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

Figure 1(a) shows in solid green the linear growth of the y-component of the mean magnetic field 〈 B 〉 in shearing simulation S1200 (see Table 1). The x-component remains constant at a value of B₀ and the z (out of plane) component is zero, as expected in our shearing setup. In Figure 1(b) we see that, due to the growth of ∣〈 B 〉∣, the electron temperatures perpendicular and parallel to 〈 B 〉, Θ_e,⊥ (≡k_B T_e,⊥/m_e c²; solid black) and Θ_e,∥ (≡k_B T_e,∥/m_e c²; solid red), grow and decrease, respectively, as expected from their initially adiabatic evolutions. Indeed, in the initial regime, Θ_e,⊥ and Θ_e,∥ evolve according to the adiabatic Chew–Goldberg–Low (CGL) equation of state (Chew et al. 1956), shown by the dashed black and dashed red lines, respectively. ⁸ The departure from the CGL evolution at t · s ≈ 1.4 coincides with the rapid growth and saturation of δ B (≡ B − 〈 B 〉), as shown by the solid red line in Figure 1(a). This shows that the growth of the temperature anisotropy is ultimately limited by temperature-anisotropy-unstable modes, which can break the adiabatic evolution of the electron temperatures by providing efficient pitch-angle scattering.

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A 2D view of the relevant unstable modes at t · s = 1.6 (right after the saturation of δ B ) is given by Figure 2, which shows the three components of the magnetic and electric fluctuations δ B and δ E . ⁹ 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 〈 B 〉, 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 · s = 3.1. In this case, the waves propagate along the background magnetic field, implying that quasiparallel modes dominate both the electric and magnetic fluctuations.

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In order to determine the transition time from the dominance of oblique to quasiparallel modes, we calculate the magnetic energy contained in each type of modes. For this, we define the modes as "oblique" or "quasiparallel" depending on whether the angle between their wavevector k and 〈 B 〉 is larger or smaller than 20°, respectively. The magnetic energies in oblique and quasiparallel modes are shown in Figure 1(a) using dashed black and solid black lines and are denoted by $\langle \delta {B}_{\mathrm{ob}}^{2}\rangle$ and $\langle \delta {B}_{\mathrm{qp}}^{2}\rangle$, respectively. We see that the oblique modes dominate until t · s ≈ 2.2. After that, the quasiparallel 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 suggest the presence of a significant electrostatic component in the electric field δ E . These density fluctuations are less prominent in the case that is dominated by quasiparallel modes, as can be seen from Figure 3(h), which shows that at t · s = 3.1, the electrostatic electric fields are weaker than at t · s = 1.6. This change in the δ E behavior can be seen more clearly in Figure 1(c), which shows the evolution of the energy contained in the electric field fluctuations δ 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 ($\delta \tilde{{\boldsymbol{E}}}$) that satisfy ${\boldsymbol{k}}\times \delta \tilde{{\boldsymbol{E}}}=0$ (electrostatic part) and ${\boldsymbol{k}}\cdot \delta \tilde{{\boldsymbol{E}}}=0$ (electromagnetic part). In the oblique regime (t · s ≲ 2.2), the electric field energy is mainly dominated by its electrostatic part ($\delta {E}_{\mathrm{es}}^{2}$), and after this, it gradually becomes dominated by its electromagnetic part ($\delta {E}_{\mathrm{em}}^{2}$). Figure 1(c) also shows that this transition occurs when the instantaneous parameter f_e (shown by the upper horizontal axis) is f_e ∼ 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 a mainly electrostatic electric field (hereafter, oblique electrostatic modes) to the dominance of quasiparallel modes with a mainly electromagnetic electric field (hereafter, quasiparallel electromagnetic modes) in S1200 is consistent with linear Vlasov theory. Indeed, a previous study by Gary & Cairns (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 quasiparallel electromagnetic (oblique electrostatic) modes dominate in our run. For this we use the NHDS solver of Verscharen & Chandran (2018) to calculate the temperature anisotropy threshold Θ_e,⊥/Θ_e,∥ − 1 needed for the growth of these modes, with a given growth rate γ_g , assuming different values of f_e and Θ_e,∥.

