Locality of MHD Turbulence in Isothermal Disks

📝 Abstract

We numerically evolve turbulence driven by the magnetorotational instability (MRI) in a 3D, unstratified shearing box and study its structure using two-point correlation functions. We confirm Fromang and Papaloizou’s result that shearing box models with zero net magnetic flux are not converged; the dimensionless shear stress $\alpha$ is proportional to the grid scale. We find that the two-point correlation of the magnetic field shows that it is composed of narrow filaments that are swept back by differential rotation into a trailing spiral. The correlation lengths along each of the correlation function principal axes decrease monotonically with the grid scale. For mean azimuthal field models, which we argue are more relevant to astrophysical disks than the zero net field models, we find that: $\alpha$ increases weakly with increasing resolution at fixed box size; $\alpha$ increases slightly as the box size is increased; $\alpha$ increases linearly with net field strength, confirming earlier results; the two-point correlation function of the magnetic field is resolved and converged, and is composed of narrow filaments swept back by the shear; the major axis of the two-point increases slightly as the box size is increased; these results are code independent, based on a comparison of ATHENA and ZEUS runs. The velocity, density, and magnetic fields decorrelate over scales larger than $\sim H $, as do the dynamical terms in the magnetic energy evolution equations. We conclude that MHD turbulence in disks is localized, subject to the limitations imposed by the absence of vertical stratification, the use of an isothermal equation of state, finite box size, finite run time, and finite resolution

💡 Analysis

We numerically evolve turbulence driven by the magnetorotational instability (MRI) in a 3D, unstratified shearing box and study its structure using two-point correlation functions. We confirm Fromang and Papaloizou’s result that shearing box models with zero net magnetic flux are not converged; the dimensionless shear stress $\alpha$ is proportional to the grid scale. We find that the two-point correlation of the magnetic field shows that it is composed of narrow filaments that are swept back by differential rotation into a trailing spiral. The correlation lengths along each of the correlation function principal axes decrease monotonically with the grid scale. For mean azimuthal field models, which we argue are more relevant to astrophysical disks than the zero net field models, we find that: $\alpha$ increases weakly with increasing resolution at fixed box size; $\alpha$ increases slightly as the box size is increased; $\alpha$ increases linearly with net field strength, confirming earlier results; the two-point correlation function of the magnetic field is resolved and converged, and is composed of narrow filaments swept back by the shear; the major axis of the two-point increases slightly as the box size is increased; these results are code independent, based on a comparison of ATHENA and ZEUS runs. The velocity, density, and magnetic fields decorrelate over scales larger than $\sim H $, as do the dynamical terms in the magnetic energy evolution equations. We conclude that MHD turbulence in disks is localized, subject to the limitations imposed by the absence of vertical stratification, the use of an isothermal equation of state, finite box size, finite run time, and finite resolution

📄 Content

arXiv:0901.0273v1 [astro-ph.GA] 2 Jan 2009 Locality of MHD Turbulence in Isothermal Disks Xiaoyue Guan and Charles F. Gammie1 Astronomy Department, University of Illinois, 1002 West Green St., Urbana, IL 61801, USA Jacob B. Simon Astronomy Department, University of Virginia, Box 400325, Charlottesville, VA 22904, USA and Bryan M. Johnson Lawrence Livermore National Laboratory, L-023, 7000 East Avenue, Livermore, CA 94550 ABSTRACT We numerically evolve turbulence driven by the magnetorotational instability (MRI) in a 3D, unstratified shearing box and study its structure using two-point correlation functions. We confirm Fromang & Papaloizou’s result that shearing box models with zero net magnetic flux are not converged; the dimensionless shear stress α is proportional to the grid scale. We find that the two-point correlation of B shows that it is composed of narrow filaments that are swept back by differential rotation into a trailing spiral. The correlation lengths along each of the correlation function principal axes decrease monotonically with the grid scale. For mean azimuthal field models, which we argue are more relevant to astrophysical disks than the zero net field models, we find that: α increases weakly with increasing resolution at fixed box size; α increases slightly as the box size is increased; α increases linearly with net field strength, confirming earlier results; the two-point correlation function of the magnetic field is resolved and converged, and is composed of narrow filaments swept back by the shear; the major axis of the two-point increases slightly as the box size is increased; these results are code independent, based on a comparison of ATHENA and ZEUS runs. The velocity, density, and magnetic fields decorrelate over scales larger than 1Physics Department, University of Illinois – 2 – ∼H, as do the dynamical terms in the magnetic energy evolution equations. We conclude that MHD turbulence in disks is localized, subject to the limitations imposed by the absence of vertical stratification, the use of an isothermal equation of state, finite box size, finite run time, and finite resolution. Subject headings: accretion, accretion disks, magnetohydrodynamics 1. Introduction Astrophysical disks appear to redistribute angular momentum rapidly, much more rapidly than one would expect based on estimates of the molecular viscosity. Classical thin accretion disk theories (Shakura & Sunyaev 1973; Lynden-Bell & Pringle 1974) solved this problem by appealing to turbulence, and modeled the effects of this turbulence as an “anomalous vis- cosity.” The idea that turbulence plays a key role was placed on firmer foundations with the (re)discovery of the magnetorotational instability (MRI; Balbus & Hawley 1991) and subsequent numerical investigations (see Balbus & Hawley 1998 for a review). Winds or gravitational instability may drive disk evolution in certain cases, but MRI-initiated MHD turbulence appears capable of driving disk evolution in a wide variety of astrophysical disks. We still do not know, however, whether the effects of MHD turbulence on disks are localized. It is possible that structures develop that are large compared to a scale height H ≡ cs/Ω, and that these structures are associated with nonlocal energy and angular momentum transport. If so, disk evolution would not be well described by a theory, such as the α model, in which the shear stress depends only on the local surface density and temperature. A related possibility, which we will not examine here, is that the time-averaged turbulent stresses W rφ might satisfy ∂W rφ/∂Σ < 0 (Σ ≡surface density; see Piran (1978) for a discussion). That is, the disk might be “viscously” unstable. This could cause the disk to break up into rings or even—to use a term of art—“blobs.” Such an outcome would be awkward for the classic phenomenological steady disk and disk evolution models, which have had some success in modeling cataclysmic variable disks and black hole x-ray binary disks in a high, soft state (e.g. Belloni et al. 1997; Lasota 2001). How can one probe the locality of MHD turbulence in disks? We will use the two- point correlation function of the magnetic field, velocity field, and density as determined by numerical experiments. Nonlocal transport would likely be associated with features in the two-point correlation function, as would viscous instabilities. For example, turbulence might excite waves (wakes) that carry energy and angular momentum over many H in radius before damping. These wakes would appear as extended features in the two-point correlation – 3 – function. The two-point correlation function and the power spectrum contain the same infor- mation since they are related by a Fourier transform. But they do not convey the same impression and they have different noise properties. For a one dimensional function sampled at N points over an interval L half the sample points in the power spectrum lie between the Nyquist frequency (πN/L) and half the Nyqu

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