Stability of Rotating Magnetized Configurations

A rotating magnetized equilibrium can fail through several mechanisms that draw energy from different reservoirs. A classical bar mode draws on rapid global rotation; a low-$T/\lvert W\rvert$ shear mode uses differential rotation and corotation; the magnetorotational instability taps decreasing angular velocity; and Tayler or Parker modes reorganize magnetic fields. Thresholds depend on the rotation law, field geometry, stratification, dissipation, symmetry, and duration of the calculation. This page therefore organizes stability claims by mechanism rather than assigning one critical number to every star.

Required background. Killing Fields and Isometry Groups supplies stationary and axial symmetries and the energy and angular-momentum charges they define.

Helpful background. Flat-Background Hilbert Stress-Energy from Metric Variation supplies fluid and electromagnetic stresses; The Einstein Equation supplies self-gravity; Relativistic Matter and Compact Objects supplies EOS and MHD closure; Formulations of General Relativity supplies evolution and constraint language; Computational Relativity supplies resolution and nonlinear-stability checks.

Rotational free energy and bar-like modes

Define $\beta=T/\lvert W\rvert$, with the relativistic definitions of kinetic and binding energy stated for the chosen equilibrium family. Uniform Newtonian incompressible models motivate a dynamical bar threshold near $\beta\simeq0.27$ and a secular threshold near $0.14$, but these are reference values, not relativistic constants. Compressibility, compactness, differential rotation, viscosity, gravitational radiation, and magnetic stress shift the onset and growth rate.

For a nonaxisymmetric disturbance proportional to $e^{i(m\phi-\omega t)}$ with $m\ne0$, the pattern speed is $\Omega_p=\operatorname{Re}\omega/m$ and the growth rate is $\operatorname{Im}\omega$. A differentially rotating star has a corotation point where $\Omega(\varpi_c)=\Omega_p$. Wave exchange there can drive one-armed or bar-like shear instabilities at much smaller global $\beta$. Watts, Andersson, and Jones set out the corotation interpretation and its supporting shell model in Watts, Andersson, and Jones 2005, pp. L37–L40 and Figs. 1–2. Consequently, being below the classical bar threshold does not establish stability when substantial shear is present.

Magnetic and magnetorotational channels

The simplest MRI calculation uses a local Newtonian patch with differential rotation $\Omega(\varpi)$, negligible buoyancy, ideal MHD, a weak uniform vertical field, and incompressible axisymmetric disturbances proportional to $e^{i(kz-\omega t)}$. Magnetic tension couples fluid elements at different radii: an outward-displaced element can gain angular momentum from its inner neighbor and move farther outward instead of returning.

The local dispersion relation is

$$\omega^4-(\kappa^2+2k^2v_A^2)\omega^2 +k^2v_A^2\left(k^2v_A^2+\frac{d\Omega^2}{d\ln\varpi}\right)=0, \qquad \kappa^2=4\Omega^2+\frac{d\Omega^2}{d\ln\varpi},$$

where $\kappa$ is the radial epicyclic frequency and $v_A$ is the Alfvén speed along the field. This is the unstratified, vertical-wavevector limit of Balbus and Hawley 1991, §§2.2–2.3, Eqs. (2.9)–(2.14). For a hydrodynamically stable rotation law with $\kappa^2\gt0$, the quartic, regarded as a quadratic in $\omega^2$, has a negative root when

$$0\lt k^2v_A^2\lt-\frac{d\Omega^2}{d\ln\varpi}.$$

Thus decreasing angular velocity permits MRI even when specific angular momentum increases outward and the hydrodynamic Rayleigh criterion is satisfied. The wavelength must also fit inside the local region; a decreasing $\Omega$ alone does not guarantee an allowed unstable wavelength in a finite star.

For $\Omega\propto\varpi^{-q}$ with $0\lt q\lt2$, maximizing the growth rate gives $\gamma_{\max}=q\Omega/2$ and $k_{\max}v_A=\Omega\sqrt{q(4-q)}/2$. The characteristic wavelength is therefore of order $2\pi v_A/\Omega$. A global simulation that does not resolve it cannot measure MRI saturation. Relativistic stratification, neutrino transport, field geometry, and finite resistivity change the applicable local problem and must be assessed separately.

Toroidal fields can suffer kink-like Tayler modes and buoyant Parker modes; poloidal fields can be unstable near neutral lines. Three-dimensional relativistic evolutions of toroidal-field equilibria found Parker-driven nonaxisymmetric growth and persistent turbulent reorganization in Kiuchi, Yoshida, and Shibata 2011, §§4–5 and Figs. 3–11. Linear Newtonian perturbation evolutions of poloidal equilibria, holding the gravitational potential fixed in the Cowling approximation, found growth localized around the neutral line in Lander and Jones 2011, §§2–3 and Figs. 2–6. Those linear calculations measure onset and growth, not nonlinear saturation.

A recent high-resolution comparison sharpens, but does not close, the saturation question. Cook and collaborators evolved magnetized and unmagnetized controls at four finest-grid spacings from $230$ to $29\,\mathrm m$; the $29\,\mathrm m$ run lasted $100\,\mathrm{ms}$, whereas the $1.2\,\mathrm s$ evolution used the coarsest grid. They resolved a magnetically driven turbulent cascade and nonlinear field rearrangement in Cook et al. 2026, §§2–4, Figs. 2–6, and Appendix C. Its formulation holds the spacetime fixed in the Cowling approximation and uses a static, nonrotating, unstratified $\Gamma=2$ ideal-MHD star with an initial poloidal field. The late-time toroidal-energy fraction from the long coarse run is therefore evidence for that setup, not a resolution-closed universal saturation level for rotating, stratified neutron stars.

