Stochastic Electron Acceleration by Temperature Anisotropy Instabilities
Under Solar Flare Plasma Conditions

A 2D view of the relevant unstable modes at $t\cdot s=1.6$ (right after the saturation of $\delta\textbf{{B}}$) is given by Figure 2, which shows the three components of the magnetic and electric fluctuations $\delta\textbf{{B}}$ and $\delta\textbf{{E}}$.⁴⁴4$\delta\textbf{{E}}$ is simply equal to E, since $\langle\textbf{{E}}\rangle=0$ in our shearing coordinate setup. Both the magnetic and electric fluctuations show that the dominant modes have an oblique wavevector with respect to the direction of the mean magnetic field $\langle\textbf{{B}}\rangle$, which is shown by the black arrows in all the panels. The dominance of the oblique modes stops at a later time, as can be seen from Figure 3, which shows the same quantities as Figure 2 but at $t\cdot s=3.1$. In this case the waves propagate along the background magnetic field, implying that quasi-parallel modes dominate both the electric and magnetic fluctuations.

In order to determine the transition time from the dominance of oblique to quasi-parallel modes, we calculate the magnetic energy contained in each type of modes. For that, we define the modes as “oblique” or “quasi-parallel” depending on whether the angle between their wave vector k and $\langle\textbf{{B}}\rangle$ is larger or smaller than 20^∘, respectively. The magnetic energies in oblique and quasi-parallel modes are shown in Figure 1$a$ using dashed-black and solid-black lines and are denoted by $\langle\delta B_{ob}^{2}\rangle$ and $\langle\delta B_{qp}^{2}\rangle$, respectively. We see that the oblique modes dominate until $t\cdot s\approx 2.2$. After that, the quasi-parallel modes contribute most of the energy of the magnetic fluctuations, which is consistent with the two regimes shown in Figures 2 and 3.

An interesting characteristic of the oblique modes is the notorious fluctuations in the electron density $n_{e}$, as shown in Figure 2$h$, which suggests the presence of a significant electrostatic component in the electric field $\delta\textbf{{E}}$. These density fluctuations are less prominent in the case dominated by quasi-parallel modes, as can be seen from Figure 3$h$, which shows that at $t\cdot s=3.1$ the electrostatic electric fields are weaker than at $t\cdot s=1.6$. This change in the $\delta\textbf{{E}}$ behavior can be seen more clearly in Figure 1$c$, which shows the evolution of the energy contained in the electric field fluctuations $\delta\textbf{{E}}$, dividing it into its electrostatic and electromagnetic components (blue and black lines, respectively). This separation is achieved by distinguishing the contributions to the Fourier transform of the electric field fluctuations ($\delta\tilde{\textbf{{E}}}$) that satisfy $\textbf{{k}}\times\delta\tilde{\textbf{{E}}}=0$ (electrostatic part) and $\textbf{{k}}\cdot\delta\tilde{\textbf{{E}}}=0$ (electromagnetic part). In the oblique regime ($t\cdot s\lesssim 2.2$) the electric field energy is mainly dominated by its electrostatic part ($\delta E^{2}_{es}$), and after that it gradually becomes dominated by its electromagnetic part ($\delta E^{2}_{em}$). Figure 1$c$ also shows that this transition occurs when the instantaneous parameter $f_{e}$ (shown by the upper horizontal axis) is $f_{e}\sim 1.2-1.5$. We see in the next section that this transition is fairly consistent with linear Vlasov theory.

In this section, we show that the transition from the dominance of oblique modes with mainly electrostatic electric field (hereafter, oblique electrostatic modes) to the dominance of quasi-parallel modes with mainly electromagnetic electric field (hereafter, quasi-parallel electromagnetic modes) in S1200 is consistent with linear Vlasov theory. Indeed, a previous study by Gary et al. (1999) predicts that for the range of electron conditions considered in our study, and assuming a bi-Maxwellian electron velocity distribution, three types of modes are relevant: parallel, electromagnetic whistler (PEMW) modes; parallel, electromagnetic z (PEMZ) modes; and oblique, quasi-electrostatic (OQES) modes, where the latter are dominated by electrostatic electric fields. In this section we use linear Vlasov theory to check whether the PEMW or PEMZ (OQES) modes are theoretically the most unstable when the quasi-parallel electromagnetic (oblique electrostatic) modes dominate in our run. For this we use the NHDS solver of Verscharen et al (2018) to calculate the temperature anisotropy threshold $\Theta_{e,\perp}/\Theta_{e,\parallel}-1$ needed for the growth of these modes with a given growth rate $\gamma_{g}$, assuming different values of $f_{e}$ and $\Theta_{e,\parallel}$.

