Fig. 3: Current-sheet flapping, ion distribution functions and dispersion analysis.

a–e, The current sheet at X = −14 R_E in the YZ plane at simulation times t = 1,340 s (a), 1,360 s (b), 1,380 s (c), 1,400 s (d) and 1,447 s (e). Colouring gives the current density J, and the red crosses mark the positions at which the ion velocity distributions from the current sheet in panels f–i are given. f–i, Ion velocity distributions in the current sheet at the position shown with a red cross in panels a–d at simulation times t = 1,340 s (f), t = 1,360 s (g), t = 1,380 s (h) and t = 1,400 s (i). The distributions are plotted in the same plane as the current sheet plots. The thick cross and red circle in panels g–i refer to the drift and thermal velocities that are required in the instability analysis in Methods. j, The positions of the flapping wave extrema as a function of run time shown as coloured dots in panels a–e. Wave maxima and minima are followed in time as the flapping waves evolve. The temporal separation of the wave extrema in panel j gives half of the wave period, while the spatial separation gives half of the wavelength. Using the average positions of the wave extrema in time and space, we get their frequency and wavelength, which are plotted in panel k as coloured dots. The horizontal error bars represent the standard error of the mean (SEM) between adjacent extrema along the time axis, while the vertical error bars represent the SEM evaluated using the adjacent extrema along the Y axis. The sample sizes to evaluate the SEMs are given in the vicinity of each point (see the legends in panel k). The red solid, dashed and dotted lines correspond to the real frequency of the instability analysis using the ion distributions in panels g–i. The black solid, dashed and dotted lines are the growth rate of the drift kink instability obtained using linear kinetic theory (see Methods for details), using the distributions in panels g–i.