Resonant Excitation of Disk Oscillations in Deformed Disks VII: Stability Criterion in MHD Systems

Abstract

In a disk with an oscillatory deformation from an axisymmetric state with frequency $\omega_{\rm D}$ and azimuthal wavenumber $m_{\rm D}$, a set of two normal mode oscillations with frequency and azimuthal wavenumber being ($\omega_1$, $m_1$) and ($\omega_2$, $m_2$) resonantly couple through the disk deformation, when the resonant conditions ($\omega_1+\omega_2+\omega_{\rm D}=0$ and $m_1+m_2+m_{\rm D}=0$) are satisfied. In the case of hydrodynamical disks, the resonance amplifies the set of the oscillations if $(E_1/\omega_1)(E_2/\omega_2)>0$ (Kato 2013b), where $E_1$ and $E_2$ are wave energies of the two oscillations with $\omega_1$ and $\omega_2$, 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.