Anisotropic Electron Heating in Turbulence-driven Magnetic Reconnection in the Near-Sun Solar Wind

We let the plasma system evolve from its initial condition until a turbulent energy cascade has fully developed and starts to slowly decay. This occurs when the rms value of the current density, J , reaches a maximum/plateau (e.g., Franci et al. 2015a, 2017). Figure 1 shows the time evolution of some fundamental space-averaged quantities that allow us to follow the development of the turbulent cascade, identifying different phases. Figure 1(a) shows the rms of the components of J both parallel and perpendicular to the ambient magnetic field B ₀. The former provides the dominant contribution to the total current and reaches a maximum at ${t}_{\mathrm{peak}}=11.4\,{{\rm{\Omega }}}_{i}^{-1}$, marked by a green dashed vertical line. The following analysis will therefore be performed at this time. The other component, ${{\boldsymbol{J}}}_{\perp }^{\mathrm{rms}}$, is initially almost zero and quickly grows, yet remaining much smaller than its parallel counterpart at all times. This is related to the rapid development of parallel magnetic fluctuations, as shown below. The eddy turnover time corresponding to the initial injection scale, ${k}^{\mathrm{inj}}{d}_{i}\simeq ({k}_{x}^{2}+{k}_{y}^{2})\,{d}_{i}\simeq 1.1$, is the time associated with nonlinear energy transfers at t = 0 and can be estimated as ${t}_{\mathrm{NL}}^{\mathrm{inj}}\sim {[{k}^{\mathrm{inj}}\delta {B}_{k}^{\mathrm{inj}}]}^{-1}\sim {[{k}^{\mathrm{inj}}\delta {B}^{\mathrm{rms}}]}^{-1}\sim 2.3\,{{\rm{\Omega }}}_{i}^{-1}$. A turbulent energy cascade develops fully from the injection scale down to the electron scales in a few nonlinear times, as t_peak ∼ 5 t_NL. This is similar to what was previously observed in hybrid simulations with similar initial conditions (e.g., Franci et al. 2015a, 2015b), where about 10 nonlinear times were required.

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Figure 1(b) shows the time evolution of the maximum of the magnitude of the current density computed over the whole simulation box. This starts increasing after about one t_NL and then reaches a first maximum at ${t}_{\mathrm{rec}}\simeq 4.9\,{{\rm{\Omega }}}_{i}^{-1}$, which is after about 2 t_NL. In Franci et al. (2017), we have provided numerical evidence of the link between the time of the first maximum of ∣ J ∣ and the onset of magnetic reconnection events. Here we observe the same, as we see clear signs of reconnection in the magnetic and current structures starting at t ≃ t_rec (not shown). Figure 1(c) shows the rms of the in-plane and out-of-plane components of the magnetic field (red solid and dashed, respectively) and of the ion bulk velocity (light blue). The behavior of ${{\boldsymbol{B}}}_{\perp }^{\mathrm{rms}}$ and ${{\boldsymbol{u}}}_{i,\perp }^{\mathrm{rms}}$ is qualitatively the same as previously observed in hybrid simulations (see Figure 1 of Franci et al. 2015a): after a quick readjustment of the initial conditions, the former slightly increases before slowly decreasing, while the latter decreases during the whole evolution. As a result, the initial small excess of magnetic energy further increases, reaching a maximum at about half the simulation and then remains almost constant. The parallel components $\delta {{\boldsymbol{B}}}_{\parallel }^{\mathrm{rms}}$ and ${{\boldsymbol{u}}}_{i,\parallel }^{\mathrm{rms}}$, which are initially zero, quickly start to increase until they reach an almost constant and comparable value, which is a few times smaller than their perpendicular counterparts. This indicates that the levels of compressibility and magnetic compressibility that spontaneously form are relatively small but not negligible. It is interesting to note that $\delta {{\boldsymbol{B}}}_{\perp }^{\mathrm{rms}}$ reaches a maximum at $t=5\,{{\rm{\Omega }}}_{i}^{-1}\simeq {t}_{\mathrm{rec}}$ and starts decreasing when reconnection events start occurring.

