MHD simulations of the magnetorotational instability in a shearing box with zero net flux - I. The issue of convergence

By Sébastien Fromang, J. Papaloizou. 2007

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Aims. We study the properties of MHD turbulence driven by the magnetorotational instability (MRI) in accretion disks. To do this we perform a series of numerical simulations for which the resolution is gradually increased. Methods. We adopt the local shearing box model and focus on the special case for which the initial magnetic flux threading the disk vanishes. We employ the finite difference code ZEUS to evolve the ideal MHD equations. Results. Performing a set of numerical simulations in a fixed computational domain with increasing resolution, we demonstrate that turbulent activity decreases as resolution increases. The highest resolution considered is 256 grid cells per scale height. We quantify the turbulent activity by measuring the rate of angular momentum transport through evaluating the standard α parameter. We find α = 0.004 when $(N_x,N_y,N_z) = (64,100,64)$, α = 0.002 when $(N_x,N_y,N_z) = (128,200,128)$ and α = 0.001 when $(N_x,N_y,N_z) = (256,400,256)$. This steady decline is an indication that numerical dissipation, occurring at the grid scale is an important determinant of the saturated form of the MHD turbulence. Analysing the results in Fourier space, we demonstrate that this is due to the MRI forcing significant flow energy all the way down to the grid dissipation scale. We also use our results to study the properties of the numerical dissipation in ZEUS. Its amplitude is characterised by the magnitude of an effective magnetic Reynolds number $\rm Re_{M}$ which increases from 10 4 to 10 5 as the number of grid points is increased from 64 to 256 per scale height. Conclusions. The simulations we have carried out do not produce results that are independent of the numerical dissipation scale, even at the highest resolution studied. Thus it is important to use physical dissipation, both viscous and resistive, and to quantify contributions from numerical effects, when performing numerical simulations of MHD turbulence with zero net flux in accretion disks at the resolutions normally considered.

Subfields: Astronomy and Astrophysics

View Paper: https://www.semanticscholar.org/paper/63f08f538c864c4a270f3ed2ffd4a43d4318f708

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Citations in the corpus

SUSTAINED MAGNETOROTATIONAL TURBULENCE IN LOCAL SIMULATIONS OF STRATIFIED DISKS WITH ZERO NET MAGNETIC FLUX (2009, methodology): “Interestingly, in this case we find quite different results compared to the unstratified boxes studied by Fromang & Papaloizou (2007).”
Semi-global simulations of the magneto-rotational instability in core collapse supernovae (2008, background): “A careful analysis is necessary to disentangle the respective influence (see e.g., Pessah et al. 2007; Fromang & Papaloizou 2007; Regev & Umurhan 2008).”
EMPIRICAL CONSTRAINTS ON TURBULENCE IN PROTOPLANETARY ACCRETION DISKS (2010, background): “It is clear that shearing box simulations of MHD turbulence with zero net magnetic field flux do not give reliable values of the viscosity parameter α due to numerical dissipation, which results in values of α that depend on resolution (Pessah et al. 2007; Fromang & Papaloizou 2007).”
MHD simulations of the magnetorotational instability in a shearing box with zero net flux II. The effect of transport coefficients (2007, methodology): “In Paper I (Fromang & Papaloizou 2007), we used one of these numerical codes, ZEUS (Hawley & Stone 1995), to study the convergence of the results as the grid resolution is increased.”
ZONAL FLOWS AND LONG-LIVED AXISYMMETRIC PRESSURE BUMPS IN MAGNETOROTATIONAL TURBULENCE (2008, background): “The saturated state of non-stratified zero net vertical flux magnetorotational turbulence has been shown to decline with increasing resolution (Fromang & Papaloizou 2007), due to the change in numerical dissipation as the resolution increases.”
Thanatology in protoplanetary discs - The combined influence of Ohmic, Hall, and ambipolar diffusion on dead zones (2014, background): “…configuration: The zero vertical-net-flux configuration has been largely explored in the past, both with unstratified (Hawley et al. 1996; Fromang & Papaloizou 2007) and stratified simulations (Brandenburg et al. 1995; Stone et al. 1996; Fleming & Stone 2003; Simon et al. 2012, 2013b).”
EFFECT OF AMBIPOLAR DIFFUSION ON THE NONLINEAR EVOLUTION OF MAGNETOROTATIONAL INSTABILITY IN WEAKLY IONIZED DISKS (2011, background): “Since all our simulations contain net magnetic flux, they are not subject to the issue of convergence found by (Fromang & Papaloizou 2007) in zero net-flux simulations, and numerical convergence is confirmed in our test simulations (and see Section 3.2.2 for the case of net toroidal flux).”
Dynamics of Protoplanetary Disks (2010, background): “(This curious behavior is seen in MHD simulations in small domains that omit explicit dissipation, e.g. Fromang & Papaloizou 2007; Guan et al. 2009; Pessah, Chan & Psaltis 2007; Simon, Hawley & Beckwith 2009).”
LOCAL STUDY OF ACCRETION DISKS WITH A STRONG VERTICAL MAGNETIC FIELD: MAGNETOROTATIONAL INSTABILITY AND DISK OUTFLOW (2012, background): “…MRI simulations provide important benchmarks on the physical properties of the MRI turbulence (e.g., Hawley et al. 1996; Sano et al. 2004; Fromang & Papaloizou 2007; Electronic address: xbai@cfa.harvard.edu 1 Department of Astrophysical Sciences, Peyton Hall, Princeton University,…”
Gaps, Rings, and Non-Axisymmetric Structures in Protoplanetary Disks - From Simulations to ALMA Observations (2014, methodology): “First, we used a vertical net flux field in which convergence is found (Fromang & Papaloizou 2007; Simon et al. 2013).”
ASSESSING QUANTITATIVE RESULTS IN ACCRETION SIMULATIONS: FROM LOCAL TO GLOBAL (2011, unknown): “…Brandenburg et al. 1995; Stone et al. 1996; Fleming et al. 2000; Miller & Stone 2000; Sano & Inutsuka 2001; Sano & Stone 2002; Sano et al. 2004; Fromang & Papaloizou 2007; Simon et al. 2009) The first shearing box simulations employed comparatively few total zones (e.g., 32×64×32 zones in x,…”
NONSTEADY ACCRETION IN PROTOSTARS (2008, background): “Recent work (Fromang & Papaloizou 2007), however, shows that αM does not converge in the sense that αM → 0 as the numerical resolution increases.”
Breaking the chains: hot super-Earth systems from migration and disruption of compact resonant chains (2017, background): “Still, the typical values of α in MRI active zones is in the range of 10−3 and 10−1 (Fromang & Papaloizou 2007), which is in great agreement with our chosen value.”
Comprehensive nucleosynthesis analysis for ejecta of compact binary mergers (2014, background): “…discs is an active field of research on its own with several fundamental issues being still vividly debated, e.g. numerical convergence (e.g. Fromang & Papaloizou 2007; Sorathia et al. 2012; Hawley et al. 2013), impact of the global magnetic field topology (e.g. Beckwith, Hawley & Krolik…”
Hot Accretion Flows Around Black Holes (2014, background): “However, when the net magnetic flux is zero, increasing the resolution actually causes α to decrease (Fromang & Papaloizou 2007, Fromang et al. 2007), with α going to zero in the limit of infinite resolution.”