Figure 4(a) shows the anisotropy thresholds for γ_g /ω_ce = 10⁻², which is appropriate for run S1200. We estimate the growth rate of δ B in this run from its exponential growth regime in Figure 1(a) (t · s ∼ 1.2−1.3), which is γ_g ∼ 10s. This implies that ${\gamma }_{g}\sim {10}^{-2}{\omega }_{\mathrm{ce}}^{\mathrm{init}}$, given that in run S1200, ${\omega }_{\mathrm{ce}}^{\mathrm{init}}/s=1200$. The dashed lines in Figure 4(a) consider the thresholds only for parallel modes (i.e., they consider 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 Θ_e,∥ = 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 Θ_e,∥ = 0.002 and 0.006 occurs when 0.4 ≲ f_e ≲ 1.8 and 0.8 ≲ f_e ≲ 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 these regimes. ¹⁰

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Thus, the dominance of oblique electrostatic modes in run S1200 from the moment when the instability sets in (f_e ∼ 0.8) until f_e ∼ 1.2–1.5, suggests that, in order to be consistent with linear theory, Θ_e,∥ should be in the range ∼ 0.002–0.006 after the growth of the instabilities (f_e ≳ 0.8). This is indeed what is shown by Figure 1(b), where Θ_e,∥ (solid red line) appears in the range Θ_e,∥ ∼ 0.0025–0.005 when f_e ≳ 0.8.

These results show that the transition between the oblique, electrostatic to quasiparallel, electromagnetic regimes at f_e ∼ 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 ∼ 1.3–1.8. Therefore, hereafter we refer to the f_e ≲ 1.2–1.5 and f_e ≳ 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 ${{dn}}_{e}/d\mathrm{ln}({\gamma }_{e}-1)$ for different values of t · s (γ_e is the electron Lorentz factor). This plot shows the rapid growth of a nonthermal tail starting at t · s ≈ 2.5. After t · s ≈ 3.5, this tail can be approximated as a power law of index α_s ≈ 2.9 (${\alpha }_{s}\equiv d\mathrm{ln}({n}_{e})/d\mathrm{ln}({\gamma }_{e}-1)$), plus a high-energy bump that reaches γ_e − 1 ∼ 0.6 (∼300 keV). Most of the nonthermal behavior of the spectrum starts at t · s ≈ 2.5, which is right after the quasiparallel electromagnetic modes become dominant, as shown by Figure 1(c).

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Figure 5(b) shows a numerical convergence test of the final spectrum at t · s = 4. It compares run S1200 (${d}_{e}^{\mathrm{init}}/{\rm{\Delta }}x=35$, N_epc = 100, and $L/{R}_{{Le}}^{\mathrm{init}}=140$) with a run with ${d}_{e}^{\mathrm{init}}/{\rm{\Delta }}x=25$ (run S1200a, dotted blue line), a run with N_epc = 50 (run S1200b, dotted red line), and a run with $L/{R}_{{Le}}^{\mathrm{init}}=70$ (run S1200c, dotted green 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)

$$\begin{eqnarray}&&\displaystyle \frac{{{dU}}_{e}}{{dt}}=r{\rm{\Delta }}{p}_{e},\end{eqnarray} \tag{ 2 }$$

where r is the growth rate of the field (in our setup, r = −sB_x B_y /B²) and Δp_e is the difference between the perpendicular and parallel electron pressures, Δp_e = p_e,⊥ − p_e,∥. In run S1200, Equation (2) reproduces the evolution of the overall electron energy gain well. This can be seen from Figure 5(c), where the time derivative of the average electron internal energy d〈U_e 〉/dt (blue) coincides well with r〈Δp_e 〉 (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 with 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 · s = 4. These populations are listed below.

