The Super-Alfvénic Rotational Instability in Accretion Disks about Black Holes

Abstract

The theory of instability of accretion disks about black holes, neutron stars, or protoplanets 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 Alfvén spectra Ω A ± ≡ m Ω ± ω A , where ω A is the static Alfvén frequency. It is crucial to take nonaxisymmetry (m ≠ 0) and super-Alfvénic rotation of the Doppler frames (∣mΩ∣ ≫ ∣ω A∣) into account. The continua Ω A + and Ω A − then overlap, ejecting a plethora of super-Alfvénic rotational instabilities (SARIs). In-depth analysis for small inhomogeneity shows that the two Alfvén 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∣ ≫ ∣m∣, any complex ω 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 com∣ ≤ 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 magnetorotational instabilities but by the excitation of modes from these two-dimensional continua of quasi-discrete nonaxisymmetric SARIs.

A Nonlocal Magneto-curvature Instability in a Differentially Rotating Disk

“…These are the points where the magnitude of the Dopplershifted frequency of the mode is equal to its Alfvén frequency and are always centered about the point of corotation, where ω r = Ω(r). In particular, localized MRI and curvature modes in systems with weak magnetic fields (i.e., super-Alfvénic flows) feature structures localized in the inner and outer parts of the system as also obtained by Ogilvie & Pringle (1996) and Goedbloed & Keppens (2022), due to the differing locations of the Alfvén singularities. Inner (MRI) modes are located between the conducting boundary at r 1 and the singularity r out ; outer (curvature) modes are between the singularity r in and the conducting boundary at r 2 (Figures 5(a) and (b)).…”

“…This can be seen in Figure 6 with m = 10 and k = 80k 1 . The complex eigenvalues are similar to their approximate counterparts in Goedbloed & Keppens (2022). Regardless of the use of a global treatment, these extremely localized instabilities only occupy a narrow region of parameter space, as they are only unstable for very weak magnetic fields (i.e., super-Alfvénic flows).…”

“…This can be seen in Figure 4(b), where the MRI mode's frequency in the low-B limit converges to Ω 0 , the system's angular velocity at the inner wall, and the ( ) (0.0894Ω 0 in our system), the angular velocity at the outer wall. We also search in the limit of super-Alfvénic flows where ω A = mΩ as studied by Ogilvie & Pringle (1996) and Goedbloed & Keppens (2022). In this region, the high-m, high-k nonaxisymmetric Alfvénic resonance instability is extremely localized due to being confined between two very close-by Alfvénic resonances.…”

“…Localized nonaxisymmetric MRI mode structures were found to be confined between the Alfvénic resonant points (where the magnitude of the Doppler-shifted wave frequency is equal to the Alfvén frequency; Matsumoto & Tajima 1995). In cylindrical shear flows (Ogilvie & Pringle 1996) and in the compressible limit (Goedbloed & Keppens 2022), it was shown that the forward and backward overlapping of these Alfvén continua results in discrete localized nonaxisymmetric modes at large axial and azimuthal mode numbers. The question arises whether, in a real domain with spatial curvature, global nonaxisymmetric modes with real frequencies can persist.…”

Magnetic Flux Transport in Radiatively Inefficient Accretion Flows and the Pathway towards a Magnetically Arrested Disk

“…Shear in the accretion flow converts poloidal field into the toroidal field, while MRI amplifies the poloidal field. Therefore, both poloidal and toroidal fields grow exponentially in a dynamical time (t dyn ≈ 1/Ω ∝ R 3/2 ) and after few dynamical time the system likely enters the non-linear regime under the influence of parasitic instabilities (Goodman & Xu 1994) or due to different super-Alfvénic rotational instabilities (SARIs; Goedbloed & Keppens 2022).…”