Figure 4$a$ shows the anisotropy thresholds for $\gamma_{g}/\omega_{ce}=10^{-2}$, which is appropriate for run S1200. We estimate the growth rate of $\delta\textbf{{B}}$ in this run from its exponential growth regime in Figure 1$a$ ($t\cdot s\sim 1.2-1.3$), which is $\gamma_{g}\sim 10s$. This implies that $\gamma_{g}\sim 10^{-2}\omega_{ce}^{\textrm{init}}$, given that in run S1200, $\omega_{ce}^{\textrm{init}}/s=1200$. The dashed lines in Fig. 4$a$ consider the thresholds only for parallel modes (i.e., considering the lowest threshold between PEMW and PEMZ modes), while the solid lines consider the lowest threshold between modes with all propagation angles. We find that for $\Theta_{e,\parallel}=0.002$ and 0.006, there are values of $f_{e}$ where the solid-red and solid-green lines separate from the corresponding dashed lines. These values of $f_{e}$, therefore, correspond to where the unstable modes are dominated by OQES modes, which, for $\Theta_{e,\parallel}=0.002$ and 0.006, occurs when $0.4\lesssim f_{e}\lesssim 1.8$ and $0.8\lesssim f_{e}\lesssim 1.3$, respectively. For values of $f_{e}$ above and below these ranges, linear theory predicts that the most unstable modes correspond to PEMZ and PEMW modes, respectively, which is consistent with the merging of the dashed and solid lines in those regimes.⁵⁵5The predictions shown in Figure 4$a$ are in good agreement with Figure 4$c$ of Gary et al. (1999)

Thus, the dominance of oblique electrostatic modes in run S1200 from the moment when the instability sets in ($f_{e}\sim 0.8$) until $f_{e}\sim 1.2-1.5$, suggests that, in order to be consistent with linear theory, $\Theta_{e,\parallel}$ should be in the range $\sim 0.002-0.006$ after the growth of the instabilities ($f_{e}\gtrsim 0.8$). This is indeed what is shown by Figure 1$b$, where $\Theta_{e,\parallel}$ (solid-red line) appears in the range $\Theta_{e,\parallel}\sim 0.0025-0.005$ when $f_{e}\gtrsim 0.8$.

These results show that the transition between the oblique, electrostatic to quasi-parallel, electromagnetic regimes at $f_{e}\sim 1.2-1.5$ in run S1200 is consistent with linear Vlasov theory, which predicts a transition from OQES to PEMZ modes at $f_{e}\sim 1.3-1.8$. Therefore, hereafter we refer to the $f_{e}\lesssim 1.2-1.5$ and $f_{e}\gtrsim 1.2-1.5$ regimes as OQES dominated and PEMZ dominated regimes, respectively.

The electron energy spectrum evolution for run S1200 can be seen from Figure 5$a$, which shows $dn_{e}/d\ln(\gamma_{e}-1)$ for different values of $t\cdot s$ ($\gamma_{e}$ is the electron Lorentz factor). This plot shows the rapid growth of a nonthermal tail starting at $t\cdot s\approx 2.5$. After $t\cdot s\approx 3.5$ this tail can be approximated as a power-law of index $\alpha_{s}\approx 2.9$ ($\alpha_{s}\equiv d\ln(n_{e})/d\ln(\gamma_{e}-1)$), plus a high energy bump that reaches $\gamma_{e}-1\sim 0.6$ ($\sim 300$ keV). Most of the nonthermal behavior of the spectrum starts at $t\cdot s\approx 2.5$, which is right after the PEMZ modes become dominant, as shown by Figure 1$c$.

