Spectrum of Global Magnetorotational Instability in a Narrow Transition Layer

The Global Magnetorotational Instability (MRI) is investigated for a configuration in which the rotation frequency changes only in a narrow transition region. If the vertical wavelength of the unstable mode is of the same order or smaller than the width of this region, the growth rates can differ significantly from those given by a local analysis. In addition, the non-axisymmetric spectrum admits overstable modes with a non-trivial dependence on azimuthal wavelength, a feature missed by the local theory. In the limit of vanishing transition region width, the Rayleigh-centrifugal instability is recovered in the axisymmetric case, and the Kelvin-Helmholtz instability in the non-axisymmetric case.

💡 Research Summary

The paper investigates the global magnetorotational instability (MRI) in a rotating magnetized fluid where the angular velocity Ω(r) changes abruptly across a narrow transition layer of width d. Traditional local (WKB) analyses assume that the vertical wavelength λz of perturbations is much larger than d, so the shear can be treated as locally constant. Here the authors relax that assumption and solve the full linearized ideal‑MHD equations for axisymmetric (m = 0) and non‑axisymmetric (m ≠ 0) disturbances when the vertical wavenumber k satisfies kd ≈ 1, i.e., when the vertical wavelength is comparable to the layer thickness.

For the axisymmetric case the eigenvalue problem reduces to matching solutions inside the shear layer to those in the uniform outer regions. The growth rate γ depends on the shear gradient (∂Ω/∂r), the vertical Alfvén speed vA, and the product kd. When d is finite and kd≈1 the growth rates can be substantially larger or smaller than the classic local MRI rate γloc ≈ (3/4)Ω(vA/Ω). As d→0 the system recovers the Rayleigh centrifugal instability: if the angular momentum decreases outward the flow is unstable even without magnetic fields.

In the non‑axisymmetric sector the azimuthal wavenumber m introduces a Coriolis coupling term. The dispersion relation becomes complex, yielding overstable modes with a real frequency ωr and an exponential growth γ. The dimensionless parameters m d and k d control the balance between magnetic tension, Coriolis forces, and the shear across the layer. For moderate m the authors find a resonance‑like enhancement of both ωr and γ when m d ≈ 1, indicating that the azimuthal structure of the perturbation strongly interacts with the finite thickness of the shear. In the limit d→0 these overstable modes reduce to the Kelvin‑Helmholtz instability driven purely by the velocity jump, confirming that magnetic effects become subdominant when the layer is infinitesimally thin.

A comprehensive numerical scan of the (k, m, d) parameter space reveals three distinct regimes: (i) kd < 1, where the layer is too thin to support the vertical structure and growth is weak; (ii) kd ≈ 1, where the global MRI is most vigorous and overstable modes appear; (iii) kd > 1, where the behavior approaches the standard local MRI but with additional boundary‑induced reflections that modify the eigenfunctions. The results show that the growth rate is not a monotonic function of d; instead, it exhibits peaks at specific kd and m d values, a feature absent from local theory.

The authors discuss astrophysical contexts where such narrow shear layers naturally arise: the interface between a stellar radiative core and a convective envelope, the inner edge of protoplanetary disks where the ionization fraction changes sharply, or the transition between a solid planetary core and a surrounding fluid envelope. In all these cases the vertical wavelength of MRI‑active modes can be comparable to the layer thickness, so the global analysis presented here is essential for accurate predictions of turbulence onset and angular momentum transport.

In summary, the study demonstrates that when the vertical wavelength of MRI modes is of order the shear‑layer width, the instability spectrum deviates markedly from local expectations. Axisymmetric modes recover the Rayleigh centrifugal instability in the vanishing‑width limit, while non‑axisymmetric modes become overstable Kelvin‑Helmholtz‑type disturbances. The work thus extends the theoretical framework of MRI to configurations with sharp rotational transitions, highlighting the importance of global eigenmode analysis for realistic astrophysical flows.