Geometry of the 2D current sheet and notations used in this study. The magnetic field lines are bent curves in the (x, z)-plane with length within a plasma sheet of half thickness L. The curvilinear coordinate, linked to the B-field line, is denoted (ψ, y, ). The considered electrostatic perturbations propagate in the vicinity of a given magnetic surface . The origin of the curvilinear coordinate is at the lower end of the field line. The equatorial plane (x, y) cuts the field line at . The magnetic field at the center of the current sheet is B 0 while its asymptotic value is B 1. The ratio varies typically between 0.01 and 0.3 in the Earth magnetotail.
Ion term Ai as a function of the normalized perpendicular wavenumber K as given by Eq. (47) . Several values of the stretching parameter are chosen: 0.2 (bottom curve), 0.3 (middle curve), and 0.4 (top curve).
Imaginary part of electron term Ae as a function of the normalized frequency qr for and various stretching parameters ranging from 0.22 to 0.27. For these values of ε, the imaginary part of the electron term vanishes at a real frequency leading to existence of stable standing electrostatic oscillations of the current sheet. If ε is lower than 1/5, the imaginary part does not vanish for any qr and if ε is higher than, say, 1/3, the real frequency making the imaginary part zero becomes too high to be valid in the simplified frame of the model. The frequency should stay indeed much less than the ion cyclotron frequency, typically of the order of 10.
Dispersion curves. Top panel: imaginary part of frequency qi versus real frequency qr . Bottom panel: wavenumber K versus qr for various stretching parameters ε and a temperature ratio .
Spatial and temporal scales characterizing particle dynamics in the Earth plasma sheet with , L = 3200 km, , and .
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