Figure 5$b$ shows a numerical convergence test of the final spectrum at $t\cdot s=4$. It compares run S1200 ($d_{e}^{\textrm{init}}/\Delta x=35$, $N_{epc}=100$ and $L/R_{Le}^{\textrm{init}}=140$) with a run with $d_{e}^{\textrm{init}}/\Delta x=25$ (run S1200a, in blue-dotted line), a run with $N_{epc}=50$ (run S1200b, in red-dotted line) and a run with $L/R_{Le}^{\textrm{init}}=70$ (run S1200c, in green-dotted line). No significant difference can be seen between the different spectra, implying that our results are fairly converged numerically.

In order to identify the energy source for the nonthermal electron acceleration, we explore first the overall energy source for electrons (thermal and nonthermal). It is well known that the presence of a temperature anisotropy in a shearing, collisionless plasma gives rise to particle heating due to the so called “anisotropic viscosity” (AV). This viscosity can give rise to an overall electron heating, for which the time derivative of the electron internal energy, $U_{e}$, is (Kulsrud 1983; Snyder et al. 1997):

where $r$ is the growth rate of the field (in our setup $r=-sB_{x}B_{y}/B^{2}$) and $\Delta p_{e}$ is the difference between the perpendicular and parallel electron pressures, $\Delta p_{e}=p_{e,\perp}-p_{e,\parallel}$. In run S1200, Equation 2 reproduces well the evolution of the overall electron energy gain. This can be seen from Figure 5$c$, where the time derivative of the average electron internal energy $d\langle U_{e}\rangle/dt$ (blue) coincides well with $r\langle\Delta p_{e}\rangle$ (green). This result shows that in our shearing setup, the heating by AV essentially explains all of the electron energization.

Although the total electron energy gain is dominated by AV, the work done by the electric field $\delta\vec{E}$ associated to the unstable modes, $W_{E}(\equiv\int e\delta\vec{E}\cdot d\vec{r}$, where $\vec{r}$ is the electron position), can differ significantly between electrons from different parts of the spectrum, making $W_{E}$ play a key role in producing the nonthermal tail by transferring energy from the thermal to the nonthermal part of the spectrum (Riquelme et al., 2017; Ley et al., 2019). We check this by analyzing the contributions of AV and $W_{E}$ to the energy gain of three different electron populations, defined by their final energy at $t\cdot s=4$. These populations are:

1. 1.

   Thermal electrons: their final energy satisfies $\gamma_{e}-1<0.05$, which corresponds to the energy range marked by the light-blue background in Figure 5$a$.

2. 2.

   Low-energy nonthermal electrons: their final energies satisfy $0.05<\gamma_{e}-1<0.2$, where the nonthermal tail roughly behaves as a power-law of index $\alpha_{s}\approx 2.9$. This energy range is marked by the white background in Figure 5$a$.

3. 3.

   High-energy nonthermal electrons: corresponding to the high-energy bump in the spectrum, defined by $0.2<\gamma_{e}-1$. This energy range is marked by the grey background in Figure 5$a$.

Figures 6$a$, 6$b$ and 6$c$ show the energy evolution for the thermal, low-energy nonthermal and high-energy nonthermal electrons, respectively. In each case, we show the average initial kinetic energy of each population, $\langle\gamma_{e}\rangle_{0}-1$, plus:

1. 1.

   the work done by the electric field of the unstable modes, $W_{E}$, shown by the blue lines.

2. 2.

   the energy gain by the anisotropic viscosity, AV, shown by the green lines.

3. 3.

   the total energy gain, shown by the black lines.

The green lines show that for the three populations there is a positive energy gain due to AV. On the other hand, the blue lines in Figure 6$a$ shows that, in the case of the thermal electrons, $W_{E}$ produces a decrease in the electron energies. This illustrates that the thermal electrons transfer a significant part ($\sim 50\%$) of their initial energy to the unstable modes. However, the overall heating of the thermal electrons is positive and, by the end of the simulation, reaches a factor $\sim 2$ increase in their thermal energy. Figures 6$b$ and 6$c$ show that for the low- and high-energy nonthermal electrons there is a significant growth in the electrons energy due to $W_{E}$, suggesting that the unstable modes transfer energy to the nonthermal particles. For the low-energy nonthermal electrons, the AV still dominates the heating, whereas for the high-energy nonthermal electrons, AV is subdominant and most of the electron energization is due to $W_{E}$. The red lines in Figures 6$a$, 6$b$ and 6$c$ show the overall heating of thermal electrons due to adding AV and $W_{E}$. The red line reproduces reasonably well the evolution of the total energy $\langle\gamma_{e}\rangle-1$ (black line) of the three electron populations.