Rotation and stable composition stratification can slow or redirect magnetic displacements, and mixed poloidal–toroidal fields may survive longer than pure components in some calculations. Competing evidence is that apparent long-lived mixed states can depend on imposed symmetry, atmosphere, numerical resistivity, current prescription, or evolution duration. Equilibrium construction alone cannot decide the nonlinear attractor.

Comparing results and unresolved questions

Evidence cutoff: 2026-08-18. A meaningful comparison records the EOS and thermal profile; uniform or differential rotation law; magnetic topology and energy; stratification and superconducting assumptions; Newtonian or relativistic gravity; ideal or resistive closure; dimensionality and imposed symmetries; grid and atmosphere; perturbation; run duration in rotation and Alfvén times; diagnostic; conservation error; and independent replication. Paschalidis and Stergioulas collect the rotational mode classes and simulation regimes in Paschalidis and Stergioulas 2017, §§4.1–4.5.

Evidence for a mode requires convergent exponential growth with a spatial eigenfunction and frequency consistent across perturbation amplitudes. A Fourier peak appearing once is weaker: surface shocks, atmosphere interaction, grid imprinting, or a constraint-violating transient can imitate it. Evidence for saturation further requires that the saturated amplitude and energy transfer converge with resolution and effective dissipation. Suppression over five growth times is stronger than nonappearance over less than one.

The main unresolved questions are the saturation amplitudes in realistic multiphysics remnants, the survival range of mixed fields, the role of stable stratification and superconducting cores, and which instability dominates when their timescales overlap. Formation also matters: a formally stable equilibrium may not be reached from collapse or merger initial data.

The interpretation could change with a new three-dimensional simulation suite that resolves the MRI and Alfvén scales with converged conservation, a controlled resistive or superconducting calculation, a theorem extending a stability boundary, or an independently reproduced nonlinear saturation result that changes one of these comparisons. A new equilibrium family without an evolution or perturbation test does not by itself revise a stability conclusion.

Timescale and limiting-case checks

Set differential rotation to zero: corotation shear and MRI free energy from $d\Omega/d\varpi$ disappear, although a rapidly rotating bar mode and magnetic instabilities may remain. Set the magnetic field to zero: Tayler, Parker, and MRI modes disappear, but hydrodynamic shear and bar modes do not. Suppress nonaxisymmetric degrees of freedom and every $m\ne0$ instability is excluded by construction rather than shown to be stable.

Compare the evolution duration with the predicted rotation, Alfvén, viscous, and resistive times. Magnetic energy and angular momentum should close to the reported fluxes and numerical loss. Constraint norms, mode growth rates, and saturation amplitudes require resolution sequences. These checks separate a physical absence from a simulation that was too symmetric, short, or coarse.

Check your understanding

In a local patch let $\Omega(\varpi)=K/\varpi$ and suppose a weak field is present. Evaluate $d\Omega^2/d\ln\varpi$. Does the simple MRI criterion flag the profile, and would $\beta\lt0.14$ rule the instability out?

Solution

Because $\Omega^2=K^2\varpi^{-2}$,

$$\frac{d\Omega^2}{d\ln\varpi} =\varpi\frac{d\Omega^2}{d\varpi} =-2\Omega^2\lt0.$$

The local criterion therefore flags MRI susceptibility. A small global $\beta$ does not remove it: the classical secular and dynamical bar thresholds refer to different mechanisms. One must still verify that the magnetic field and numerical grid resolve the relevant MRI wavelength and that the more complete stratified relativistic model retains the instability.

References

• Balbus, Steven A., and John F. Hawley. “A Powerful Local Shear Instability in Weakly Magnetized Disks. I. Linear Analysis.” The Astrophysical Journal 376 (1991): 214–222. DOI. Open PDF.
• Cook, William, Raj Kishor Joshi, Sebastiano Bernuzzi, Brynmor Haskell, and Jacob Fields. “Turbulence in Magnetized Neutron Stars.” Monthly Notices of the Royal Astronomical Society 547 (2026): stag136. DOI.
• Kiuchi, Kenta, Shijun Yoshida, and Masaru Shibata. “Nonaxisymmetric Instabilities of Neutron Star with Toroidal Magnetic Fields.” Astronomy and Astrophysics 532 (2011): A30. DOI.
• Lander, Samuel K., and David I. Jones. “Oscillations and Instabilities in Neutron Stars with Poloidal Magnetic Fields.” Monthly Notices of the Royal Astronomical Society 412 (2011): 1730–1740. DOI.
• Paschalidis, Vasileios, and Nikolaos Stergioulas. “Rotating Stars in Relativity.” Living Reviews in Relativity 20 (2017): 7. DOI.
• Watts, Anna L., Nils Andersson, and David I. Jones. “The Nature of Low $T/\lvert W\rvert$ Dynamical Instabilities in Differentially Rotating Stars.” The Astrophysical Journal Letters 618 (2005): L37–L40. DOI.