Abstract
We report our results of a study on non-axisymmetric oscillation modes of a torus with non-Keplerian rotational profile around Schwarzschild black hole. We study a simple toy equilibrium model of incompressible torus whose vertical structure is neglected. The master equation of the oscillation in this case becomes a second-order ordinary differential equation in the cylindrical radial coordinate. When the eigenvalue is assumed to be real, the equation has a singular point where the pattern speed of the mode in the azimuthal direction coincides with the local angular frequency of the flow (co-rotation of modes). This singularity leads to the appearance of continuous spectrum in twofold ways. One of them has its origin in an effect of general relativistic gravity. When the eigenvalue problem is extended to complex domain of frequency with finite imaginary part, the master equation no longer has a co-rotation singularity. Thus we only find a discrete spectrum. We see dynamically unstable mode outside the co-rotation, but do not see those in the co-rotation region for the particular cases studied here.
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