• 1.  
  Thermal electrons: their final energy satisfies γ_e − 1 <0.05, which corresponds to the energy range marked by the light blue background in Figure 5(a).
• 2.  
  Low -energy nonthermal electrons: their final energies satisfy 0.05 < γ_e − 1 < 0.2, where the nonthermal tail roughly behaves as a power law of index α_s ≈ 2.9. This energy range is marked by the white background in Figure 5(a).
• 3.  
  High -energy nonthermal electrons: corresponding to the high-energy bump in the spectrum, defined by 0.2 < γ_e − 1. This energy range is marked by the gray 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, 〈γ_e 〉₀ − 1, plus

• 1.  
  the work done by the electric field of the unstable modes, W_E , shown by the blue lines,
• 2.  
  the energy gain by the anisotropic viscosity, AV, shown by the green lines,
• 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) show 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 (∼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 an increase of a factor ∼2 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 electron 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 the evolution of the total energy 〈γ_e 〉 − 1 (black line) of the three electron populations reasonably well.

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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 electric field of the waves. 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 · s ∼ 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 loop tops 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 loop tops to collapse into a more stable configuration.

We thus estimate s by dividing the typical Alfvén velocity, v_A, in the loop tops (v_A should be close to the speed at which the newly reconnected loops are ejected from their current sheet) by the typical length scale of the contracting loop tops, L_LT. Using our fiducial parameters n_e ∼ 10⁹ cm⁻³ and B ∼ 100 G, and estimating L_LT ∼ 10⁹ cm (e.g., Chen et al. 2020), we obtain s ∼ 1 s⁻¹ and ω_ce/s ∼ 10⁹. This value of ω_ce/s is several orders of magnitude higher than what can be achieved in our simulations. Therefore, two important questions arise. The first question is whether for realistic values of ω_ce/s, the dominance of PEMZ and OQES modes should occur for the same regimes as 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 ω_ce/s.

Since in Section 3.3 we showed that linear Vlasov theory is suitable for predicting the dominance of the different modes, in this section, we use linear theory to show that increasing ${\omega }_{\mathrm{ce}}^{\mathrm{init}}/s$ by several orders of magnitude should not modify the relative importance of the modes in the different plasma regimes. First, we note that the growth rate γ_g of the unstable modes should be proportional to s, implying that ${\omega }_{\mathrm{ce}}^{\mathrm{init}}/s\propto {\omega }_{\mathrm{ce}}^{\mathrm{init}}/{\gamma }_{g}$. This proportionality is physically expected because 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 about 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 δ B divided into oblique and quasiparallel modes (dashed and solid lines, respectively) for runs with ${\omega }_{\mathrm{ce}}^{\mathrm{init}}/s=600$, 1200, and 2400 (runs S600, S1200, and S2400, respectively). We see that the three runs show essentially the same ratio γ_g /s ∼ 10. This means that, in order to find out which instabilities would dominate for realistically high values of ${\omega }_{\mathrm{ce}}^{\mathrm{init}}/s$, we must calculate the instability thresholds for a comparatively high value of ω_ce/γ_g . Following this criterion, Figure 4(b) shows these thresholds using the same values of f_e and Θ_e,∥ as were used in Figure 4(a), but assuming γ_g /ω_ce = 10⁻⁶ instead of 10⁻². We see that for γ_g /ω_ce = 10⁻⁶, the PEMW, PEMZ, and OQES modes dominate for values of f_e and Θ_e,∥ that are very similar to those shown for γ_g /ω_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.

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We now investigate whether increasing ${\omega }_{\mathrm{ce}}^{\mathrm{init}}/s$ affects the spectral evolution of the electrons. This is done in Figure 7(b), where the final spectra (t · s = 4) are shown for runs with ${\omega }_{\mathrm{ce}}^{\mathrm{init}}/s=300,600$, 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 ${\omega }_{\mathrm{ce}}^{\mathrm{init}}/s$ increases, which does not seem significant when comparing the cases ${\omega }_{\mathrm{ce}}^{\mathrm{init}}/s=$ 2400 and 1200. This shows that the magnetization parameter ${\omega }_{\mathrm{ce}}^{\mathrm{init}}/s$ does not play a significant role in the efficiency of the electron nonthermal acceleration.

