Fig. 3

The resonance overlap illustrated by the energetic particle distribution \(f_{total}\) with detailed time evolution. The time evolution of \(f_{total}\) along (a) the left (or higher) \(E'\) line of Fig. 2 and (b) the vertical line of Fig. 2 with a \(P_\phi /e_{EP}\psi _{max}\) value of 0.778, are shown in details. The particle \(\mu\) value is \(7.45~{\mathrm{keV/T}}\). These three horizontal lines from top to bottom represent three resonant layers of \(L_{Alv}=1\), \(L_{EGAM}=1\), and \(L_{Alv}=0\), respectively. To better understand the physical pictures described above, the bird’s-eye view 3-dimensional sub-figures are also presented. The sub-figures (c) and (d) correspond to the sub-figures (a) and (b), respectively, and their vertical axes represent \(f_{total}\). The sub-figures (e–g) show the \(f_{total}\) along \(P_\phi = const.\) at different times, the \(P_\phi /e_{EP}\psi _{max}\) values are 0.994, 0.778, and 0.804, the maximum values of the vertical axis are 6.5, 3.5 and 3.5, and the particle \(\mu\) values are 5.51, 7.45 and \(9.39~{\mathrm{keV/T}}\), respectively. The vertical dotted lines represent the resonant layers of \(L_{EGAM}=1\).