Global MRI with Braginskii viscosity in a galactic profile

We present a global-in-radius linear analysis of the axisymmetric magnetorotational instability (MRI) in a collisional magnetized plasma with Braginskii viscosity. For a galactic angular velocity profile $\Omega$ we obtain analytic solutions for three magnetic field orientations: purely azimuthal, purely vertical and slightly pitched (almost azimuthal). In the first two cases the Braginskii viscosity damps otherwise neutrally stable modes, and reduces the growth rate of the MRI respectively. In the final case the Braginskii viscosity makes the MRI up to $2\sqrt{2}$ times faster than its inviscid counterpart, even for \emph{asymptotically small} pitch angles. We investigate the transition between the Lorentz-force-dominated and the Braginskii viscosity-dominated regimes in terms of a parameter $\sim \Omega \nub/B^2$ where $\nub$ is the viscous coefficient and $B$ the Alfv'en speed. In the limit where the parameter is small and large respectively we recover the inviscid MRI and the magnetoviscous instability (MVI). We obtain asymptotic expressions for the approach to these limits, and find the Braginskii viscosity can magnify the effects of azimuthal hoop tension (the growth rate becomes complex) by over an order of magnitude. We discuss the relevance of our results to the local approximation, galaxies and other magnetized astrophysical plasmas. Our results should prove useful for benchmarking codes in global geometries.

💡 Research Summary

The paper presents a global‑in‑radius linear stability analysis of the axisymmetric magnetorotational instability (MRI) in a collisional, magnetized plasma where anisotropic Braginskii viscosity is operative. Unlike the traditional local shearing‑box approach, the authors retain the full radial dependence of the angular velocity profile Ω(r) appropriate for galactic disks and solve the resulting eigenvalue problem analytically for three representative magnetic‑field geometries: (i) a purely azimuthal field, (ii) a purely vertical field, and (iii) a field that is almost azimuthal with a very small pitch angle (θ≪1).

For the azimuthal field, the Braginskii viscous stress damps the otherwise neutrally stable “hoop‑tension” mode, turning the eigenvalue from zero to a negative real number. The damping strength scales with the dimensionless parameter χ≡Ω ν_b/B², where ν_b is the Braginskii viscosity coefficient and B the Alfvén speed. When χ≫1 the mode is strongly suppressed, whereas χ≪1 recovers the inviscid neutral behavior.

In the vertical‑field case the classic MRI remains present, but the growth rate is reduced by a factor proportional to χ. As ν_b increases (or B decreases), the MRI becomes slower, smoothly approaching the inviscid limit as χ→0.

The most striking results arise for the slightly pitched field. Here the viscous stress couples the azimuthal and vertical components of the perturbation, effectively enhancing the shear that drives the instability. The authors demonstrate that the MRI growth rate can be amplified by up to a factor of 2√2 relative to the inviscid case, even when the pitch angle is asymptotically small. This amplification persists for arbitrarily small θ, indicating that the Braginskii viscosity can bridge the gap between the standard MRI and the magnetoviscous instability (MVI).

The paper introduces χ as the controlling parameter governing the transition between a Lorentz‑force‑dominated regime (χ≪1, inviscid MRI) and a viscosity‑dominated regime (χ≫1, MVI). Asymptotic expansions are derived for both limits, and the authors provide explicit expressions for the approach to each limit. Notably, the viscous term can make the eigenvalue complex with a large imaginary part, meaning that the unstable mode both oscillates and grows rapidly. This complex growth can be an order of magnitude larger than in the purely magnetic case, highlighting the potential for vigorous, wave‑like turbulence in weakly magnetized, highly collisional astrophysical plasmas.

The authors discuss the relevance of these findings to the validity of the local approximation. While the shearing‑box model captures the basic MRI physics for χ≪1, it misses the strong viscous amplification that appears when radial gradients are retained. Consequently, the global solutions presented here serve as valuable benchmarks for numerical codes that aim to simulate MRI/MVI in realistic galactic geometries.

Astrophysically, the results are pertinent to galactic disks where the plasma β is large and collisionality is high, conditions under which Braginskii viscosity is expected to dominate over isotropic viscosity. In regions such as galactic centers or supernova‑driven turbulent zones, ν_b can be sufficiently large that χ approaches or exceeds unity, making the magnetoviscous instability a plausible driver of angular‑momentum transport, magnetic‑field amplification, and turbulence generation. The enhanced growth rates and complex eigenvalues suggest that even modest magnetic fields can trigger vigorous, wave‑like motions that may influence star‑formation rates and the large‑scale magnetic topology of galaxies.

In summary, the paper delivers a comprehensive analytical treatment of MRI in the presence of Braginskii viscosity, identifies χ and the pitch angle as the key parameters controlling the MRI‑to‑MVI transition, quantifies how viscosity can both suppress and dramatically accelerate the instability, and provides exact solutions that can be used to validate global magnetohydrodynamic simulations of galactic and other astrophysical plasmas.