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

$$\begin{eqnarray}&&\displaystyle \frac{d{{\rm{\Theta }}}_{e,\parallel }}{{dt}}+2{{\rm{\Theta }}}_{e,\parallel }\hat{{\boldsymbol{b}}}\hat{{\boldsymbol{b}}}:{\rm{\nabla }}{\boldsymbol{v}}={\nu }_{\mathrm{eff}}\displaystyle \frac{2}{3}{\rm{\Delta }}{{\rm{\Theta }}}_{e},\end{eqnarray} \tag{ 3 }$$

where v is the plasma shear velocity, $\hat{{\boldsymbol{b}}}=\hat{{\boldsymbol{B}}}/B$, and ΔΘ_e = Θ_e,⊥ − Θ_e,∥. Considering that ${\boldsymbol{v}}=-{sx}\hat{y}$, we can rewrite Equation (3) as

$$\begin{eqnarray}&&\displaystyle \frac{d{{\rm{\Theta }}}_{e,\parallel }}{d({ts})}+2{{\rm{\Theta }}}_{e,\parallel }{\hat{b}}_{x}{\hat{b}}_{y}=\displaystyle \frac{2}{3}{\rm{\Delta }}{{\rm{\Theta }}}_{e}\displaystyle \frac{{\nu }_{\mathrm{eff}}}{s}.\end{eqnarray} \tag{ 4 }$$

Figure 8(a) shows the evolution of 〈ΔΘ_e 〉 for the runs with ${\omega }_{\mathrm{ce}}^{\mathrm{init}}/s=600,1200$, and 2400 (runs S600, S1200, and S2400). When t · s ≳ 2.2, the factor ΔΘ_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 Θ_e,∥, which in the t · s ≳ 2.2 regime is also similar in the three runs, as can be seen from Figure 8(b). This means that for t · s ≳ 2.2, the ratio ν_eff/s in Equation (4) is fairly constant (within about ∼ 10%) for these three values of ${\omega }_{\mathrm{ce}}^{\mathrm{init}}/s$. Thus, for simulations with a fixed value of ${\omega }_{\mathrm{ce}}^{\mathrm{init}}$ but different s, the behaviors of 〈ΔΘ_e 〉 and 〈Θ_e,∥〉 for t · s ≳ 2.2 imply that ν_eff should be approximately proportional to s. Because of this, the number of times that the electrons are strongly deflected in a fixed interval of t · s should be largely independent of ${\omega }_{\mathrm{ce}}^{\mathrm{init}}/s$ on average. 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 ${\omega }_{\mathrm{ce}}^{\mathrm{init}}/s$. Indeed, as long as ${\omega }_{\mathrm{ce}}^{\mathrm{init}}\gg s$, the mode propagation and oscillations should occur rapidly (on timescales of $\sim {\omega }_{\mathrm{ce}}^{-1}$), and should not be affected by the slowly evolving background (on timescales of ∼s⁻¹). These arguments thus imply that the acceleration efficiency should be largely independent of ${\omega }_{\mathrm{ce}}^{\mathrm{init}}/s$.

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We note that Figures 8(a) and 8(b) show that in the t · s ≲ 2.2 regime, 〈ΔΘ_e 〉 and 〈Θ_e,∥〉 tend to depend more strongly on ${\omega }_{\mathrm{ce}}^{\mathrm{init}}/s$, so the electron acceleration efficiency should differ significantly during this period. However, Figure 7(a) shows that for t · s ≲ 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 Section 4.1), the different evolutions of 〈ΔΘ_e 〉 and 〈Θ_e,∥〉 for t · s ≲ 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 ${\omega }_{\mathrm{ce}}^{\mathrm{init}}/s$ tested by our simulations, we expect the ν_eff ∝ s relation to hold even for realistic values of ${\omega }_{\mathrm{ce}}^{\mathrm{init}}/s$. This is based on comparing the linear theory thresholds presented in Figures 4(a) and 4(b), which predict ΔΘ_e /Θ_e,∥ to have essentially the same evolution (only differing by an overall factor ∼3) when decreasing γ_g /ω_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 loop tops and precipitate toward 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,⊥/p_e,∥ 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 Section 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,⊥/p_e,∥ 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 Section 3, occurs at t · s ∼ 1.4. After this, the pitch-angle scattering tends to reduce p_e,⊥/p_e,∥ for the three populations, which occurs more abruptly for the low- and high-energy nonthermal electrons, whose pitch angles by t · s ∼ 1.7 become ∼2–3 times smaller than those of the thermal electrons. This shows that even though the OQES modes (which dominate until t · s ≈ 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 this, in the PEMZ-dominated regime, the anisotropy evolves in the opposite way. In this case, the p_e,⊥/p_e,∥ 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,⊥/p_e,∥ reaches values ∼2–3 times higher than for thermal electrons by the end of the run.

