tokens: (electron + proton) fluid, we find that the post-shock proton temperature in the direction perpendicular to the field, which we take as the initial condition in our periodic boxes, will beThe parallel proton and electron temperatures are the same as in the pre-shock region; in particular, With regard to the initialization of electrons, we assume that the electron anisotropy is rapidly erased due to the rapid development of electron-scale instabilities, as it indeed occurs in the shock ramp. 8 8 Despite the fact that we now assume isotropic electrons, the ansatz of a Γ = 2 gas that we employed to calculate the jump conditions will still hold because the post-shock pressure is mostly contributed by protons (see Section 5 ). If this happens without any energy exchange with the protons, the isotropic electron temperature isFinally, the initial electron plasma beta can be computed from flux freezing as The physical and numerical parameters used for the periodic box experiments of this section are summarized in Table 2 (the run names have the suffix “nc” to indicate that the boxes are not compressing). We indicate the Mach number M s and plasma beta β p 0 of the corresponding shock simulations, the initialization values of , , , and employed in the periodic boxes, the number of computational particles per cell (), and the box length L y along the y -direction of the background large-scale field, in units of the proton skin depth . We employ a reduced mass ratio . From Table 2 , it is apparent that the Mach number M s has a pronounced effect on the initial proton anisotropy. In fact, in the limit of , we have and , so that from Equations ( 12 ) and ( 13 ). Table 2 Parameters for the Anisotropic Protons Box Experiments Described in Section 4 Note. We employ a fixed mass ratio and resolve the electron skin depth with seven cells, which is sufficient to capture the electron debye length. 4.2. Dependence on M<sub>s</sub> In this subsection, we compare the periodic box simulations , , , , that correspond to shocks with fixed but different M s , ranging from 2 to 5. Our results are shown in Figure 5 . The case corresponding to M s = 2 is plotted with a dashed line to emphasize that its time axis has been reduced by a factor of two, to facilitate comparison with the other cases. Figure 5. Dependence on M s of various quantities in periodic box experiments with anisotropic protons (runs , , , ), at fixed . As a function of time in units of , we plot: (a) energy in magnetic field fluctuations, normalized to the energy of the initial field in the box (b) proton temperature anisotropy; (c) magnetic pressure in electron-scale fluctuations, normalized to the electron parallel pressure; (d) electron temperature anisotropy (solid lines for the box-averaged values, shaded regions for the 50–90 percentiles) and threshold of the electron whistler instability (dotted lines with the same color coding as the solid lines); (e) rate of violation of the electron
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