Figure 2 shows the isocontours of the energy in the magnetic fluctuations, $| \delta {\boldsymbol{B}}{| }^{2}=| {\boldsymbol{B}}-{{\boldsymbol{B}}}_{0}{| }^{2}=\delta {{\boldsymbol{B}}}_{x}^{2}+\delta {{\boldsymbol{B}}}_{y}^{2}+\delta {{\boldsymbol{B}}}_{z}^{2}$ (panels (a) and (b)), and in the current density, ∣ J ∣², both in the whole 32 d_i × 32 d_i simulation domain (panels (c) and (d)) and in an 8 d_i × 8 d_i subdomain (bottom panels). Coherent magnetic structures, i.e., vortices or islands, are embedded in a more chaotic environment where stretched and twisted shapes emerge. The vortices have radii from a few times d_i down to a few times d_e . Gradients of the magnetic field occur at smaller scales, as the strongest current structures have a width of the order of d_e . For an easy comparison by eye, we have also drawn circles with radius r = d_i and r = d_e in Figures 2(a)–(c) and lines with length l = d_i and l = d_e in Figures 2(b)–(d).

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Figure 2 has shown that the turbulent structures are randomly oriented, as expected given that the ambient magnetic field is orthogonal to the simulation plane so there is no privileged direction in the plane. We can then reasonably assume the 2D spectra of the turbulent fluctuations to be statistically isotropic and analyze the spectral properties of each field by computing its omnidirectional power spectrum P. The top panel of Figure 3(a) shows the power spectra of the fluctuations of the magnetic field B , of the electric field E , and of the ion and electron bulk velocities u _i and u _e at t = t^peak. Black vertical dashed lines mark the wavenumbers corresponding to the ion and electron inertial length, d_i,e , and gyroradius, ρ_i,e . For each field, a dashed line with the same color indicates the noise level, estimated by computing the power spectrum at t = 0 when only the large-scale initial fluctuations should be present. The bottom panel of Figure 3(a) shows the signal-to-noise ratio (S/N) for each field. In the following, we will consider the power spectra reliable (meaning that their shape is not affected by the noise) at those scales where S/N > 3. This holds for k_⊥ d_i ≳ 20 (or, equivalently, k_⊥ d_e ≳ 2) for P _E and ${P}_{{{\boldsymbol{u}}}_{i}}$ and for k_⊥ d_i ≳ 45 (k_⊥ d_e ≳ 4.5) for P _B and ${P}_{{{\boldsymbol{u}}}_{e}}$. This ensures that our modeling of the magnetic field power spectrum is meaningful around the electron scales and for a further half a decade in wavenumber above k_⊥ d_e = 1. The top panel of Figure 3(a) suggests that both P _B and ${P}_{{{\boldsymbol{u}}}_{e}}$ may exhibit a power-law behavior with different spectral indices in different ranges above and below the electron scales. The power spectrum of the ion bulk velocity, ${P}_{{{\boldsymbol{u}}}_{i}}$, follows the magnetic field's at large scales, with a slightly smaller level that is compatible with the excess of magnetic over kinetic energy (residual energy) typically observed in the inertial range. Around k_⊥ ρ_i ≃ 1, it starts diverging from P _B and gets steeper. Compensating by different powers of k_⊥ (not shown), we find that it exhibits a power-law behavior with a spectral index of −4.5 in the range 4 ≲ k_⊥ d_i ≲ 10. At smaller scales, it starts flattening just before the noise level is reached. Finally, P _E at sub-ion scales is much less steep than P _B , so that the level of the electric field fluctuations is much larger than the magnetic field's for k_⊥ d_i ≳ 5. In order to provide a more quantitative characterization of P _B and ${P}_{{{\boldsymbol{u}}}_{e}}$, Figure 3(b) shows them compensated by different powers of k_⊥. The top panel of Figure 3(b) shows that the magnetic field power spectrum can be modeled by ${P}_{{\boldsymbol{B}}}\propto {k}_{\perp }^{{\alpha }_{B}}$ with α_B ≃ −11/3 for almost a decade at k_⊥ d_e < 1 and −5 for a shorter interval at k_⊥ d_e > 1. Correspondingly, we observe ${P}_{{{\boldsymbol{u}}}_{e}}\propto {k}_{\perp }^{{\alpha }_{e}}$ with α_e ≃ −5/3 and −3 in the two intervals of scales, respectively. The relation α_e = α_B + 2 can be easily explained by considering the definition of the current density, J = n_i u _i − n_e u _e , and Ampère's law, which links it to magnetic field, J = ∇ × B (where the displacement current has been neglected). Due to charge neutrality, n_i = n_e = n and, assuming that n = n₀ + δ n with δ n ≪ n₀, we get u _e ∝ ∇ × B . In Fourier space this reads u _e,k ∝ k × B _k , from which we obtain ${P}_{{{\boldsymbol{u}}}_{e}}\propto {k}_{\perp }^{2}{P}_{{\boldsymbol{B}}}$ (since in 2D k = k _⊥). At smaller scales, k_⊥ d_e ≳ 2.5, there is a hint of another possible power-law range in ${P}_{{{\boldsymbol{u}}}_{e}}$, with α_e ≃ −5. This interval is very narrow, just about a factor of 2 in wavenumber, as the power spectrum quickly reaches the noise level. Although we cannot consider this power-law behavior and its slope reliable enough, the further steepening of the power spectrum is evident from the top panel of Figure 3(a). In the same interval, the magnetic field power spectrum also steepens accordingly, although α_e = α_B + 2 does not seem to hold perfectly anymore, possibly due to the effect of the large noise in the density due to which the assumption δ n ≪ n₀ might not hold anymore and/or to the fact that at large wavenumbers (and frequencies) we cannot neglect the displacement current anymore. The fact that at the smallest scales ${P}_{{{\boldsymbol{u}}}_{e}}$ seems to exhibit still a power-law behavior ${P}_{{{\boldsymbol{u}}}_{e}}$ while P _B seems to follow approximately might represent a hint that the turbulent cascade is still proceeding at k_⊥ ρ_e ≳ 1, but the dynamics might be dominated by the electron processes rather than by the magnetic fluctuations. Investigating this will require new simulations with an even higher resolution and larger number of electrons per cell, in order to further improve the accuracy of the power spectra below the electron scales.

