Anomalous scaling of the ion beam energy in the current sheet

The scaling of the energy of ion beams (beamlets) $W_N\sim N^A$ in the $N$ resonant regions of the current sheet in data from the SC-1 and SC-4 CLUSTER satellites has been analyzed. The case on February 5, 2003, for energy dispersive small-scale substructures, which are signatures of $1$–$20$ keV beamlets in the auroral magnetosphere at geocentric distances $(4.5{-}5.3)R_{\mathrm{E}}$, where $R_{\mathrm{E}}$ is the radius of the Earth, has been studied. This case is anomalous, since the energies of beamlets in the resonant regions (seven regions $N = 1{-}7$ with resonances $R = 1{-}7$ are identified and the region with $R = 7$ is located in the highest latitude auroral region) do not obey a single scaling law. The exponents $A$ are $0.04$ and $0.40$ for the regions with resonances $R = 1{-}4$ and are $0.83$ and $1.14$ for the regions with $R = 5{-}7$ according to the SC-1 and SC-4 data, respectively. The exponents obtained from the CLUSTER satellite data differ from the theoretically predicted value $A = 1.33$ [L.M. Zeleny et al., JETP Lett. 85, 187 (2007)]. The beamlet energy scales for the regions with $R = 5{-}7$ can be explained by taking into account the electric field $E_z$ perpendicular to the plane of the current sheet. The observed exponents $A$ in the regions $N = 1{-}4$ can occur because the spatial decrease in the normal component $B_z$ of the magnetic field, which controls the increment of the energy of ion beams in the current sheet, in these resonant regions is smaller than that in the regions $N = 5{-}7$.