effects of air perturbation and of preionization are combined, we observe a reversed trend compared to in uniform air because of the significant growth of the electron density in the vicinity of the streamer head in perturbed air even for small levels of preionization. Instead of increasing, the number of runaway electrons in perturbed air decreases with the preionization level. For n_pre,0 ≈ 10¹⁰ m⁻³, ${\max }_{t}\left({k}_{{RE}}\right)$ is approximately 10⁹ m⁻¹ for all levels air perturbation; for larger n_pre,0, the number of runaway electrons in non-perturbed air exceeds that in perturbed air. Although the runaway rate decreases for n_pre,0 = 10¹³ m⁻³, ${\max }_{t}\left({k}_{{RE}}\right)$ is some orders of magnitude higher in perturbed air than in uniform and non-ionized air. Ultimately, the number of runaway electrons is highest for ξ = 0.5 in non-ionized air and for n_pre,0 ≈ 10¹¹ m⁻³ in uniform air with ξ = 0.

Figure 8(b) compares the maximum electron energy in perturbed air with that in non-perturbed air for n_pre,0. For a perturbation level of ξ = 0.25, the maximum electron energy varies from approximately 4–13 keV for n_pre,0 < 10¹¹ m⁻³ which is one order of magnitude larger than in uniform air and high enough to start runaway electron avalanches. Above ${n}_{{\rm{pre}},0}\gtrsim {10}^{11}$ m⁻³, the maximum electron energy varies between approximately 13 and 1 keV which is comparable to ${\max }_{t}\left({E}_{{\rm{kin}}}\right)$ in non-perturbed air.

Zoom In Zoom Out Reset image size

For ξ = 0.5, the maximum electron energy is highest and varies between 200 and 25 keV which is significant enough to trigger a relativistic avalanche for any level of preionization. In contrast to ξ = 0 and ξ = 0.25, there is not a distinct maximum between n_pre,0 = 10¹¹ and 10¹² m⁻³.

Conclusively, we identify three regimes: (i) for air perturbations below $\xi =0.5$ and for preionization levels below n_pre,0 = 10¹¹ m⁻³, the air perturbation determines the maximum electron energy whereas (ii) for ${n}_{{\rm{pre}},0}\gtrsim {10}^{11}$ m⁻³ the influence of the air perturbation is negligible and the maximum electron energy is determined by n_pre,0; (iii) for air perturbations as large as ξ = 0.5, the maximum electron energy is mainly determined by the air perturbation with minor effect of the preionization level.

So far, we have discussed the streamer velocity and the effect of preionizing and perturbing air on the production of runaway electrons in an ambient field of 1.56E_k. However, the multitude of streamers adjacent to leader stepping also effects the electric field distribution in the corona. As an example, we therefore now turn to the production of runaway electrons in a subbreakdown field of 0.5E_k. Note that the value of this field refers to the electric field strengths in uniform air. Thus, E_amb = 0.5E_k means 0.5 times the breakdown field in uniform air with density n₀. Hence, if ξ = 0.5, a field of 0.5E_k is the same field strength as the breakdown field strength in unperturbed air for r = 0 decreasing for r > 0. For ξ < 0.5 the electric field strength would be below the breakdown field value in the whole simulation domain and therefore we only consider ξ = 0.5 and E_amb = 0.5E_k.

For this particular set-up, figure 9 shows the streamer velocities at the negative (a) and positive (b) front as a function of time for different levels of preionization. We observe a similar dependency on preionization as in E_amb = 1.56E_k. For time steps $\lesssim 15-20$ ns, the preionization effect is negligible; for larger time steps, streamers accelerate more efficiently when wave fronts grow into preionized air with larger n_pre,0. Such waves create more space charges, thus induce higher self-consistent electric fields and lead to more efficient streamer acceleration.

