Resonant Excitation of Disk Oscillations in Deformed Disks VI: Stability Criterion Revisited

We re-examine excitation of a set of disk oscillations in a deformed disk by a resonant process. We assume that the disk is deformed from an axisymmetric steady state by an oscillatory deformation with frequency ${\omega }_{\mathrm{D}}$ and azimuthal wavenumber $m_{\mathrm{D}}$. Then, we consider two normal mode oscillations with a set of frequencies and azimuthal wavenumber being (${\omega }_1$, $m_1$) and (${\omega }_2$, $m_2$) and satisfying the resonant conditions (${\omega }_1+{\omega }_2+{\omega }_{\mathrm{D}}=0$ and $m_1+m_2+m_{\mathrm{D}}=0$). These oscillations are resonantly excited if $(E_1\mathrm{/}{\omega }_1)(E_2\mathrm{/}{\omega }_2)>0$, where $E_1$ and $E_2$ are wave energies of the above two oscillations, when the deformation is maintained by external forces or has a large amplitude compared with the oscillations. This instability condition is rather general as long as unperturbed density and pressure vanish on the surface of the system. Possibility of application to superhump and negative superhump in superoutburst state of dwarf novae are briefly discussed.