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As mentioned in Section 1, there is no consensus in the literature on the shape of the magnetic field power spectrum across the electron characteristic scales and, more precisely, on whether it exhibits a double power-law behavior or rather an exponential decay. So far, we have discussed how two power laws with a spectral index of −11/3 and −5 seem to provide a good modeling for P _B over a little more than a full decade across the electron scales, more specifically for 0.25 ≲ k_⊥ d_e ≲ 4.

We have also compensated the magnetic field power spectrum by the inverse of an exponential cutoff of the form ${P}_{{\boldsymbol{B}}}\propto {k}_{\perp }^{-8/3}\exp (-c\,{k}_{\perp }{\rho }_{e})$, following the empirical model first suggested by Alexandrova et al. (2012). The result is shown by the green line in the top panel of Figure 3(b), where we have set c = 1. This seems to provide a good approximation for P _B , as it is almost a horizontal line for a little more than a decade in wavenumber, although with significant oscillations. The value of c that we have set here, however, differs from both c = 1.4 found by Alexandrova et al. (2012) based on Cluster observations at 1 au and c = 1.8 found by Alexandrova et al. (2021) based on Helios observations from 0.3 to 0.9 au. As the authors of these two studies suggest, the constant in front of ρ_e in the exponential cutoff seems to be weakly dependent on the radial distance from the Sun. In our simulation, the different value of c could be due either to the unrealistically small ion-to-electron mass ratio, which causes the electron scales to be too close to the ion scales, or to the specific plasma conditions. In this regard, it is worth noting that almost all the intervals analyzed in Alexandrova et al. (2021) have a larger ion beta than our simulation, and they all have a smaller beta (see Figures 3(g) and (h) therein). Properly testing the empirical exponential cutoff model would require an extensive parameter space exploration and goes beyond the scope of this paper. Here, we only intend to show that our simulation is able to model the turbulent cascade accurately down to scales smaller than the electron gyroradius, regardless of the exact shape of the power spectrum.

