Kinetic energy partitions in electron–ion PIC simulations of ABC fields

Various astrophysical environments involve extremely rapid and very luminous gamma-ray flares, such as the blazars and the Crab pulsar wind nebula (PWN). Particle acceleration mechanisms are needed to explain how magnetic energy is released and converted to kinetic energy. Currently, magnetic reconnection is one of the very promising candidates (for a review, see references [1, 2]).

The essential condition for triggering magnetic reconnection includes oppositely directed magnetic fields and a very thin current sheet that can be produced by certain plasma instabilities. The traditional magnetic field configuration is the Harris-layer-type kinetic equilibrium [3], in which the kinetic-scale current layer limits the outflow (reconnection rate) to the order of ∂(0.1); hence, the magnetic dissipation efficiency is also limited. In order to understand the rapid and efficient conversion of electromagnetic energy into radiation, the Arnold–Beltrami–Childress (ABC) fields [4, 5] have recently been proposed as an alternative candidate [6–8].

Owing to the improvements in computational power, we can study the properties of magnetic reconnection by practicing numerical simulation. Based on the ABC fields, we particularly focus on the difference in the behaviors of electrons and ions, especially their kinetic energy gains.

We performed particle-in-cell (PIC) numerical simulations using the Zeltron code⁽¹⁾ [9] of 2-D periodic magnetic equilibria known as the ABC fields [10–12]:

1 Bx=B1[sin⁡(α0(x+y))+sin⁡(α0(x−y))]/2By=B0[sin⁡(α0(x−y))−sin⁡(α0(x+y))]/2Bz=B0[cos⁡(α0(x+y))−cos⁡(α0(x−y))] \begin{aligned} & B_x=B_1\left[\sin \left(\alpha_0(x+y)\right)+\sin \left(\alpha_0(x-y)\right)\right] / \sqrt{2} \\ & B_y=B_0\left[\sin \left(\alpha_0(x-y)\right)-\sin \left(\alpha_0(x+y)\right)\right] / \sqrt{2} \\ & B_z=B_0\left[\cos \left(\alpha_0(x+y)\right)-\cos \left(\alpha_0(x-y)\right)\right] \end{aligned}

where α₀ = 2πk_eff/L for the linear domain size L and keff =2$${k_{{\rm{eff}}}} = \sqrt 2 $$ is the effective wavenumber.

The particle density is given by the expression

2 n=32α0B04πea~1⟨β⟩ $$n = {{3\sqrt 2 {\alpha _0}{B_0}} \over {4\pi e{{\tilde a}_1}\left\langle \beta \right\rangle }}$$

where 〈β〉 is the average dimensionless speed of ions and electrons, and a˜1≤½${\mathop a\limits^ \sim _1} \le {\raise0.5ex\hbox{$\scriptstyle 1$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 2$}}$ is a constant that normalizes the dipole moment of the local particle distribution. The initial total mean magnetization is given as follows:

3 ⟨σini⟩=22a~1⟨β⟩3[Cw,i(Θi)μ+Cw,e(Θe)]α0ρe $$\left\langle {{\sigma _{{\rm{ini}}}}} \right\rangle = {{2\sqrt 2 {{\tilde a}_1}\left\langle \beta \right\rangle } \over {3\left[ {{C_{{\rm{w}},{\rm{i}}}}({{\rm{\Theta }}_i})\mu + {C_{{\rm{w}},{\rm{e}}}}({{\rm{\Theta }}_{\rm{e}}})} \right]{\alpha _0}{\rho _{\rm{e}}}}}$$

where C_w(Θ) = 4Θ + K₁(1/Θ)/K₂(1/Θ) is a function of the dimensionless temperature Θ = kT/mc² and K_n(x) is the modified Bessel function of the second kind with n = 0, 1, 2. In this paper, we use the subscript “i” to represent the parameters related to ions and “e” to represent electrons. Further, ρ_e = m_ec²/eB₀ is the electron gyroradius.

We investigate a wide range of relativistic ion temperatures as 10⁻² ≤ Θ_i ≤ 10² and electron temperatures as 10⁻² ≤ Θ_e ≤ 10³. We set B₀ = 1. In order to resolve the degeneration of 〈σ_ini〉 with the temperatures Θ_i and Θ_e, we probe a˜1$${\mathop a\limits^ \sim _1}$$ as 0.1, 0.25 (default), and 0.4. We use numerical grids of 2000² and 4608² cells. The particle number per cell per species is 64. The domain size is L = 1920ρ₀, where ρ₀ = min[Θ_em_ec²/(eB₀), μΘ_im_ec²/(eB₀)] is the nominal gyroradius. Each simulation runs at least 51000 numerical steps.

For kinetic simulations, it is necessary to resolve the gyroradii of both ions and electrons at the same time. Limited by the grid sizes and resolutions of both species of particles, we choose mass ratios of μ = 10,100 instead of the real ion:electron mass ratio 1836. Previous studies configured with mass ratios either as 1836 [13] or a sequence such as 10, 25, 50, and 1836 [14]. Differences caused by the choices of the mass ratio can be ignored by simultaneously increasing the electron temperatures, because the plasma magnetization 〈σ_ini〉∝Θ_e/μ in the limit Θ_e >> 1.

In conclusion, our configurations yield initial plasma magnetiza tion 〈σ_ini〉 ≲ <14.38.

In this contribution, we show that the initial ion–electron enthalpy ratio determines the partition of kinetic energy between the particle species, such that the species dominating the initial enthalpy also dominates the peak of the spectra and the kinetic energy gains. However, because both species reach a similar level of the maximum particle energy by the end of the simulations, the species not dominating the initial enthalpy dominates the nonthermal particle number fractions and the nonthermal energy fractions. These results differ from the findings of previous studies, because not all our results follow their predicted trends. One most likely reason is that the majority of the previous papers’ choices of the ion–electron temperature combinations are limited in or near the relation Θ_i = Θ_e/μ, but the cases other than this relation are not taken into consideration. On the other hand, our results cover relatively larger temperature combinations. Meanwhile, we are currently analyzing whether these differences may also result from using different initial magnetic field configurations.