| In this paper, we derive the dispersion equations for field-aligned cyclotron waves in an axisymmetric dipole magnetospheric plasmas with both the bi-Maxwellian and bi-Lorentzian distribution functions. To evaluate the contribution of trapped particles to the transverse current density components the Vlasov equation is solved using a standard method of switching to new variables associated with conservation integrals; new time-like variable is introduced (instead of the geomagnetic latitude angle) to describe the bounce-motion of trapped particles along the geomagnetic field; the perturbed electric field and current density components are Fourier-decomposed over the length of the geomagnetic field lines. As a result, the transverse permittivity elements are expressed by summation of bounce-resonant terms including the double integration in velocity space, the resonant denominators, and the corresponding phase coefficients. Due to geomagnetic field nonuniformity, the wave-particle resonance conditions in magnetospheric plasmas are entirely different from ones in the straight magnetic field; the all spectrum of the electric field is present in the given current density harmonic; the left- and right-hand polarized waves are coupled. To have some analogy with the linear theory of cyclotron waves in the straight magnetic field, we assume that the n-th harmonic of the electric field gives the main contribution to the n-th harmonic of the current density, and the connection of the left- and right-hand polarized waves is small. In this case, the dispersion equations for cyclotron waves have the simplest form and are suitable to analyze the instabilities of both the electron- and ion-cyclotron waves accounting for the bounce resonance effects. |
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