Stability criterion in MHD systems

In a disk with an oscillatory deformation from an axisymmetric state with frequency ωD and azimuthal wavenumber mD, two normal mode oscillations with a set of frequency and azimuthal wavenumber being (ω1, m1) and (ω2, m2) are resonantly coupled to each other through the disk deformation when the resonant conditions (ω1 + ω2 + ωD = 0 and m1 + m2 + mD = 0) are satisfied. In the case of hydrodynamical disks, the resonance amplifies the set of the oscillations if (E1/ω1)(E2/ω2) > 0 (Kato 2013b, PASJ, 65, 75), where E1 and E2 are wave energies of the two oscillations with ω1 and ω2, respectively. In this paper we show that this instability criterion is still valid even when the oscillations are ideal MHD ones in magnetized disks, if the displacements associated with the oscillations vanish on the boundary of the system.