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This increase in the electron pitch angle of the highest energy electrons due to PEMZ-mode scattering is consistent with quasilinear 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 toward the flare footpoints, the high-energy nonthermal electrons produced by PEMZ-mode scattering should tend to be more confined to the loop top 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 ${\omega }_{\mathrm{ce}}^{\mathrm{init}}/s=600$ and 2400 (runs S600 and S2400). After the triggering of the instabilities (t · s ∼ 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 ${\omega }_{\mathrm{ce}}^{\mathrm{init}}/s$.

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 〈γ_e 〉₀ − 1 ∼ 0.01, ∼0.025, and ∼0.04, respectively. This energy correlation is also expected from the quasilinear 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 higher initial energies (assuming a fixed value of f_e ).

Additionally, Figure 9(b) shows that the initial values of p_e,⊥/p_e,∥ are higher for electrons that end up being more energetic, with the initial p_e,⊥/p_e,∥ being ∼0.9, ∼1.1, and ∼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 · s ≲ 2.2), the three electron populations gain their energy mainly due to AV, as can be seen from the three panels in Figure 6. This means that during this period, energy is gained more efficiently by electron populations with larger average pitch angle. After this, 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,⊥/p_e,∥, but to the more efficient acceleration that the PEMZ modes cause on initially more energetic electrons.

Figure 10(a) shows in solid green the growth of the x-component of 〈 B 〉 for simulation C2400 (${\omega }_{\mathrm{ce}}^{\mathrm{init}}/q=2400$), which evolves as ∣B_x ∣ = ∣〈 B 〉∣ = B₀(1 + qt)². Because of this magnetic amplification, initially, Θ_e,⊥ and Θ_e,∥ grow and stay constant, respectively, as seen from the solid black and solid red lines in Figure 10(b). For t · q ≲ 1, Θ_e,⊥ and Θ_e,∥ coincide with the adiabatic CGL evolution shown by the dashed black and dashed red lines, respectively. The departure from the CGL behavior at t · q ≈ 1 is coincident with the growth and saturation of δ B , shown by the solid red line in Figure 10(a). This shows that Θ_e,⊥ and Θ_e,∥ are regulated by the pitch-angle scattering provided by temperature-anisotropy-unstable modes, as is the case for the shearing simulations shown in Section 3.

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Figures 11(a) and 11(b) show a 2D view of δ B_z for the unstable modes in run C2400 at t · q = 1.5 (after the saturation of δ B ) and t · q = 3, respectively. Although the box initially has 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 · q = 1.5 have wavevectors that are oblique with respect to the direction of the mean magnetic field 〈 B 〉, which is shown by the black arrows. At t · q = 3, the waves propagate mainly along 〈 B 〉, which shows that by the end of the simulation, the modes are quasiparallel. The time for the transition between the dominance of oblique to quasiparallel modes can be obtained by calculating the magnetic energy contained in each type of mode. As for the shearing simulations shown in Section 3, we define the modes as oblique (quasiparallel) when the angle between their wavevector k and 〈 B 〉 is larger (smaller) than 20°. Figure 10(a) shows the magnetic energies in oblique and quasiparallel modes using dashed black and solid black lines, which are denoted by $\langle \delta {B}_{\mathrm{ob}}^{2}\rangle$ and $\langle \delta {B}_{\mathrm{qp}}^{2}\rangle$, respectively. The oblique modes dominate until t · s ≈ 2, which corresponds to f_e ≈ 1.6. After this, the energy in quasiparallel modes dominates the magnetic fluctuations. The existence of these oblique and quasiparallel regimes, and the fact that the transition occurs when f_e ≈ 1.6, implies that dominance of the different modes is essentially determined by the value of f_e , as was found in the case of the shearing simulations.