A limitation of this fully kinetic simulation, as for most simulations employing the same model, is the fact that the MHD scales are not retained. Usually, especially when explicit numerical codes are used, this is due to the fact that all characteristic spatial scales need to be resolved, including the Debye length. This is not a requirement in our case, as the iPIC3D code employs the implicit moment method. Here, however, the box size is still limited, as we decided to prioritize employing a very large number of ppc to reduce the numerical noise. While this choice assures that the power spectra are more accurate at large wavenumbers, missing completely the MHD range makes the turbulent cascade start developing at the ion scales. Some important properties (e.g., the plasma and magnetic compressibility, the residual energy) at ion scales are set by our initial conditions rather than self-developing from the larger scales. One could then wonder whether this sub-ion-scale turbulence is representative of the plasma dynamics in a larger plasma system or is instead sensibly constrained or affected by the box size. In order to investigate this and validate our results, we have run a hybrid simulation with the same physical conditions but a much larger box, able to model the turbulent cascade over two decades in wavenumber, one above and one below the ion scales. This simulation has the same parameters and initial conditions as the one described in Franci et al. (2020a), so that further information can be found in Appendix A therein. The only (minor) differences are that the grid points are 4000 × 4000 instead of 4096 × 4096 (and correspondingly the box size is 250 d_i instead of 256) and 2048 ppc instead of 1024. It is worth recalling here that the injection scale for this hybrid simulation is k_⊥ d_i ≃ 0.4, about a factor of 3 larger than for the fully kinetic simulation. Its initial level of magnetic fluctuations is also larger, i.e., δ B^ms ≃ 0.44B₀. As a result, the two simulations end up having a similar amplitude of the magnetic fluctuations in the inertial range.

In Figure 4 we compare the power spectra of different fields for the fully kinetic simulation, P^kin, and for the hybrid simulation, P^hyb. For the former, we only draw the portion of the power spectra where the corresponding S/N is above 3. For the latter, since we have only the initial power spectra for u _i and n_i , we apply the S/N criterion by using the noise threshold S/N( u _i ) < 3 for ${P}_{{{\boldsymbol{u}}}_{i}}$ and S/N(n_i ) < 3 for all other power spectra. Figure 4(a) compares the spectral behavior of the magnetic fluctuations. ${P}_{{\boldsymbol{B}}}^{\mathrm{hyb}}$ is more extended at large scales, since the hybrid simulation box is larger. This allows for the development of a Kolmogorov-like power law with a spectral index of −5/3 for k_⊥ d_i ≲ 3.5, which then steepens to a power law with a slope compatible with −11/3, as already observed in Franci et al. (2020a). ${P}_{{\boldsymbol{B}}}^{\mathrm{kin}}$, instead, completely lacks the −5/3 range and is higher at the largest scales. This is likely due to two main reasons. First, the injection scale is just in the middle of that range, and probably the fluctuations at the largest scales did not have time to fully partake in the cascade. Second, in the fully kinetic simulation the mechanisms leading to a fully developed inertial range are missing, and therefore those underlying the sub-ion-scale turbulence may start determining the dynamics at slightly larger scales. Between the ion-scale break of the hybrid simulation (the end of MHD inertial range) and the electron scales, i.e., in the range 3.5 ≲ k_⊥ d_i ≲ 10, ${P}_{{\boldsymbol{B}}}^{\mathrm{hyb}}$ and ${P}_{{\boldsymbol{B}}}^{\mathrm{kin}}$ overlap. Finally, at k_⊥ d_i ≃ 10, i.e., k_⊥ d_e ∼ k_⊥ ρ_e ∼ 1, when ${P}_{{\boldsymbol{B}}}^{\mathrm{kin}}$ steepens as the electron physics kicks in, ${P}_{{\boldsymbol{B}}}^{\mathrm{hyb}}$ maintains the same slope until the noise level is reached. In Figure 4(b), ${P}_{{\boldsymbol{E}}}^{\mathrm{hyb}}$ and ${P}_{{\boldsymbol{E}}}^{\mathrm{kin}}$ are compared. These are quite overlapped in the range 0.6 ≲ k_⊥ d_i ≲ 6, whereas at smaller scales, the former flattens while the latter keeps going down with a more or less constant slope. This difference in behavior may be partially due to the different level of noise in the two simulations, as Δx^hyb = 4 Δx^kin and ppc^hyb = 1/4 ppc^kin. The latter condition is related to the fact that in the hybrid case the electrons are treated as an isothermal fluid, so that P_e ∝ β_e /2 n_i and the number of ions per cell (2048) determines the noise in the electric field, while in the fully kinetic case P_e ∝ n_e T_e , so the number of electrons per cell (8192) is the determining one. As a consequence, a subdomain of a given size contains 16 times more particles in the fully kinetic simulation with respect to the hybrid one. It is, however, reasonable to ascribe the flattening at scales much larger than the spatial resolution to different physics, given that in the hybrid model we approximate ∇ · P_e ∼ β_e /2∇ n_i , so any possible effect due to electron temperature anisotropy and nongyrotropy, which can be expected to become important when the electron scales are approached, is not retained in the hybrid model. In Figure 4(c) we compare the power spectra of the ion bulk velocity. These are very close to each other at all the scales where the noise is negligible, with just a minor difference when reaching the electron scales. From this comparison, we can conclude that the spectral behavior of the ion velocity seems to be the same in the two simulations; thus, it is not significantly affected by the presence of kinetic electrons and related processes. The power spectra of the electron bulk velocity, shown in Figure 4(d), start overlapping at k_⊥ d_i ≃ 0.5, and they proceed together for about a decade, up to k_⊥ d_e ≃ 0.5, where ${P}_{{{\boldsymbol{u}}}_{e}}^{\mathrm{hyb}}$ further steepens, accordingly with ${P}_{{\boldsymbol{B}}}^{\mathrm{hyb}}$ as explained above. Figure 4(e) compares the power spectra of the ion density fluctuations. We note here that for the fully kinetic simulation we have verified that the spectrum of the ion density, ${P}_{{n}_{i}}$, and that of the electron density, ${P}_{{n}_{e}}$, are exactly identical up to k_⊥ d_i ≃ 40, where they start being affected by the two different levels of noise (due to the different number of pcc for the two species). We observe that ${P}_{{n}_{i}}^{\mathrm{hyb}}$ and ${P}_{{n}_{i}}^{\mathrm{kin}}$ almost perfectly overlap in the whole range k_⊥ d_i ≃ 1 and k_⊥ ρ_e ≃ 1, with ${P}_{{n}_{i}}^{\mathrm{hyb}}$ getting larger just below the electron scales. Finally, Figure 4(f) compares the power spectra of the parallel magnetic fluctuations. Interestingly, ${P}_{{{\boldsymbol{B}}}_{z}}^{\mathrm{hyb}}$ is slightly larger than ${P}_{{{\boldsymbol{B}}}_{z}}^{\mathrm{kin}}$ at all scales, especially below the ion-scale break, where they exhibit the same spectral slope, therefore keeping a constant differing factor. This could be somehow related to the different initial amplitude of the turbulent fluctuations with respect to the ambient field, given that the ratio ${P}_{{{\boldsymbol{B}}}_{z}}^{\mathrm{hyb}}/{P}_{{{\boldsymbol{B}}}_{z}}^{\mathrm{hyb}}$ is compatible with the ratio ${{\boldsymbol{B}}}_{\mathrm{rms}}^{\mathrm{hyb}}/{{\boldsymbol{B}}}_{\mathrm{rms}}^{\mathrm{kin}}\simeq 1.2$. Another possible reason for the difference in the magnetic compressibility could be related to the plasma betas: although we start the two simulations with the same values of β_i and β_e , these evolve differently and reach different values when the turbulence is fully developed (in particular, β_e does not change in the hybrid case by definition). This difference, however, is very small and as such cannot be considered as an indication of a different regime or different underlying physics.

