Particle Energization During Solar Maximum: Diffusive Shock Acceleration at Multiple Shocks

Abstract

We present a model for the acceleration of particles at multiple shocks using an approach related to box models. A distribution of particles is diffusively accelerated inside the box while simultaneously experiencing decompression through adiabatic expansion and losses from the convection and diffusion of particles out of the box by either the method used in Melrose & Pope and Pope & Melrose or by the approach introduced in Zank et al. where we solve the transport equation by a method analogous to operator splitting. The second method incorporates the additional loss terms of convection and diffusion and allows for the use of a variable time between shocks. We use a maximum injection energy (E max ) appropriate for quasi-parallel and quasi-perpendicular shocks. We provide a preliminary application of the diffusive acceleration of particles by multiple shocks with frequencies appropriate for solar maximum.

Numerical Test of Different Approximations Used in the Transport Theory of Energetic Particles

“…3. In Matthaeus et al (2003) it was suggested to approximate fourth-order correlations involving particle velocity vector components and magnetic field vector components by a product of two second-order correlations (see Equation (21) of the current paper). As shown in Figures 1-5, this approximation is not valid.…”

“…By combining Equations (13) and 14, we can compute the coefficient k xx GC . In linear and nonlinear analytical treatments of perpendicular diffusion, one usually starts with the following equation of motion (see, e.g., Matthaeus et al 2003):…”

“…In the second line we can see the emergence of a fourth-order correlation function involving magnetic fields and particle velocities. In Matthaeus et al (2003) such fourth-order correlations were approximated by a product of two second-order correlations…”

“…Analytical theory is required in order to achieve a complete understanding of particle motion in plasmas. Furthermore, simple analytical forms of the different diffusion parameters are necessary in investigations of solar modulation (see, e.g., Engelbrecht & Burger 2014;Siluszyk et al 2014;Zhao et al 2014;Ahluwalia & Ygbuhay 2015;Manuel et al 2015) and different shock acceleration scenarios (see, e.g., Dosch & Shalchi 2010;Li et al 2012;Ferrand et al 2014;Neergaard Parker & Zank 2014). Those can only be obtained from analytical treatments of the transport.…”

“…As shown in Shalchi (2015b), quasi-linear theory works only for perpendicular diffusion if pitch-angle scattering is suppressed and if the Kubo number is small. Therefore, nonlinear theories for perpendicular diffusion were developed such as the nonlinear guiding center (NLGC) theory of Matthaeus et al (2003). The latter theory is based on a certain set of approximations and assumptions.…”