tokens: the importance of the presented ion acceleration in these systems. In terms of the long-term evolution of the acceleration process, in the turbulent environment of accretion disks, we expect many successive δB / B ∼ 1 amplifications and decreases of the field (see also Verscharen et al. 2016 ). So a more realistic picture of this process should consider the acceleration presented in this work occurring many times as the plasma is gradually accreted. We will study these aspects of the acceleration process in future investigations. Appendix A 1D Shear Setup The goal of our 1D shear runs is to simulate a shearing plasma, assuming that its properties depend only on the direction parallel to the mean background magnetic field . This is equivalent to assuming that the wavevectors captured in the simulations satisfy . Since, in a shearing plasma, the direction of evolves with time, the orientation of the wavevectors that can be consistently resolved needs to evolve accordingly. Our 1D setup is built upon the “shearing coordinates” setup presented by Riquelme et al. ( 2012 ). This setup was designed so that the simulation domain follows the shearing flow of the plasma, which is given by the shear velocity (). Figures 1 (a) and (b) illustrate the way the shape of the domain evolves as seen by an inertial observer at t = 0 and t > 0, respectively. Formally, the shearing coordinates ( x ′, y ′, z ′, t ′) are defined in terms of the regular, inertial coordinates ( x , y , z , t ) aswhere , , 19 19 Notice that, in the limit , the expression is equivalent to v = − sx , as we assume in Figures 1 (a) and (b). Since ensures that , this expression was adopted in Riquelme et al. ( 2012 ) in order to deduce the equations that describe the dynamics of the plasma in the shearing coordinate system. However, both in Riquelme et al. ( 2012 ) and in this paper, we are interested in the plasma dynamics in the regime . c is the speed of light, and s is the shear rate of the plasma. Our 1D domain is defined by y ′ = 0, and it is shown by the blue lines in Figures 1 (a) and (b). These figures show how the length and orientation of the 1D domain change over time. This in turn changes the orientation of the wavevectors that can be captured within the domain. In addition, there is an initially homogeneous magnetic field, , pointing parallel to the 1D domain at t = 0. Magnetic flux conservation ensures that will always be parallel to the 1D domain at t > 0, as also depicted in Figures 1 (a) and (b). Therefore, if the problem of interest is dominated by waves that propagate parallel to , our 1D simulations will be able to capture the essence of the phenomenon. In order for our 1D simulations to consistently satisfy , we need to replace x ′ with a new coordinate, , which we define asIndeed, if we assume and sy ′ ≪ c , the partial derivatives with respect to the spatial coordinates in the inertial frame areThus, if we impose the 1D condition, namely, that the fields depend only
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