The Super-Alfv\'enic Rotational Instability in accretion disks about black holes

The theory of instability of accretion disks about black holes, neutron stars or proto-planets, is revisited by means of the recent method of the Spectral Web. The cylindrical accretion disk differential equation is shown to be governed by the forward and backward Doppler-shifted continuous Alfv\'en spectra $\Omega_{\rm A}^\pm \equiv m \Omega \pm \omega_{\rm A}$, where $\omega_{\rm A}$ is the static Alfv\'en frequency. It is crucial to take non-axisymmetry ($m \ne 0$) and super-Alfv\'enic rotation of the Doppler frames ($|m\Omega| \gg |\omega_{\rm A}|$) into account. The continua $\Omega_{\rm A}^+$ and $\Omega_{\rm A}^-$ then overlap, ejecting a plethora of Super-Alfv\'enic Rotational Instabilities (SARIs). In-depth analysis for small inhomogeneity shows that the two Alfv\'en singularities reduce the extent of the modes to sizes much smaller than the width of the accretion disk. Generalization for large inhomogeneity leads to the completely unprecedented result that, for mode numbers $|k| \gg |m|$, any complex $\omega$ in a wide neighborhood of the real axis is an approximate `eigenvalue'. The difference with genuine eigenmodes is that the amount of complementary energy to excite the modes is tiny, $|W_{\rm com}| \le c$, with $c$ the machine accuracy of the computation. This yields a multitude of two-dimensional continua of quasi-discrete modes: quasi-continuum SARIs. We conjecture that the onset of 3D turbulence in magnetized accretion disks is governed, not by the excitation of discrete axisymmetric Magneto-Rotational Instabilities, but by the excitation of modes from these two-dimensional continua of quasi-discrete non-axisymmetric Super-Alfv\'enic Rotational Instabilities.