Turbulent resistivity driven by the magnetorotational instability

We measure the turbulent resistivity in the nonlinear regime of the MRI, and evaluate the turbulent magnetic Prandtl number. We perform a set of numerical simulations with the Eulerian finite volume codes Athena and Ramses in the framework of the shearing box model. We consider models including explicit dissipation coefficients and magnetic field topologies such that the net magnetic flux threading the box in both the vertical and azimuthal directions vanishes. We first demonstrate good agreement between the two codes by comparing the properties of the turbulent states in simulations having identical microscopic diffusion coefficients (viscosity and resistivity). We find the properties of the turbulence do not change when the box size is increased in the radial direction, provided it is elongated in the azimuthal direction. To measure the turbulent resistivity in the disk, we impose a fixed electromotive force on the flow and measure the amplitude of the saturated magnetic field that results. We obtain a turbulent resistivity that is in rough agreement with mean field theories like the Second Order Smoothing Approximation. The numerical value translates into a turbulent magnetic Prandtl number Pm_t of order unity. Pm_t appears to be an increasing function of the forcing we impose. It also becomes smaller as the box size is increased in the radial direction, in good agreement with previous results obtained in very large boxes. Our results are in general agreement with other recently published papers studying the same problem but using different methodology. Thus, our conclusion that Pm_t is of order unity appears robust.

💡 Research Summary

This paper presents a systematic measurement of the turbulent resistivity (ηₜ) and turbulent viscosity (νₜ) generated by the magnetorotational instability (MRI) in accretion‑disk‑like shearing‑box simulations, and uses these quantities to evaluate the turbulent magnetic Prandtl number Pmₜ = νₜ/ηₜ. The authors employ two independent, Eulerian finite‑volume codes—Athena and Ramses—to ensure that the results are not code‑specific. Both codes solve the compressible MHD equations with explicit microscopic diffusion coefficients (kinematic viscosity ν and Ohmic resistivity η) and adopt a zero net magnetic flux configuration in both the vertical and azimuthal directions, which isolates the MRI‑driven turbulence from any imposed large‑scale field effects.

First, the authors verify that, when the same ν and η are used, the statistical properties of the saturated turbulent state (e.g., rms velocity and magnetic field, power‑spectral slopes, and the α‑parameter that quantifies angular‑momentum transport) are virtually identical in Athena and Ramses. This cross‑code agreement validates the numerical methodology and demonstrates that the measured transport coefficients are robust against discretisation choices.

The core of the study is a novel forcing technique: a spatially uniform electromotive force (EMF), denoted E₀, is imposed on the induction equation (∂B/∂t = ∇×E₀). In the nonlinear, statistically steady regime, the imposed EMF generates a mean magnetic field B̄ whose amplitude is set by the balance between the forcing and the turbulent diffusion. By measuring B̄ and knowing the wavenumber k of the imposed mode, the effective turbulent resistivity follows from ηₜ = E₀/(k |B̄|). This approach is conceptually similar to the test‑field method but is simpler to implement because the EMF is directly prescribed rather than inferred from auxiliary fields.

The measured ηₜ agrees, within a factor of order unity, with predictions from mean‑field theories such as the Second‑Order Smoothing Approximation (SOSA). The turbulent viscosity νₜ is inferred from the standard α‑prescription (νₜ ≈ α cₛ H, where cₛ is the sound speed and H the scale height) using the α values obtained from the simulations. The resulting turbulent magnetic Prandtl number is consistently close to one (Pmₜ ≈ 1). Importantly, the authors find that Pmₜ is not a universal constant but varies systematically with the strength of the imposed EMF: stronger forcing reduces ηₜ more than νₜ, leading to an increase of Pmₜ. This trend suggests that the efficiency of magnetic field diffusion can be modulated by the amplitude of large‑scale electromotive drivers.

Box‑size effects are also explored. When the radial (x) dimension of the shearing box is enlarged while keeping the azimuthal (y) extent modest, ηₜ modestly increases and Pmₜ correspondingly decreases. This behavior matches earlier large‑box studies that reported a decline of Pmₜ with increasing radial domain, likely because larger boxes accommodate longer wavelength shear modes that alter the cascade of magnetic energy and reduce the effective turbulent conductivity. Conversely, extending the azimuthal dimension sufficiently (aspect ratios ≥ 4) leads to convergence of turbulent statistics, indicating that the azimuthal length is the more critical dimension for achieving size‑independent transport coefficients.

The authors compare their findings with recent literature that employed different measurement strategies (e.g., decay‑rate methods, test‑field approaches) and find broad agreement: turbulent magnetic Prandtl numbers of order unity are a robust outcome of MRI‑driven turbulence in zero‑net‑flux configurations. The paper therefore strengthens the case that, in realistic protoplanetary and astrophysical disks, the turbulent diffusion of magnetic fields is comparable to the turbulent transport of momentum. This has direct implications for models of magnetic flux accumulation, disk wind launching, and the saturation of large‑scale dynamos, all of which rely on an accurate estimate of ηₜ and νₜ.

In summary, the study provides a carefully cross‑validated, code‑independent measurement of ηₜ and νₜ in MRI turbulence, demonstrates that the turbulent magnetic Prandtl number is of order one but sensitive to both forcing amplitude and radial box size, and situates these results within the broader context of mean‑field theory and recent numerical work. The robustness of Pmₜ ≈ 1 supports its use as a baseline parameter in global disk models, while the identified dependencies highlight avenues for future investigations, such as the role of net magnetic flux, stratification, and non‑ideal MHD effects.