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Figure 4 globally returns a picture where the MHD-scale fluctuations and their properties and evolution do not seem to have significant effects on those below the ion scales. This is supported in a particular mode by the spectral behavior of the density fluctuations, since these are zero at the beginning (apart from the small-scale noise due to the finite number of ppc) and only develop self-consistently through plasma processes. This further validates the findings by Franci et al. (2020b) on the kinetic turbulent cascade: the plasma dynamics in the kinetic range is mainly controlled by a few fundamental physical parameters, so that the properties of the fluctuations at sub-ion scales are independent of the presence and extension of an inertial range.

Since in the fully kinetic simulation β_e = 0.5, then ${\rho }_{e}=\sqrt{{\beta }_{e}}\,{d}_{i}\simeq 0.7\,{d}_{i}$, so the two electron characteristic scales are very close to each other. It is not possible to infer whether one of the two or a combination of them determines the steepening of the power spectra around the electron scales. Preliminary analysis on another fully kinetic simulation (not shown here) with β_e = 0.04, where ρ_e = 0.2d_e , seems to suggest that the electron-scale break is related to k_⊥ ρ_e rather than to k_⊥ d_e . At this level, however, we cannot exclude that the break might occur at either of the two scales depending on the value of β_e , just like it is observed to happen for the role of β_i for the ion-scale break (Chen et al. 2014; Franci et al. 2016).

