Ballooning instability in the dipole magnetosphere: The finite transverse wavelength influence

This paper is concerned with the condition for the development of ballooning instability in the dipole Earth's magnetosphere. Under investigation is the dependence of the ratio of the radial and azimuthal wave vector components squared on the wave frequency squared ω 2. It is shown that this dependence is depicted by continuous curve (the dispersion curve). The part of the dispersion curve with ω 2 > 0 corresponds to the slow magnetosonic mode modified by the coupling with the Alfvén mode, while the part with ω 2 < 0 corresponds to the ballooning instability. The instability appears at some critical ratio of the radial and azimuthal wave vector components, the growth rate reaches the maximum value where this ratio is zero. The instability threshold is determined depending on the β parameter and the pressure gradient when the coupling of the slow and Alfvén modes is taken into account. On a given magnetic shell, at a given β value, large pressure gradient favors the instability, and vice versa, at a given pressure gradient, a large β favors instability. It is shown that in the unstable mode, the compressional magnetic field component has a phase shift with respect to the radial component. This can be used to detect unstable ballooning modes during observations in the Earth's magnetosphere.