Sensitivity of the magnetorotational instability to the shear parameter in stratified simulations

Abstract

The magnetorotational instability (MRI) is a shear instability and thus its sensitivity to the shear parameter q = −d ln Ω/d ln r is of interest to investigate. Motivated by astrophysical disks, most (but not all) previous MRI studies have focused on the Keplerian value of q = 1.5. Using simulation with 8 vertical density scale heights, we contribute to the subset of studies addressing the the effect of varying q in stratified numerical simulations. We discuss why shearing boxes cannot easily be used to study q > 2 and thus focus on q < 2. As per previous simulations, which were either unstratified or stratified with a smaller vertical domain, we find that the q dependence of stress for the stratified case is not linear, contrary to the Shakura-Sunyaev model. We find that the scaling agrees with Abramowicz et al. (1996) who found it to be proportional to the shear to vorticity ratio q/(2 − q). We also find however, that the shape of the magnetic and kinetic energy spectra are relatively insensitive to q and that the ratio of Maxwell stress to magnetic energy ratio also remains nearly independent of q. This is consistent with a theoretical argument in which the rate of amplification of the azimuthal field depends linearly on q and the turbulent correlation time τ depends inversely on q. As such, we measure the correlation time of the turbulence and find that indeed it is inversely proportional to q.

Motivation and challenge to capture both large-scale and local transport in next generation accretion theory

“…There have been specific attempts to improve the closure (Ogilvie 2003;Pessah, Chan & Psaltis 2006), and more recent direct evidence that the MRI stress does not agree with the SS73 formulation (Pessah, Chan & Psaltis 2008;Nauman & Blackman 2015). As discussed further below, our interpretation of numerical evidence further suggests that the transport from the MRI is actually dominated by large scales.…”

“…Using the above procedure for a q = 3/2 case from a run with 32 zones/H and domain size H × 2H × 4H (Nauman & Blackman 2015), we find that the ratio of the stresses from the mean field to that of the total to be T M,xy /T xy 0.24 and the ratio of magnetic energy from the mean field to that of the total to be 0.47. These indicate a significant contribution from the mean field, bolstering the overall message that large-scale fields and transport are important.…”

“…For c s ∼ ΩH in hydrostatic equilibrium, we then have (4.10) for k z H ∼ 1. Empirically, α ss ∝ q/(2 − q) (Abramowicz et al 1996;Pessah et al 2008;Nauman & Blackman 2015) so tan θ M 1.5α ss,0 /(2 − q), where α ss,0 ≡ α ss (q = 3/2).…”

3D global simulations of RIAFs: convergence, effects of azimuthal extent, and dynamo

“…Recently Nauman & Blackman (2015) and Gressel & Pessah (2015) found that the dynamo period becomes shorter and less well-defined (e. g. see the butterfly diagram for q = 1.8 in Fig. 6 in Nauman & Blackman (2015)) when the shear parameter q is increased from its Keplerian value. Interestingly, in the quasi-steady state of our simulations the angular velocity is sub-Keplerian with Ω = v φ r ∝ r −1.7 (see equation 42).…”

On properties of Velikhov–Chandrasekhar MRI in ideal and non-ideal plasma

“…Hawley, Gammie & Balbus 1995;Sorathia et al 2012;Hawley et al 2013) confirmed that MRI can sustain turbulence and dynamo in accretion discs. However, semi-analytical and numerical simulations (see, for example, Masada & Sano 2008;Stone 2011;Hawley et al 2013;Suzuki & Inutsuka 2014;Nauman & Blackman 2015) suggest that the total (Reynolds + Maxwell) stresses due to MRI are insufficient to cause the effective angular momentum transfer in accretion discs in terms of the phenomenological alpha-parameter α S S (Shakura & Sunyaev 1973), giving rather low values α S S ∼ 0.01 − 0.03. Note that from the observational point of view, the alpha-parameter can be reliably evaluated, e.g., from the analysis of non-stationary accretion discs in X-ray novae (Suleimanov, Lipunova & Shakura 2008), dwarf-nova and AM CVn stars (Kotko & Lasota 2012), and turns out to be an order of magnitude higher than typically found in the numerical MRI simulations.…”