Zoom In Zoom Out Reset image size

For comparison, the dotted line shows the front velocity in non-perturbed air with ${n}_{{\rm{pre}},0}\equiv 0$, $\xi \equiv 0$ and E_amb = E_k. This comparison reveals that the streamer fronts move significantly slower for ξ = 0.5 and E_amb = 0.5E_k than for ξ = 0 and E_amb = E_k because of the non-uniformity of the air perturbation. Only at the symmetry axis where n_air = 0.5n₀ is the reduced electric field E/n_air comparably large as E_k/n₀ in non-perturbed air. Since the reduced electric field decreases with r, the streamer motion is damped for r > 0, and the streamers move slower in perturbed air.

This effect of air perturbations on the reduced electric field is visualized in figure 10. It shows the electron density for ξ = 0 and E_amb = E_k and for ξ = 0.5 and E_amb = 0.5E_k. It shows that after 5 ns the streamer in uniform air moves faster and is thicker and more diffuse than in perturbed air. The reason for this is again that the reduced electric field in perturbed air decreases as a function of r and thus generates a quenching effect on the streamer in E_amb = 0.5E_k.

Zoom In Zoom Out Reset image size

Panels (a) and (b) of figure 8 compare the maximum runaway production rate and the maximum electron energy for ξ = 0.5 and E_amb = 0.5E_k with the rates and energies of electrons from streamers in E_amb = 1.56E_k. There are two effects contributing to the production runaway electrons in subbreakdown fields in our simulations: firstly, the air perturbation level ξ = 0.5 ensures that for r = 0 the reduced electric field is effectively as large as the breakdown field in unperturbed air; secondly, as we have seen in section 3.3, levels of preionization between 10¹⁰ and 10¹³ m⁻³ enhance the production of runaway electrons. For ${n}_{{\rm{pre}},0}\lesssim {10}^{12}$ m⁻³ the runaway rates and the maximum electron energy are smaller than in uniform air. For ${n}_{{\rm{pre}},0}\gtrsim {10}^{12}$ m⁻³, however, ${\max }_{t}\left({k}_{{RE}}\right)$ becomes comparable to the one in uniform air in 1.56E_k. The maximum electron energy in this set-up increases with n_pre,0 and reaches approximately 3 keV which allows the formation of runaway electron avalanches in subbreakdown fields.

Whereas the previous results were obtained for λ_pre = λ₀, we now discuss how the streamer velocities and the number of runaway electrons change for λ_pre = 0.5λ₀ and for λ_pre = 1.5λ₀.

Note that the absence of any preionization is equivalent to ${\lambda }_{{\rm{pre}}}\to 0$ and the presence of uniform background ionization in the complete simulation domain is equivalent to ${\lambda }_{{\rm{pre}}}\to \infty$. In our simulations we have observed that in non-perturbed air, both the positive and the negative streamer front move faster for small channel radii and slower for large channel radii. In contrast, for ξ = 0.5, there is no significant difference between λ_pre = 0.5λ₀ and λ_pre = λ₀; yet, for a wide channel with λ_pre = 1.5λ₀ the streamer moves slower than in small channels. This is consistent with modeling results of streamer heads under different levels of uniform background ionization [43] which have shown that streamers move faster in the absence of any background ionization than in the presence of uniform background ionization.

Figure 11 compares ${\max }_{t}\left({k}_{{RE}}\right)$ and ${\max }_{t}\left({E}_{{\rm{kin}}}\right)$ for different n_pre,0 and λ_pre. Note that there is no channel for n_pre,0 = 0, thus we only use the values for n_pre,0 = 0 as comparison. These panels show that for ${\lambda }_{{\rm{pre}}}\ne {\lambda }_{0}$, the maximum generation rate of runaway electrons is slightly higher than in preionized air with λ_pre = λ₀. Still, the overall trend is the same regardless of λ_pre. In non-perturbed air, the maximum generation rate ${\max }_{t}\left({k}_{{RE}}\right)$ of runaway electrons reaches its maximum for ${\lambda }_{{\rm{pre}}}\ne {\lambda }_{0}$ at n_pre,0 = 10¹⁰ m⁻³, and decreases for larger preionization levels. In perturbed air, $\max \left({k}_{{RE}}\right)$ is maximal for n_pre,0 = 0 and decreases for n_pre,0 > 0.

Zoom In Zoom Out Reset image size