The way AV and $W_{E}$ contribute to the energization of thermal and nonthermal electrons suggests that the formation of an electron nonthermal tail is caused by the transfer of energy from the thermal to the nonthermal electrons, which is mediated by the waves electric field. This characteristic of the formation of a nonthermal tail is in line with previous results where the acceleration of ions and electrons by temperature anisotropy instabilities was studied in nearly relativistic plasmas (Riquelme et al., 2017; Ley et al., 2019). Also, Figures 6$a$, 6$b$ and 6$c$ show that most of the energy transfer from thermal to nonthermal electrons occurs after $t\cdot s\sim 2.2$, which corresponds to the regime dominated by PEMZ modes, implying a subdominant contribution to the acceleration by the initially dominant OQES modes.

The shear parameter $s$ is a measure of the rate at which temperature anisotropy growth is driven in our simulations. We can thus estimate the corresponding parameter $s$ in the contracting looptops of solar flares as the rate at which temperature anisotropy grows in these environments. This can be obtained by estimating the inverse of the time that it takes for the contracting looptops to collapse into a more stable configuration.

We thus estimate $s$ by dividing the typical Alfvén velocity, $v_{A}$, in the looptops ($v_{A}$ should be close to the speed at which the newly reconnected loops get ejected from their current sheet) by the typical lengthscale of the contracting looptops, $L_{\textrm{LT}}$. Using our fiducial parameters $n_{e}\sim 10^{9}$ cm^-3 and $B\sim 100$ G, and estimating $L_{\textrm{LT}}\sim 10^{9}$ cm (e.g., Chen et al., 2020), we obtain $s\sim 1$ sec^-1 and $\omega_{ce}/s\sim 10^{9}$. This value of $\omega_{ce}/s$ is several orders of magnitude larger than what can be achieved in our simulations. Therefore, two important questions arise. The first one is whether, for realistic values of $\omega_{ce}/s$, the dominance of PEMZ and OQES modes should occur for the same regimes observed in run S1200. And, if that is the case, the second question is whether the effect on electron acceleration of these modes remains the same, independently of $\omega_{ce}/s$.

Since in §3.3 we showed the suitability of linear Vlasov theory to predict the dominance of the different modes, in this section we use linear theory to show that increasing $\omega_{ce}^{\textrm{init}}/s$ by several orders of magnitude should not modify the relative importance of the modes in the different plasma regimes. First, we notice that the growth rate $\gamma_{g}$ of the unstable modes should be proportional to $s$, implying that $\omega_{ce}^{\textrm{init}}/s\propto\omega_{ce}^{\textrm{init}}/\gamma_{g}$. This proportionality is physically expected, since $s^{-1}$ sets the timescale for the evolution of the macroscopic plasma conditions. Thus, the different modes that dominate in different simulation stages need to have a growth rate of the order of $s$ to have time to set in and regulate the electron temperature anisotropy. This can also be seen from Figure 7$a$, which shows the evolution of the energy in $\delta\textbf{{B}}$ divided into oblique and quasi-parallel modes (dashed and solid lines, respectively) for runs with $\omega_{ce}^{\textrm{init}}/s=600$, 1200 and 2400 (runs S600, S1200 and S2400, respectively). We see that the three runs show essentially the same ratio $\gamma_{g}/s\sim 10$. This means that, in order to find out which instabilities would dominate for realistically large values of $\omega_{ce}^{\textrm{init}}/s$, we must calculate the instability thresholds for a comparatively large value of $\omega_{ce}/\gamma_{g}$. Following that criterion, Figure 4$b$ shows these thresholds using the same values of $f_{e}$ and $\Theta_{e,\parallel}$ used in Figure 4$a$, but assuming $\gamma_{g}/\omega_{ce}=10^{-6}$ instead of $10^{-2}$. We see that for $\gamma_{g}/\omega_{ce}=10^{-6}$ the PEMW, PEMZ and OQES modes dominate for values of $f_{e}$ and $\Theta_{e,\parallel}$ very similar to the ones shown for $\gamma_{g}/\omega_{ce}=10^{-2}$. This implies that the role played by the different modes in controlling the electron temperature anisotropy in the different plasma regimes should not change significantly for realistic values of the shearing parameter $s$.

