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Register a new userGlobal simulations of magnetorotational turbulence I: convergence and the quasi-steady state
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Magnetorotational turbulence provides a viable mechanism for angular momentum transport in accretion disks. We present global, three dimensional (3D), MHD accretion disk simulations that investigate the dependence of the turbulent stresses on resolution. Convergence in the time-and-volume-averaged stress-to-gas-pressure ratio, at a value of $sim0.04$, is found for a model with radial, vertical, and azimuthal resolution of 12-51, 27, and 12.5 cells per scale-height (the simulation mesh is such that cells per scale-height varies in the radial direction). A control volume analysis is performed on the main body of the disk (|z|<2H) to examine the production and removal of magnetic energy. Maxwell stresses in combination with the mean disk rotation are mainly responsible for magnetic energy production, whereas turbulent dissipation (facilitated by numerical resistivity) predominantly removes magnetic energy from the disk. Re-casting the magnetic energy equation in terms of the power injected by Maxwell stresses on the boundaries of, and by Lorentz forces within, the control volume highlights the importance of the boundary conditions (of the control volume). The different convergence properties of shearing-box and global accretion disk simulations can be readily understood on the basis of choice of boundary conditions and the magnetic field configuration. Periodic boundary conditions restrict the establishment of large-scale gradients in the magnetic field, limiting the power that can be delivered to the disk by Lorentz forces and by stresses at the surfaces. The factor of three lower resolution required for convergence in turbulent stresses for our global disk models compared to stratified shearing-boxes is explained by this finding. (Abridged)
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Magnetorotational turbulence draws its energy from gravity and ultimately releases it via dissipation. However, the quantitative details of this energy flow have not been assessed for global disk models. In this work we examine the energetics of a well-resolved, three-dimensional, global magnetohydrodynamic accretion disk simulation by evaluating statistically-averaged mean-field equations for magnetic, kinetic, and internal energy using simulation data. The results reveal that turbulent magnetic (kinetic) energy is primarily injected by the correlation between Maxwell (Reynolds) stresses and shear in the (almost Keplerian) mean flow, and removed by dissipation. This finding differs from previous work using local (shearing-box) models, which indicated that turbulent kinetic energy was primarily sourced from the magnetic energy reservoir. Lorentz forces provide the bridge between the magnetic and kinetic energy reservoirs, converting ~ 1/5 of the total turbulent magnetic power input into turbulent kinetic energy. The turbulent energies (both magnetic and kinetic) are mainly driven by terms associated with the turbulent fields, with only a minor influence from mean magnetic fields. The interaction between mean and turbulent fields is most evident in the induction equation, with the mean radial magnetic field being strongly influenced by the turbulent electromotive force (EMF). During the quasi-steady turbulent state roughly 2/3 of the Poynting flux travels into the corona, with the remainder transporting magnetic energy in the radial direction. In contrast to previous studies, the stress-related part of the Poynting flux is found to dominate, which may have important implications for reflection models of Seyfert galaxy coronae that typically invoke a picture of buoyant rising of magnetic flux tubes via advection.
The stresses produced by magnetorotational turbulence can provide effective angular momentum transport in accretion disks. However, questions remain about the ability of simulated disks to reproduce observationally inferred stress-to-gas-pressure ratios. In this paper we present a set of high resolution global magnetohydrodynamic disk simulations which are initialised with different field configurations: purely toroidal, vertical field lines, and nested poloidal loops. A mass source term is included which allows the total disk mass to equilibrate in simulations with long run times, and also enables the impact of rapid mass injection to be explored. Notably different levels of angular momentum transport are observed during the early-time transient disk evolution. However, given sufficient time to relax, the different models evolve to a statistically similar quasi-steady state with a stress-to-gas-pressure ratio, $alpha sim 0.032-0.036$. The indication from our results is that {it steady, isolated} disks may be unable to maintain a large-scale magnetic field or produce values for the stress-to-gas-pressure ratio implied by some observations. Supplementary simulations exploring the influence of trapping magnetic field, injecting vertical field, and rapidly injecting additional mass into the disk show that large stresses ($alpha sim 0.1-0.25$) can be induced by these mechanisms. The simulations highlight the common late-time evolution and characteristics of turbulent disks for which the magnetic field is allowed to evolve freely. If the boundaries of the disk, the rate of injection of magnetic field, or the rate of mass replenishment are modified to mimic astrophysical disks, markedly different disk evolution occurs.
