The structure of low-frequency standing Alfvén waves in the box model of the magnetosphere with magnetic field shear

The paper is concerned with the influence of magnetic field shear on the structure of Alfvén waves standing along field lines in the one-dimensionally inhomogeneous box model of the magnetosphere, enclosed between two parallel. infinitely conducting planes (ionospheres). We consider the transverse small-scale Alfvén waves whose azimuthal component of the wave vector ky satisfies the condition kyl >> 1, where l is the distance between the ionospheres. For this model, the Alfvén resonance condition has been established. It is shown that resonance can also occur at a constant Alfvén velocity if the field-line inclination to the ionosphere is changed. On resonant magnetic shells there occurs a singularity of the wave field of the same kind as in the absence of shear. Moreover there are found many resemblances between Alfvén-wave behavior in our one-dimensionally inhomogeneous model and in two-dimensional inhomogeneous models with plasma and magnetic field parallel inhomogeneity taken into account. Thus, the presence of shear leads to a difference of the frequencies of poloidal and toroidal oscillations of field lines, and to the dependence of the wave's frequency on the transversal components of wave vector. Then. in the sheared magnetic field with highly conductive boundaries the source excites multiple standing Alfvén harmonics at different locations. In general, the localisation regions of different longitudinal harmonics overlap. However, in the small but finite shear limit, a total wave field represents a set of mutually isolated transparent regions corresponding to different harmonic numbers. In each of these regions the waves are found to be travelling across the magnetic shell, and the transparent region is limited in the coordinate x by two turning points, at one of which the mode is poloidally polarised. and the other point it is toroidally polarized (it is at this latter point where Alfvén resonance occurs). Furthermore, the phase velocity of the wave is directed toward the poloidal point, and the group velocity is directed at the toroidal point.