We now investigate whether increasing $\omega_{ce}^{\textrm{init}}/s$ affects the spectral evolution of the electrons. This is done in Figure 7$b$, where the final spectra ($t\cdot s=4$) are shown for runs with $\omega_{ce}^{\textrm{init}}/s=300,600$, 1200 and 2400 (runs S300, S600, S1200 and S24000, respectively). No significant differences are seen in the final spectra, except for a slight hardening as $\omega_{ce}^{\textrm{init}}/s$ increases, which does not seem significant when comparing the cases $\omega_{ce}^{\textrm{init}}/s=$ 2400 and 1200. This shows that the magnetization parameter $\omega_{ce}^{\textrm{init}}/s$ does not play a significant role in the efficiency of the electron nonthermal acceleration.

The lack of dependence of the acceleration on $\omega_{ce}^{\textrm{init}}/s$ can be physically understood in terms of the relation between the effective pitch-angle scattering rate due to the instabilities, $\nu_{\textrm{eff}}$, and $s$. We estimate $\nu_{\textrm{eff}}$ using the evolution of $\Theta_{e,\parallel}$, which in a slowly evolving and homogeneous shearing plasma is given by (Sharma et al., 2007):

where v is the plasma shear velocity, $\hat{\textbf{{b}}}=\hat{\textbf{{B}}}/B$ and $\Delta\Theta_{e}=\Theta_{e,\perp}-\Theta_{e,\parallel}$. Considering that $\textbf{{v}}=-sx\hat{y}$, we can rewrite Equation 3 as:

Figure 8$a$ shows the evolution of $\langle\Delta\Theta_{e}\rangle$ for the runs with $\omega_{ce}^{\textrm{init}}/s=600,1200$ and 2400 (runs S600, S1200 and S2400). When $t\cdot s\gtrsim 2.2$, the factor $\Delta\Theta_{e}$ that appears on the right hand side of Equation 4 is very similar in the three runs. The left hand side of Equation 4 depends on the evolution of $\Theta_{e,\parallel}$, which, in the $t\cdot s\gtrsim 2.2$ regime, is also similar in the three runs as can be seen from Figure 8$b$. This means that, for $t\cdot s\gtrsim 2.2$, the ratio $\nu_{\textrm{eff}}/s$ in Equation 4 is fairly constant (within about $\sim 10\%$) for these three values of $\omega_{ce}^{\textrm{init}}/s$. Thus, for simulations with a fixed value of $\omega_{ce}^{\textrm{init}}$ but different $s$, the behaviors of $\langle\Delta\Theta_{e}\rangle$ and $\langle\Theta_{e,\parallel}\rangle$ for $t\cdot s\gtrsim 2.2$ imply that $\nu_{\textrm{eff}}$ should be approximately proportional to $s$. Because of that, on average, the number of times that the electrons are strongly deflected in a fixed interval of $t\cdot s$ should be largely independent of $\omega_{ce}^{\textrm{init}}/s$. Thus, in a stochastic acceleration scenario, we expect the acceleration effect during a fixed number of shear times ($s^{-1}$) to only depend on the dispersive properties of the unstable modes (see, e.g., Summers et al, 1998), which are not expected to depend on the value of $\omega_{ce}^{\textrm{init}}/s$. Indeed, as long as $\omega_{ce}^{\textrm{init}}\gg s$, the modes propagation and oscillations should occur rapidly (on timescales of $\sim\omega_{ce}^{-1}$), and should not be affected by the slowly evolving background (on timescales of $\sim s^{-1}$). These arguments thus imply that the acceleration efficiency should be largely independent of $\omega_{ce}^{\textrm{init}}/s$.

