Stability of MRI turbulent Accretion Disks

Based on the characteristics of the magnetorotational instability (MRI) and the MRI-driven turbulence, we construct a steady model for a geometrically thin disk using “non-standard” $\alpha$-prescription. The efficiency of the angular momentum transport depends on the magnetic Prandtl number, $Pm = \nu/\eta$, where $\nu$ and $\eta$ are the microscopic viscous and magnetic diffusivities. In our disk model, Shakura-Sunyaev’s $\alpha$-parameter has a power-law dependence on the magnetic Prandtl number, that is $\alpha \propto Pm^\delta$ where $\delta$ is the constant power-law index. Adopting Spitzer’s microscopic diffusivities, the magnetic Prandtl number becomes a decreasing function of the disk radius when $\delta > 0$. The transport efficiency of the angular momentum and the viscous heating rate are thus smaller in the outer part of the disk, while these are impacted by the size of index $\delta$. We find that the disk becomes more unstable to the gravitational instability for a larger value of index $\delta$. The most remarkable feature of our disk model is that the thermal and secular instabilities can grow in its middle part even if the radiation pressure is negligibly small in the condition $\delta > 2/3$. In the realistic disk systems, it would be difficult to maintain the steady mass accretion state unless the $Pm$-dependence of MRI-driven turbulence is relatively weak.

💡 Research Summary

The paper investigates how the efficiency of angular‑momentum transport in thin accretion disks, traditionally parametrized by the Shakura‑Sunyaev α‑prescription, is altered when the underlying magnetorotational‑instability (MRI) driven turbulence depends on the magnetic Prandtl number (Pm = ν/η). Recent local shearing‑box simulations have shown that the turbulent stress, and therefore α, scales as a power law of Pm, α ∝ Pm^δ, with δ > 0. The authors embed this relation into a global, steady, geometrically thin disk model, using Spitzer’s classical formulas for microscopic viscosity (ν ∝ ρ⁻¹ T^{5/2}) and resistivity (η ∝ T^{-3/2}) to compute Pm as a function of radius. Because temperature falls outward, Pm decreases sharply with radius, becoming much smaller than unity in the outer disk while exceeding unity only in the innermost ≈100 Schwarzschild radii.

With α = α₀ Pm^δ, the model reduces to the standard α‑disk when δ = 0. For δ = 0.25, 0.5, 1.0 the authors solve the coupled mass, angular‑momentum, energy, and vertical hydrostatic balance equations, adopting electron‑scattering opacity in the inner/middle zones and free‑free opacity in the outer zone. The resulting radial profiles show that larger δ leads to a steeper decline of α with radius, a lower viscous heating rate, higher surface density, and reduced radial inflow speed in the outer parts. The inner region remains radiation‑pressure dominated, while the middle and outer regions are gas‑pressure dominated; the transition radius where Pm = 1 (≈100 r_s) marks a sharp change in α.

Stability analysis is then performed for three classic modes: gravitational (Toomre Q), thermal (∂Q⁺/∂T > ∂Q⁻/∂T), and secular (∂(ν_t Σ)/∂Σ < 0). Because α decreases outward for δ > 0, the surface density Σ rises, lowering Q and making the disk more prone to gravitational instability, especially for larger δ. More strikingly, the authors find that when δ > 2⁄3 the thermal and secular instability criteria are satisfied in the middle part of the disk even when radiation pressure is negligible. In the standard α‑disk (δ = 0) such instabilities would require radiation pressure dominance; thus the Pm‑dependence introduces a new route to instability.

The paper applies the model to two representative systems: an X‑ray binary (M ≈ 10 M_⊙) and an active galactic nucleus (M ≈ 10⁸ M_⊙), both with an accretion rate of one Eddington. In both cases, increasing δ shifts the unstable region outward and enlarges it. The authors argue that realistic astrophysical disks, where Pm can vary from ≪1 to ≫1, would need a relatively weak dependence (δ ≈ 0) to maintain a steady accretion flow with α ≈ 0.1–0.4, as inferred from observations. A strong dependence (δ ≈ 1) would render the disk unable to sustain a steady state, leading to episodic outbursts or collapse, potentially explaining dwarf‑nova‑type variability and X‑ray transients.

In conclusion, incorporating the MRI‑driven turbulence’s Pm‑dependence into the α‑prescription fundamentally alters the radial structure and stability of thin disks. The key finding—that thermal and secular instabilities can arise without radiation pressure when δ > 2⁄3—suggests that the magnetic microphysics of the plasma may be a primary driver of observed disk variability. The authors recommend future work that includes non‑isothermal ion–electron coupling, partially ionized regions, and global 3‑D MHD simulations to test the robustness of these results.