We now focus on the electron and proton heating generated by turbulence. In Figure 5, we report the overall evolution of both ion and electron temperatures and their spatial distribution once turbulence is fully developed. Figure 5(a) shows the time evolution of the average value (over the whole simulation box) of the perpendicular, T_i,⊥, parallel, T_i,∥, and total temperature, T_i = (2 T_i,⊥ + T_i,∥)/3, and of the temperature anisotropy, A_i = T_i,⊥/T_i,∥, all normalized to their initial values. We recall here that the temperature components are defined with respect to the local mean magnetic field. At the time of maximum turbulent activity, T_i has increased by almost 8% with respect to its initial value ${T}_{i}^{\mathrm{in}}=0.1$. Such increase does not stop at t^peak, being linear in time afterward. This hints at the possibility that, in the presence of an energy injection mechanism that keeps feeding the turbulent cascade and maintains it in a quasi-steady state over a longer time, the ion temperature could increase significantly more. T_i,⊥ starts increasing as soon as the simulation begins and keeps increasing throughout the whole evolution, with some oscillations until t ≃ 5 and then a linear increase in time. T_i,∥, instead, remains constant until t ≃ 2 (comparable to the eddy turnover time at the injection scale); then decreases, reaching a minimum just before t^rec; and increases monotonically afterward. Correspondingly, A_i increases until t^rec and then decreases very slowly, reaching an almost constant value of 1.06 at t^peak. The corresponding quantities for the electrons are shown in Figure 5(b). T_e remains almost constant until about t^rec and then increases very slowly, reaching a value of $1.01\,{T}_{e}^{\mathrm{in}}$ at t^peak, with ${T}_{e}^{\mathrm{in}}=0.25$ being its initial value. Despite the quite small increase in the total electron temperature, its perpendicular and parallel components exhibit very large variations. T_e,⊥ shows a small initial decrease, then stays almost constant until just before t^rec, and then starts decreasing more rapidly, linearly in time. On the contrary, T_e,∥ increases initially more or less linearly in time, and then at around t^rec it starts increasing much faster. Correspondingly, A_e decreases monotonically throughout the simulation (faster after t^rec), reaching a value of about 0.9 at t^peak, which keeps decreasing afterward.

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Summarizing, Figures 5(a)–(b) show that at $t\simeq {t}^{\mathrm{rec}}=4.9\,{{\rm{\Omega }}}_{i}^{-1}$ (i.e., when the first maximum of ∣ J ∣ marks the onset of magnetic reconnection) there is a clear and significant change of behavior in the particle heating, characterized by the fact that (i) the rate of variation of the total ion temperature increases and becomes almost constant, (ii) the total electron temperature starts increasing linearly in time, (iii) the ion temperature anisotropy reaches a plateau and starts decreasing very slowly, and (iv) the rate of variation of the electron temperature anisotropy increases. It is then reasonable to assume that the change in particle heating is related to the onset of magnetic reconnection events.

In order to test this assumption, Figures 5(c)–(h) show the spatial distribution of the above quantities in the 2D simulation domain. In the top row (panels (c)–(e)), we show the ion perpendicular and parallel temperatures and their ratio. T_i,⊥ and T_i,∥ exhibit a very patchy behavior, with alternating regions where they are larger or smaller than their initial value. Typically, in regions where one component has increased, the other one has decreased. As a result, despite that their average values are very similar and their maximum values are also comparable, A_i is larger than 2 in many large areas, reaching a maximum as large as 3.7. Regions with values well above or below 1 are alternating, and somewhere we observe a quadrupolar configuration, both in correspondence of X-points where magnetic lines reconnect (e.g., at the point [8, 17]) and inside big vortices (e.g., at [19, 9]). The spatial distribution of the electron temperature looks quite different. T_e,⊥ is close to its initial value in most of the simulation box, exhibiting just a small decrease in correspondence with what look like reconnection outflows around X-points and a small increase at the center of vortices. On the contrary, T_e,∥ exhibits much larger variations, slightly decreasing in regions where the magnetic field lines get compressed and becoming about twice as large in the reconnection outflows. As a consequence, in the latter areas A_e gets very small, reaching values as small as 0.36. The spatial distribution of both ion and electron temperatures is consistent with what we have inferred from the time evolution of their average values and with the observed change of behavior at the time when reconnection starts occurring: reconnection is likely to be at least partially responsible for particle heating, especially for the electrons. This sounds reasonable, as magnetic reconnection is a mechanism that transfers energy from the magnetic field to the particles.