Notice that Figures 8$a$ and 8$b$ show that, in the $t\cdot s\lesssim 2.2$ regime, $\langle\Delta\Theta_{e}\rangle$ and $\langle\Theta_{e,\parallel}\rangle$ tend to depend more strongly on $\omega_{ce}^{\textrm{init}}/s$, so the electron acceleration efficiency should differ significantly during that period. However, Figure 7$a$ shows that for $t\cdot s\lesssim 2.2$ the dominant instabilities are mainly oblique modes, which is indicative of the dominance of OQES modes. Therefore, since no significant acceleration is expected to happen in that regime (as we showed in §4.1), the different evolutions of $\langle\Delta\Theta_{e}\rangle$ and $\langle\Theta_{e,\parallel}\rangle$ for $t\cdot s\lesssim 2.2$ should not have an appreciable effect on the final electron spectra. Although this analysis is valid for the rather limited range of values of $\omega_{ce}^{\textrm{init}}/s$ tested by our simulations, we expect the $\nu_{\textrm{eff}}\propto s$ relation to hold even for realistic values of $\omega_{ce}^{\textrm{init}}/s$. This is based on comparing the linear theory thresholds presented in Figures 4$a$ and 4$b$, which predict $\Delta\Theta_{e}/\Theta_{e,\parallel}$ to have essentially the same evolution (only differing by an overall factor $\sim 3$) when decreasing $\gamma_{g}/\omega_{ce}$ by four orders of magnitude.

The evolution of the electron pitch-angle is important to determine the ability of the electrons to escape the flare looptops and precipitate towards the footpoints, which is a key ingredient for solar flare emission models (e.g., Minoshima et al., 2011). In this section we investigate the way temperature anisotropy instabilities affect the pitch-angle evolution for electrons with different energies.

As a measure of the average pitch-angle for different electron populations, Figure 9$b$ shows $p_{e,\perp}/p_{e,\parallel}$ for the thermal (red), low-energy nonthermal (green) and high-energy nonthermal (blue) electrons from run S1200. These three electron populations are defined according to their final energies, as we did in §4.1 (in each of these cases, the pressures are calculated only considering the electrons in each population). For comparison, we also show as dashed-black lines the time dependence of $p_{e,\perp}/p_{e,\parallel}$ for a hypothetical, initially isotropic electron population that evolves according to the CGL equation of state. The three electron populations show evolutions similar to the CGL prediction until the onset of the OQES instability, which, as we saw in §3, occurs at $t\cdot s\sim 1.4$. After that, the pitch-angle scattering tends to reduce $p_{e,\perp}/p_{e,\parallel}$ for the three populations, which occurs more abruptly for the low- and high-energy nonthermal electrons, whose pitch-angles by $t\cdot s\sim 1.7$ becomes $\sim 2-3$ times smaller than the ones of the thermal electrons. This shows that, even though the OQES modes (which dominate until $t\cdot s\approx 2.2$) make a subdominant contribution to the nonthermal electron acceleration, they still have an important effect by significantly reducing the pitch-angle of the highest energy particles. After that, in the PEMZ dominated regime, the anisotropy evolves in the opposite way. In that case, the $p_{e,\perp}/p_{e,\parallel}$ ratios of both populations of nonthermal electrons grow more rapidly than for the thermal electrons. This is especially true for the high-energy nonthermal electrons, for which $p_{e,\perp}/p_{e,\parallel}$ reaches values $\sim 2-3$ times larger than for thermal electrons by the end of the run.

This increase in the electron pitch-angle of the highest energy electrons due to PEMZ mode scattering is consistent with quasi-linear theory results that describe the stochastic acceleration of electrons by whistler/z modes in terms of the formation of a “pancake” pitch-angle distribution for the most accelerated electrons (Summers et al, 1998). These results thus suggest that, assuming a more realistic solar flare scenario where the electrons were allowed to prescipitate towards the flare footpoints, the high-energy nonthermal electrons produced by PEMZ mode scattering should tend to be more confined to the looptop than the low-energy nonthermal and thermal electrons.

Figures 9$a$ and 9$c$ show the same quantities as Figure 9$b$ but for runs with $\omega_{ce}^{\textrm{init}}/s=600$ and 2400 (runs S600 and S2400). After the triggering of the instabilities ($t\cdot s\sim 1.4$), there are no substantial differences between the three magnetizations, suggesting that the energy dependence of the pitch-angle evolution in our runs is fairly independent of $\omega_{ce}^{\textrm{init}}/s$.