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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 δ n_e at t · q = 1.5. The fluctuations in the electron density practically disappear when the quasiparallel modes dominate, as shown by Figure 12(b), which shows δ n_e at t · q = 3. These different behaviors of δ n_e suggest the existence of a significant electrostatic component in the electric field δ E when the oblique modes dominate. This is confirmed by Figure 10(c), which shows the energy contained in the electric field fluctuations δ E as a function of time, separating it into electrostatic ($\delta {E}_{\mathrm{es}}^{2}$) and electromagnetic ($\delta {E}_{\mathrm{em}}^{2}$) components, which are shown as blue and black lines, respectively. When the oblique modes dominate (t · q ≲ 2), the electric field energy is dominated by $\delta {E}_{\mathrm{es}}^{2};$ after this, it gradually becomes dominated by $\delta {E}_{\mathrm{em}}^{2}$. Figure 10(c) also shows that this transition occurs when the instantaneous f_e ∼ 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 ∼ 1.2–1.5.

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

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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 α_s ≈ 3.7 instead of α_s ≈ 2.9). This difference is not surprising given that the overall electron energy gain when f_e evolves from f_e = 0.53 and f_e ≈ 2 is significantly higher for the compressing runs. While in the shearing case, the final value of Θ_e,⊥ is about 3 times higher than its initial value (see the solid black line in Figure 1(b)), in the compressing case, the final Θ_e,⊥ is about 10 times higher than the initial Θ_e,⊥ (see the solid black line in Figure 10(b)). Because at the end of both types of simulations Θ_e,⊥ ≫ Θ_e,∥, 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 has been shown by previous quasilinear 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 ∼2.

Figure 13(b) shows a numerical convergence test of the final spectrum at t · q = 3. It compares run C2400 (${d}_{e}^{\mathrm{init}}/{\rm{\Delta }}x=50$, N_epc = 200, and $L/{R}_{{Le}}^{\mathrm{init}}=78$) with a run with ${d}_{e}^{\mathrm{init}}/{\rm{\Delta }}x=40$ (run C2400a, dotted blue line), a run with N_epc = 100 (run C2400b, dotted red line), and a run with $L/{R}_{{Le}}^{\mathrm{init}}=39$ (run C2400c, dotted green 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 ${\omega }_{\mathrm{ce}}^{\mathrm{init}}/q$ is investigated in Figure 13(c), where the final spectra (t · q = 3) are shown for runs with ${\omega }_{\mathrm{ce}}^{\mathrm{init}}/q=300,600$, 1200, and 2400 (runs C300, C600, C1200, and C2400, respectively). As ${\omega }_{\mathrm{ce}}^{\mathrm{init}}/q$ increases, there is a hardening of the low-energy part of the nonthermal tail (0.3 ≲ γ_e − 1 ≲ 0.8), which converges toward a power-law tail of index α_s ≈ 3.7 with little difference between the cases with ${\omega }_{\mathrm{ce}}^{\mathrm{init}}/q=1200$ and 2400. Additionally, as ${\omega }_{\mathrm{ce}}^{\mathrm{init}}/q$ increases, there is a decrease in the prominence of a high-energy bump at γ_e − 1 ∼ 1, which essentially disappears for ${\omega }_{\mathrm{ce}}^{\mathrm{init}}/q=2400$. This suggests that the magnetizations ${\omega }_{\mathrm{ce}}^{\mathrm{init}}/q$ used in our compressing runs provide a reasonable approximation to the expected spectral behavior in realistic flare environments.