In order to confirm the role of reconnection in heating the particles and in generating a strong temperature anisotropy, especially for the electrons, we now provide a more quantitative characterization of the regions where reconnection events occur. Figure 2 clearly showed qualitative evidence for the presence of reconnection events in the simulation, e.g., X-points and small magnetic islands emerging in proximity of thin strong current sheets. Magnetic reconnection is known to occur in turbulent plasmas as the result of the interaction of turbulent eddies, which leads to the formation and stretching of intense current sheets. Indeed, it has been observed to occur spontaneously in numerical simulations of plasma turbulence performed with different methods (e.g., Matthaeus & Lamkin 1986; Servidio et al. 2009; Franci et al. 2015b; Wan et al. 2015; Cerri & Califano 2017; Haggerty et al. 2017), and it has been shown to provide a significant contribution to the further development of the turbulent cascade at sub-ion scales (e.g., Franci et al. 2017; Mallet et al. 2017; Papini et al. 2019). In Figure 6(a), we show a pseudocolor plot of the out-of-plane component of the vector potential, A _z , with its isocontours superimposed as black lines. Green stars mark saddle points, i.e., X-points that represent potential reconnection sites. Figure 6(b) shows arrows marking the direction of the local magnetic field in the simulation box, on top of the contour plot of the ∣δ B ∣². This clearly shows that the local magnetic field has opposite sign at the two sides of most of the X-points, as required for magnetic reconnection to occur. Recently, Agudelo Rueda et al. (2021) have investigated the spontaneous occurrence of reconnection in a 3D fully kinetic simulation of plasma turbulence by means of a set of indicators based on thresholds. Here we chose to apply the same criteria to detect and characterize reconnection events. Although our 2D setting is definitely less realistic than a 3D one, here we have the advantage that it is straightforward to identify X-points in 2D. In this sense, our analysis of reconnection events can be considered as complementary to the work by Agudelo Rueda et al. (2021), and in some sense a further validation. For a given (positive) scalar quantity, Ψ, we define a threshold ${{\rm{\Psi }}}_{\mathrm{th}}=\langle {\rm{\Psi }}\rangle +{ \mathcal N }{{\rm{\Psi }}}^{\mathrm{rms}}$ and look for regions where Ψ/Ψ_th − 1 > 0. Here, we chose to set ${ \mathcal N }=1$, as we find that this is enough for detecting the strongest reconnection events in our simulation. The criteria we have evaluated are

$$\begin{eqnarray*}\begin{array}{lr}{\rm{C}}1:| {\boldsymbol{J}}| \gt {\left|{\boldsymbol{J}}\right|}^{\mathrm{th}}\equiv \langle | {\boldsymbol{J}}| \rangle +{\left|{\boldsymbol{J}}\right|}^{\mathrm{rms}} & (\mathrm{Fig}.\,6{\rm{c}})\\ {\rm{C}}2:| {{\boldsymbol{u}}}_{i,e}| \gt {\left|{{\boldsymbol{u}}}_{i,e}\right|}^{\mathrm{th}}\equiv \langle | {{\boldsymbol{u}}}_{i,e}| \rangle +| {{\boldsymbol{u}}}_{i,e}{| }^{\mathrm{rms}} & (\mathrm{Fig}.\,6{\rm{d}},{\rm{e}})\\ {\rm{C}}3:{T}_{i,e}\gt {T}_{i,e}^{\mathrm{th}}\equiv \langle {T}_{i,e}\rangle +{T}_{i,e}^{\mathrm{rms}} & (\mathrm{Fig}.\,6{\rm{g}},{\rm{h}})\\ {\rm{C}}4:{\boldsymbol{J}}\cdot {\boldsymbol{E}}{\left|{}_{+}\gt {\boldsymbol{J}}\cdot {\boldsymbol{E}}\right|}_{+}^{\mathrm{th}}\equiv \langle {\boldsymbol{J}}\cdot {\boldsymbol{E}}{\left|{}_{+}\rangle +{\boldsymbol{J}}\cdot {\boldsymbol{E}}\right|}_{+}^{\mathrm{rms}} & (\mathrm{Fig}.\,6{\rm{f}}),\end{array}\end{eqnarray*}$$

