Particle Acceleration in the Presence of Weak Turbulence at an X-Type Neutral Point

Abstract

We simulate the likely noisy situation near a reconnection region by superposing many 2D linear reconnection eigenmodes. The superposition of modes on the steady state X-type magnetic field creates multiple X- and O-type neutral points close to the original neutral point and so increases the size of the non-adiabatic region. We study test particle trajectories of initially thermal protons in these fields. Protons become trapped in this region and are accelerated by the turbulent electric field to energies up to 1 MeV in time scales relevant to solar flares. Higher energies are achieved due to the interaction of particles with increasingly turbulent electric and magnetic fields.

Appendices

Appendix A: The Hypergeometric Function

₂ F ₁(a,b;c;z) is given by Abramowitz and Stegun (1965), Chapter 15:

$$ _2F_1(a,b;c;z)= \displaystyle\sum \limits _{i=0}^n \frac{(a)_n(b)_n}{(c)_n}\frac{z^n}{n!},$$

(8)

where (x) _n =x(x+1)(x+2)⋯(x+n−1). Equation (8) converges only for |z|<1. An efficient evaluation for |z|>1 is achieved via a transformation formula (Abramowitz and Stegun 1965):

(9)

For the calculation of the magnetic field perturbation, the derivative of the hypergeometric function is used, which is given by Abramowitz and Stegun (1965) as:

$$\frac{d}{dz}F(a,b;c;z)=\frac{ab}{c}F(a+1,b+1;c+1;z).$$

Appendix B: Explicit Forms of Electric and Magnetic Fields

The electric fields we use are the same as those given in Petkaki and MacKinnon (1997).

$$\overline{E_z}=A_0\bigl[\exp(-\kappa t)\bigl[\kappa \bigl(\cos(\omega t)f_\Re \sin(\omega t)f_\Im\bigr)+\omega \bigl(\cos(\omega t)f_\Im+\sin(\omega t)f_\Re\bigr)\bigr]\bigr],$$

(10)

(11)

(12)

A ₀ is the amplitude of the fluctuation, which we have chosen to be 1×10⁻⁴ for all modes. η is the resistivity, f is the hypergeometric function and f′ is its derivative. Bars denote quantities which are in our dimensionless units.