An interesting aspect of the electron energy evolution is the correlation between the final electron energies and $i)$ their initial energies and $ii)$ their initial pitch-angles. The correlation between the initial and final energies can be seen from Figures 6$a$, 6$b$ and 6$c$, which show that for the thermal, low-energy nonthermal and high-energy nonthermal electrons, the initial energies are given by $\langle\gamma_{e}\rangle_{0}-1\sim 0.01,\sim 0.025$ and $\sim 0.04$, respectively. This energy correlation is also expected from the quasi-linear theory results of Summers et al (1998), which show that the maximum energy that electrons can acquire due to stochastic acceleration by whistler/z modes increases for larger initial energies (assuming a fixed value of $f_{e}$).

Additionally, Figure 9$b$ shows that the initial values of $p_{e,\perp}/p_{e,\parallel}$ are larger for electrons that end up being more energetic, with the initial $p_{e,\perp}/p_{e,\parallel}$ being $\sim 0.9$, $\sim 1.1$ and $\sim 1.7$ for the thermal, low-energy nonthermal and high-energy nonthermal electrons, respectively. This is consistent with the fact that, before the PEMZ dominated regime ($t\cdot s\lesssim 2.2$), the three electrons populations gain their energy mainly due to AV, as can be seen from the three panels in Figure 6. This means that during that period, energy is gained more efficiently by electron populations with larger average pitch-angle. After that, given that the scattering by OQES waves strongly reduces the pitch-angle of the nonthermal electrons, their energy gain is no longer related to their $p_{e,\perp}/p_{e,\parallel}$, but to the more efficient acceleration that the PEMZ modes cause on initially more energetic electrons.

Figures 11$a$ and 11$b$ show a 2D view of $\delta\textit{B}_{z}$ for the unstable modes in run C2400 at $t\cdot q=1.5$ (after the saturation of $\delta\textbf{{B}}$) and $t\cdot q=3$, respectively. Although the box has initially a square shape, the effect of compression makes its $y$-size decreases with time as $1/(1+qt)$, which explains the progressively more elongated shape of the box shown by the two panels. The dominant modes at $t\cdot q=1.5$ have wavevectors that are oblique with respect to the direction of the mean magnetic field $\langle\textbf{{B}}\rangle$, which is shown by the black arrows. At $t\cdot q=3$ the waves propagate mainly along $\langle\textbf{{B}}\rangle$, which shows that by the end of the simulation the modes are quasi-parallel. The time for the transition between the dominance of oblique to quasi-parallel modes can be obtained by calculating the magnetic energy contained in each type of mode. As for the shearing simulations shown in §3, we define the modes as oblique (quasi-parallel) when the angle between their wave vector k and $\langle\textbf{{B}}\rangle$ is larger (smaller) than 20^∘. Figure 10$a$ shows the magnetic energies in oblique and quasi-parallel modes using dashed-black and solid-black lines, which are denoted by $\langle\delta B_{ob}^{2}\rangle$ and $\langle\delta B_{qp}^{2}\rangle$, respectively. The oblique modes dominate until $t\cdot s\approx 2$, which corresponds to $f_{e}\approx 1.6$. After that, the energy in quasi-parallel modes dominates the magnetic fluctuations. The existence of these oblique and quasi-parallel regimes, and the fact that the transition occurs when $f_{e}\approx 1.6$, implies that dominance of the different modes is essentially determined by the value of $f_{e}$, as it was found in the case of the shearing simulations.

Similarly to what occurs in the shearing case, the oblique modes in the compressing runs show significant fluctuations in the electron density $n_{e}$. This is shown in Figure 12$a$, which shows $\delta n_{e}$ at $t\cdot q=1.5$. The fluctuations in the electron density practically disappear when the quasi-parallel modes dominate, as shown by Figure 12$b$, which shows $\delta n_{e}$ at $t\cdot q=3$. These different behaviors of $\delta n_{e}$ suggest the existence of a significant electrostatic component in the electric field $\delta\textbf{{E}}$ when the oblique modes dominate. This is confirmed by Figure 10$c$, which shows the energy contained in the electric field fluctuations $\delta\textbf{{E}}$ as a function of time, separating it into electrostatic ($\delta E^{2}_{es}$) and electromagnetic ($\delta E^{2}_{em}$) components, which are shown in blue and black lines, respectively. When the oblique modes dominate ($t\cdot q\lesssim 2$) the electric field energy is dominated by $\delta E^{2}_{es}$; after that it gradually becomes dominated by $\delta E^{2}_{em}$. Figure 10$c$ also shows that this transition occurs when the instantaneous $f_{e}\sim 1.6-1.7$ (the instantaneous $f_{e}$ is shown by the upper horizontal axis), which is fairly similar to the transition seen in the case of shearing simulations at $f_{e}\sim 1.2-1.5$.