where for the last criterion we consider only positive values of J · E , which denote energy transfer from the fields to the particles. Figure 6(c) shows the result of indicator C1 for ∣ J ∣, revealing the presence of intense current structures. We can clearly see that all these have a width of the order of d_e and lengths of the order of a few times d_i . We count about a dozen of them, taking into account that some of them are actually portions of the same current sheet, as the simulation box is periodic. In Figures 6(d)–(e), we estimate the indicator C2 for ∣ u _i,e ∣, related to the presence of fast ions and electrons that provide evidence for the presence of a reconnection outflow (or jet). The ion bulk velocity exceeds the threshold in some regions with a width of the order of 1d_i –2d_i , some of which seem to be directly related to X-points and represent ion jets in the reconnection outflow, while others seem to be within large-scale vortices. On the contrary, the electron bulk velocity is above its threshold in all the regions directly connected to the strong current sheets observed in panel (c). These fast electron jets have a very small width, of the order of d_e , as they are confined in small regions between larger-scale structures. It is interesting to note that many of the observed reconnection events seem to be highly asymmetric, as jets are observed only on one side of the X-point and not on the other. Figure 6(f) shows the contour plot of indicator C4. J · E ∣₊ is above its threshold in a few thin regions corresponding to the most intense current sheets, providing further evidence that in the location of reconnection events magnetic energy is converted into particle energy. In Figures 6(g)–(h), we estimate indicator C3 on T_i,e , which is related to the presence of heated ions and electrons. The regions where the particle temperatures exceed their respective threshold look similar for ions and electrons, meaning that most reconnection events are eventually leading to both ion and electron heating. A major difference, however, can be appreciated by comparing with Figures 5(c)–(h): reconnection heats the electrons preferentially in the parallel direction with respect to the local magnetic field and the ions in the perpendicular direction. The latter result appears to be in agreement with the fact that a sheared in-plane ion flow tends to transfer energy to the in-plane ion pressure tensor components via the action of the symmetric part of the strain tensor (Del Sarto et al. 2016; Del Sarto & Pegoraro 2017; Yang et al. 2017; Matthaeus et al. 2020; Bandyopadhyay et al. 2021; Hellinger et al. 2022). Again, we observe a significant lack of symmetry, as particles are heated only on one side of the X-point, typically the same for both species. In Figure 6(i), we compare the indicators C2 for ions and electrons, by showing the contour plots of their perpendicular components, ∣ u _i,⊥∣² and ∣ u _e,⊥∣², on top of each other (in shades of blue and red, respectively). This allows us to appreciate whether a reconnection event produces both in-plane ion and electron jets, as in "standard" reconnection, or only an electron jet with no ion counterpart, as in "electron-only" reconnection. For some of the X-points, it is not straightforward to apply this classification, mainly because no strong ion or electron jets are observed or because the geometry is complex. We still observe about a dozen events that can be labeled as electron-only (yellow circles) and about half a dozen that are standard (light-blue circles). Summarizing Figure 6, we can conclude that as a result of the interaction between turbulent magnetic structures, thin intense current sheets form and shrink until they undergo reconnection. Such reconnection events transfer magnetic energy to the particles, heating the electrons in the direction parallel to the mean field and the ions mainly in the orthogonal direction. Thin electron jets are also observed in the reconnection outflow, with ion counterparts only in about one-third of events. The fact that standard reconnection is observed only in a minority of events might be related to the simulation box size, which is only a few tens of d_i , and to the energy injection, which occurs at the ion scales (Califano et al. 2020). These conditions strongly limit the magnetic correlation length, which characterizes the size of the interacting turbulent magnetic structures and, as consequence, also sets the length and thickness of the current sheets forming in between (e.g., Stawarz et al. 2019). In other words, there are only a few vortices with radius of the order of few times d_i , so there are only a few chances to develop strong current sheets with such a length. On the contrary, most of them are formed by the interaction of magnetic structures with smaller size and will therefore be shorter. Indeed, the ion jets observed in Figure 6(i) seem to be concentrated just next to some of the largest vortices. Quantitatively characterizing the properties of reconnection and its interplay with electron-scale turbulence goes beyond the scope of this work and will be the subject of future investigation. Here, we intended to show evidence for the coexistence of both types of reconnection events spontaneously driven by turbulence at sub-ion scales and at electron scales.

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