On the relation of standard and helical magnetorotational instability

The magnetorotational instability (MRI) plays a crucial role for cosmic structure formation by enabling turbulence in Keplerian disks which would be otherwise hydrodynamically stable. With particular focus on MRI experiments with liquid metals, which have small magnetic Prandtl numbers, it has been shown that the helical version of this instability (HMRI) has a scaling behaviour that is quite different from that of the standard MRI (SMRI). We discuss the relation of HMRI to SMRI by exploring various parameter dependencies. We identify the mechanism of transfer of instability between modes through a spectral exceptional point that explains both the transition from a stationary instability (SMRI) to an unstable travelling wave (HMRI) and the excitation of HMRI in the inductionless limit. For certain parameter regions we find new islands of the HMRI.

💡 Research Summary

The paper investigates the relationship between the standard magnetorotational instability (SMRI) and its helical counterpart (HMRI), focusing on the parameter regimes relevant to liquid‑metal experiments where the magnetic Prandtl number (Pm) is extremely small (≈10⁻⁶). The authors begin by recalling that SMRI, driven by a purely axial magnetic field, requires high Reynolds (Re) and Hartmann (Ha) numbers to become unstable in such low‑Pm fluids, making laboratory realization difficult. By contrast, when an azimuthal field component is added, forming a helical magnetic configuration, HMRI can be excited at much lower Re and Ha, a fact that has motivated numerous tabletop experiments.

To elucidate how SMRI morphs into HMRI, the authors formulate the linearized magnetohydrodynamic (MHD) equations for a cylindrical Taylor‑Couette flow under a combined axial (Bz) and azimuthal (Bφ) field. After nondimensionalisation they identify four key control parameters: Re, Ha, Pm, and the “helicity” β = Bφ/Bz. The eigenvalue problem reduces to a quartic polynomial in the complex growth rate λ, whose real part determines stability. By scanning the (Re, Ha, β) space with Pm fixed at experimentally realistic values, they map out the loci of marginal stability.

A central discovery is the presence of a spectral exceptional point (EP) where two real eigenvalues coalesce and then bifurcate into a complex‑conjugate pair. Below the EP the system exhibits a stationary instability identical to SMRI; above it the instability becomes a travelling wave, i.e., HMRI. The EP thus provides a unified framework for the transition from a non‑propagating to a propagating mode. Importantly, the EP exists even in the inductionless limit (Pm → 0) provided β is sufficiently large, explaining why HMRI can be observed in liquid‑metal experiments despite negligible magnetic induction.

The authors further reveal “islands” of HMRI that lie outside the classic HMRI domain. These islands appear at high β and low Re·Pm, where the eigenvalues bypass the EP and directly become complex. This indicates that multiple pathways connect SMRI and HMRI, and that the helical field can generate instability even when the traditional induction mechanism is absent.

From an experimental standpoint, the results suggest that precise control of the azimuthal field component allows researchers to navigate the EP and deliberately trigger either SMRI or HMRI, or to explore the newly identified islands. Consequently, the demanding requirement of extremely high rotation rates can be relaxed, making MRI studies more accessible.

In astrophysical contexts, the findings imply that in accretion disks with weak but helical magnetic fields, HMRI‑like travelling‑wave instabilities may operate even when the magnetic Reynolds number is low, potentially contributing to angular‑momentum transport in regions previously thought to be magnetically quiescent.

Overall, the paper unifies the two MRI variants through the concept of a spectral exceptional point, clarifies the scaling differences in low‑Pm fluids, and expands the known instability landscape, offering valuable guidance for both laboratory experiments and theoretical models of astrophysical disks.