As in the shearing case, PEMZ modes driven by plasma compression also accelerate electrons, although there are some significant differences. Figure 13$a$ shows $dn_{e}/d\ln(\gamma_{e}-1)$ for run C2400 at different times $t\cdot q$. We see the rapid growth of a nonthermal tail starting at $t\cdot q\approx 2$. After $t\cdot s\approx 2.5$ this tail can be approximated as a power-law of index $\alpha_{s}\approx 3.7$ with a break at $\gamma_{e}-1\sim 1$, where the spectrum becomes significantly steeper. The fact that most of the nonthermal behavior starts when the PEMZ modes become dominant ($t\cdot q\sim 2$), suggests that, as in the shearing case, the nonthermal acceleration is mainly driven by the PEMZ modes.

One significant difference between the shearing and compressing case is that the latter gives rise to a softer nonthermal component in the final electron energy spectrum than the former (a power-law of index $\alpha_{s}\approx 3.7$ instead of $\alpha_{s}\approx 2.9$). This difference is not surprising given that the overall electron energy gain when $f_{e}$ evolves from $f_{e}=0.53$ to $f_{e}\approx 2$ is significantly larger for the compressing runs. While in the shearing case the final value of $\Theta_{e,\perp}$ is about $\sim 3$ times larger than its initial value (see the solid-black line in Figure 1$b$), in the compressing case the final $\Theta_{e,\perp}$ is about $\sim 10$ times larger than the initial $\Theta_{e,\perp}$ (see the solid-black line in Figure 10$b$). Since at the end of both types of simulations $\Theta_{e,\perp}\gg\Theta_{e,\parallel}$, this difference implies that by the end of the compressing runs the electrons are significantly hotter than in the shearing runs. The different electron internal energies may affect the efficiency with which electrons gain energy from their interaction with the PEMZ modes, as it has been shown by previous quasi-linear theory studies of the stochastic acceleration of electrons by whistler/z modes (Summers et al, 1998). This implies that shearing and compressing runs that start with the same electron parameters should not necessarily produce equally hard nonthermal component in the electron spectrum after $f_{e}$ has increased from 0.53 to $\sim 2$.

Figure 13$b$ shows a numerical convergence test of the final spectrum at $t\cdot q=3$. It compares run C2400 ($d_{e}^{\textrm{init}}/\Delta x=50$, $N_{epc}=200$ and $L/R_{Le}^{\textrm{init}}=78$ as shown in Table 2) with a run with $d_{e}^{\textrm{init}}/\Delta x=40$ (run C2400a, in blue-dotted line), a run with $N_{epc}=100$ (run C2400b, in red-dotted line) and a run with $L/R_{Le}^{\textrm{init}}=39$ (run C2400c, in green-dotted line). No significant difference can be seen between the different spectra, with only a slight hardening for the highest resolution run C2400, which implies that our results are reasonably well converged numerically.

The effect of varying $\omega_{ce}^{\textrm{init}}/q$ is investigated in Figure 13$c$, where the final spectra ($t\cdot q=3$) are shown for runs with $\omega_{ce}^{\textrm{init}}/q=300,600$, 1200 and 2400 (runs C300, C600, C1200 and C2400, respectively). As $\omega_{ce}^{\textrm{init}}/q$ increases there is a hardening of the low energy part of the nonthermal tail (0.3 $\lesssim\gamma_{e}-1\lesssim 0.8$), which converges towards a power-law tail of index $\alpha_{s}\approx 3.7$ with little difference between the cases with $\omega_{ce}^{\textrm{init}}/q=1200$ and 2400. Additionally, as $\omega_{ce}^{\textrm{init}}/q$ increases, there is a decrease in the prominence of a high energy bump at $\gamma_{e}-1\sim 1$, which essentially disappears for $\omega_{ce}^{\textrm{init}}/q=2400$. This suggests that the magnetizations $\omega_{ce}^{\textrm{init}}/q$ used in our compressing runs provides a reasonable approximation to the expected spectral behavior in realistic flare environments.