The Effect of Resistivity on the Nonlinear Stage of the Magnetorotational Instability in Accretion Disks

By T. Fleming, James M. Stone, J. Hawley. 2000

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We present three-dimensional magnetohydrodynamic simulations of the nonlinear evolution of the magnetorotational instability (MRI) with a nonzero ohmic resistivity. The simulations begin from a homogeneous (unstratified) density distribution and use the local shearing-box approximation. The evolution of a variety of initial field configurations and strengths is considered for several values of the constant coefficient of resistivity η. For uniform vertical and toroidal magnetic fields, we find unstable growth consistent with the linear analyses; finite resistivity reduces growth rates and, when large enough, stabilizes the MRI. Even when unstable modes remain, resistivity has significant effects on the nonlinear state. The properties of the saturated state depend on the initial magnetic field configuration. In simulations with an initial uniform vertical field, the MRI is able to support angular momentum transport even for large resistivities through the quasi-periodic generation of axisymmetric radial channel solutions rather than through the maintenance of anisotropic turbulence. Reconnective processes rather than parasitic instabilities mediate the resurgent channel solution in this case. Simulations with zero-net flux show that the angular momentum transport and the amplitude of magnetic energy after saturation are significantly reduced by finite resistivity, even at levels where the linear modes are only slightly affected. The MRI is unable to sustain angular momentum transport and turbulent flow against diffusion for ReM ≲ 104, where the Reynolds number is defined in terms of the disk scale height and sound speed, ReM = csH/η. As this is close to the Reynolds numbers expected in low, cool states of dwarf novae, these results suggest that finite resistivity may account for the low and high angular momentum transport rates inferred for these systems.

Subfields: Astronomy and Astrophysics, Instrumentation

View Paper: https://www.semanticscholar.org/paper/ff3204169654ab8a38534512b2a3e7854ee1a2c4

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Citations in the corpus

EFFECT OF AMBIPOLAR DIFFUSION ON THE NONLINEAR EVOLUTION OF MAGNETOROTATIONAL INSTABILITY IN WEAKLY IONIZED DISKS (2011, background): “…Elsasser number equals one being the boundary between MRI permitted and suppressed regions identified in unstratified simulations (Sano et al. 1998; Fleming et al. 2000; Sano & Stone 2002b) do agree with results from stratified simulations (Fleming & Stone 2003; Turner et al. 2007; Ilgner & Nelson…”
MHD simulations of the magnetorotational instability in a shearing box with zero net flux II. The effect of transport coefficients (2007, background): “Unfortunately because of the limited computational resources available during the 90’s, these early calculations were restricted to low and moderate resolutions, having at most 64 grid cells per disk scale height (Hawley et al. 1995, 1996; Fleming et al. 2000; Sano et al. 2004).”
THE RUNTS OF THE LITTER: WHY PLANETS FORMED THROUGH GRAVITATIONAL INSTABILITY CAN ONLY BE FAILED BINARY STARS (2009, background): “More restrictively this requires that other local transport mechanisms such as the MRI be weaker than currently predicted by simulations—they produce α ∼ 10−2 at least in disks with a net magnetic flux (Fleming et al. 2000; Fromang et al. 2007; Johansen et al. 2009).”
Global simulations of protoplanetary disks with net magnetic flux: I. Non-ideal MHD case (2016, background): “Both diffusive effects can damp magnetic structures, and potentially quench the MRI (Jin 1996; Kunz & Balbus 2004), as supported by numerous non-linear simulations (e.g., Hawley & Stone 1998; Fleming et al. 2000; Bai & Stone 2011).”
SUSTAINED MAGNETOROTATIONAL TURBULENCE IN LOCAL SIMULATIONS OF STRATIFIED DISKS WITH ZERO NET MAGNETIC FLUX (2009, background): “…behavior of MRI in the linear regime (e.g., Sano & Miyama 1999; Pessah & Chan 2008), and the saturation of turbulence in unstratified shearing boxes depends on the nature of the dissipation (Sano et al. 1998; Fleming et al. 2000; Fromang et al. 2007; Lesur & Longaretti 2007; Simon & Hawley 2009).”
Turbulent Mixing and the Dead Zone in Protostellar Disks (2006, background): “Turbulence occurred only at Reynolds numbers greater than 104 with zero net magnetic flux and at Reynolds numbers as low as 100 with a net vertical magnetic flux in unstratified shearing-box calculations by Fleming et al. (2000).”
The ionization fraction in α models of protoplanetary discs (2001, unknown): “Numerical simulations including ohmic dissipation (Fleming et al. 2000) indicate that MHD turbulence cannot be sustained if ReM falls below a critical value, ReM,crit, which depends upon the field geometry.”
Enhanced Angular Momentum Transport in Accretion Disks (2003, background): “In the simulations of Fleming et al. (2000), a critical magnetic Reynolds number ReM emerged below which turbulence is suppressed.”
Thanatology in protoplanetary discs - The combined influence of Ohmic, Hall, and ambipolar diffusion on dead zones (2014, background): “S R ] 2 4 Fe b 20 14 2000; Fromang et al. 2002; Ilgner & Nelson 2006a,b) and threedimensional numerical simulations (Fleming et al. 2000; Fleming & Stone 2003; Turner et al. 2007; Ilgner & Nelson 2008).”
WIND-DRIVEN ACCRETION IN PROTOPLANETARY DISKS. I. SUPPRESSION OF THE MAGNETOROTATIONAL INSTABILITY AND LAUNCHING OF THE MAGNETOCENTRIFUGAL WIND (2013, background): “…expectations is that the field geometry is not optimal: in both the ideal MHD (e.g., Hawley et al. 1995; Bai & Stone 2013) and non-ideal MHD (e.g., Fleming et al. 2000; Bai & Stone 2011), one finds stronger turbulence in the presence of a net vertical magnetic flux, and the optimistic estimate…”
Linear Analysis of the Hall Effect in Protostellar Disks (2000, background): “Numerical simulations which include only Ohmic losses have yet to demonstrate that MHD turbulence can be sustained at the ionization fractions that may be present in these cool disks (Fleming et al. 2000, Menou 2000).”
The Effect of the Hall Term on the Nonlinear Evolution of the Magnetorotational Instability. II. Saturation Level and Critical Magnetic Reynolds Number (2002, unknown): “…Number In this subsection, we consider the critical value of the magnetic Reynolds number for significant angular momentum transport in accretion disks with uniform vertical fields (as shown by previous studies, this critical value depends on the initial field geometry, Fleming et al. 2000).”
ASSESSING QUANTITATIVE RESULTS IN ACCRETION SIMULATIONS: FROM LOCAL TO GLOBAL (2011, background): “…initial field strength and geometry, and other thermodynamic factors (e.g., Hawley et al. 1995, 1996; 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 Effect of the Hall Term on the Nonlinear Evolution of the Magnetorotational Instability. I. Local Axisymmetric Simulations (2002, methodology): “In 3D, re-emergence of the MRI on the background field (which can never be dissipated) leads to strong fluctuations in, e.g., the Maxwell stress (Fleming et al. 2000).”