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tokens: 1, 2), etc., while the (1, 1, 0) mode is not important. We should therefore conclude that magnetic relaxation towards the k = 1 ABC state is more advanced in our k ini = 4 simulation. Figure 5. Energy contribution of the dominant Fourier modes of the B x magnetic field component as function of simulation time ct / L for simulations k2_T5_1024M (left), k2_T6ic_1152P (middle), and k4_T5_1152P (right). The black lines show the total energy of the B x component, represented as . The red lines show the contribution of the initial ‘ABC’ modes: (0, k ini , 0) and (0, 0, k ini ); the green lines show the contribution of the (1, 1, 0) modes; the blue lines show the contribution of the ‘Taylor’ modes (0, 1, 0) and (0, 0, 1); and the grey lines show the residual energy contained in all other modes. Fig. 6 shows the magnetic energy spectra for simulations k2_T5_1024M, k2_T6ic_1152P, and k4_T5_1152P for a range of simulation times. The magnetic energy spectra are dominated by the white noise for for the T5 simulations and for for the k2_T6ic_1152P simulation. The inertial subrange extends over at least one order of magnitude in wavenumber . In all cases, we observe a freely decaying turbulent cascade that can be described roughly as a power law , with significant deviations in either direction. The cascade appears to be most regular for simulation k4_T5_1152P. Figure 6. Magnetic energy spectra calculated for simulations k2_T5_1024M (left), k2_T6ic_1152P (middle), and k4_T5_1152P (right). Line colours indicate the progression of simulation time from deep blue to brown within the time ranges indicated in the bottom-left corner of each panel. The thick dashed lines indicate the power-law slopes of −5/3, −2, −2.5, −3. Fourier decomposition was also attempted for electric field , and for velocity field , however, the corresponding spectra of electric and kinetic energies are dominated by the white noise. 3.4 Particle momentum distribution Fig. 7 presents the time evolution of the particle momentum distributions for all five simulations. For ultra-relativistic electrons with Θ ≫ 1, the dimensionless particle momentum is basically equivalent to the dimensionless particle energy u = γβ = p /( m e c ) ≃ γ. For all simulations, the initial momentum distribution is adopted to be the Maxwell–Jüttner distribution (Section 2). Figure 7. Particle momentum distributions u 2 N ( u ) compared for all simulations. The distributions are normalized to the peak of the initial distribution for each simulation. Line colours indicate the progression of simulation time from deep blue to brown over the range indicated in the top-left corner of each panel. We find evidence for non-thermal particle acceleration for all simulations with k ini = 2. The effective power-law indices p (such that N ( u )∝ u − p ) are estimated roughly by finding a power-law slope that makes a line parallel to the data (see Werner et al. 2016 for more elaborate fitting methods), with a different approach used