Global three dimensional magnetohydrodynamic (MHD) simulations of turbulent accretion disks are presented which start from fully equilibrium initial conditions in which the magnetic forces are accounted for and the induction equation is satisfied. The local linear theory of the magnetorotational instability (MRI) is used as a predictor of the growth of magnetic field perturbations in the global simulations. The linear growth estimates and global simulations diverge when non-linear motions - perhaps triggered by the onset of turbulence - upset the velocity perturbations used to excite the MRI. The saturated state is found to be independent of the initially excited MRI mode, showing that once the disk has expelled the initially net flux field and settled into quasi-periodic oscillations in the toroidal magnetic flux, the dynamo cycle regulates the global saturation stress level. Furthermore, time-averaged measures of converged turbulence, such as the ratio of magnetic energies, are found to be in agreement with previous works. In particular, the globally averaged stress normalized to the gas pressure, <alpha_{rm P}> = 0.034, with notably higher values achieved for simulations with higher azimuthal resolution. Supplementary tests are performed using different numerical algorithms and resolutions. Convergence with resolution during the initial linear MRI growth phase is found for 23-35 cells per scaleheight (in the vertical direction).
In biochemical networks, reactions often occur on disparate timescales and can be characterized as either fast or slow. The quasi-steady state approximation (QSSA) utilizes timescale separation to project models of biochemical networks onto lower-dimensional slow manifolds. As a result, fast elementary reactions are not modeled explicitly, and their effect is captured by non-elementary reaction rate functions (e.g. Hill functions). The accuracy of the QSSA applied to deterministic systems depends on how well timescales are separated. Recently, it has been proposed to use the non-elementary rate functions obtained via the deterministic QSSA to define propensity functions in stochastic simulations of biochemical networks. In this approach, termed the stochastic QSSA, fast reactions that are part of non-elementary reactions are not simulated, greatly reducing computation time. However, it is unclear when the stochastic QSSA provides an accurate approximation of the original stochastic simulation. We show that, unlike the deterministic QSSA, the validity of the stochastic QSSA does not follow from timescale separation alone, but also depends on the sensitivity of the non-elementary reaction rate functions to changes in the slow species. The stochastic QSSA becomes more accurate when this sensitivity is small. Different types of QSSAs result in non-elementary functions with different sensitivities, and the total QSSA results in less sensitive functions than the standard or the pre-factor QSSA. We prove that, as a result, the stochastic QSSA becomes more accurate when non-elementary reaction functions are obtained using the total QSSA. Our work provides a novel condition for the validity of the QSSA in stochastic simulations of biochemical reaction networks with disparate timescales.
The phenomenological Disc Instability Model has been successful in reproducing the observed light curves of dwarf nova outbursts by invoking an enhanced Shakura-Sunyaev $alpha$ parameter $sim0.1-0.2$ in outburst compared to a low value $sim0.01$ in quiescence. Recent thermodynamically consistent simulations of magnetorotational (MRI) turbulence with appropriate opacities and equation of state for dwarf nova accretion discs have found that thermal convection enhances $alpha$ in discs in outburst, but only near the hydrogen ionization transition. At higher temperatures, convection no longer exists and $alpha$ returns to the low value comparable to that in quiescence. In order to check whether this enhancement near the hydrogen ionization transition is sufficient to reproduce observed light curves, we incorporate this MRI-based variation in $alpha$ into the Disc Instability Model, as well as simulation-based models of turbulent dissipation and convective transport. These MRI-based models can successfully reproduce observed outburst and quiescence durations, as well as outburst amplitudes, albeit with different parameters from the standard Disc Instability Models. The MRI-based model lightcurves exhibit reflares in the decay from outburst, which are not generally observed in dwarf novae. However, we highlight the problematic aspects of the quiescence physics in the Disc Instability Model and MRI simulations that are responsible for this behavior.
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