On the Anisotropic Nature of MRI-Driven Turbulence in Astrophysical Disks

The magnetorotational instability is thought to play an important role in enabling accretion in sufficiently ionized astrophysical disks. The rate at which MRI-driven turbulence transports angular momentum is related to both the strength of the amplitudes of the fluctuations on various scales and the degree of anisotropy of the underlying turbulence. This has motivated several studies of the distribution of turbulent power in spectral space. In this paper, we investigate the anisotropic nature of MRI-driven turbulence using a pseudo-spectral code and introduce novel ways to robustly characterize the underlying turbulence. We show that the general flow properties vary in a quasi-periodic way on timescales comparable to 10 inverse angular frequencies motivating the temporal analysis of its anisotropy. We introduce a 3D tensor invariant analysis to quantify and classify the evolution of the anisotropic turbulent flow. This analysis shows a continuous high level of anisotropy, with brief sporadic transitions towards two- and three-component isotropic turbulent flow. This temporal-dependent anisotropy renders standard shell-average, especially when used simultaneously with long temporal averages, inadequate for characterizing MRI-driven turbulence. We propose an alternative way to extract spectral information from the turbulent magnetized flow, whose anisotropic character depends strongly on time. This consists of stacking 1D Fourier spectra along three orthogonal directions that exhibit maximum anisotropy in Fourier space. The resulting averaged spectra show that the power along each of the three independent directions differs by several orders of magnitude over most scales, except the largest ones. Our results suggest that a first-principles theory to describe fully developed MRI-driven turbulence will likely have to consider the anisotropic nature of the flow at a fundamental level.Comment: 13 pages, 13 figures, submitted to Ap

Spatio-Temporal Multiway Data Decomposition Using Principal Tensor Analysis on k-Modes: The R Package PTAk

The purpose of this paper is to describe the R package {PTAk and how the spatio-temporal context can be taken into account in the analyses. Essentially PTAk() is a multiway multidimensional method to decompose a multi-entries data-array, seen mathematically as a tensor of any order. This PTAk-modes method proposes a way of generalizing SVD (singular value decomposition), as well as some other well known methods included in the R package, such as PARAFAC or CANDECOMP and the PCAn-modes or Tucker-n model. The example datasets cover different domains with various spatio-temporal characteristics and issues: (i)~medical imaging in neuropsychology with a functional MRI (magnetic resonance imaging) study, (ii)~pharmaceutical research with a pharmacodynamic study with EEG (electro-encephaloegraphic) data for a central nervous system (CNS) drug, and (iii)~geographical information system (GIS) with a climatic dataset that characterizes arid and semi-arid variations. All the methods implemented in the R package PTAk also support non-identity metrics, as well as penalizations during the optimization process. As a result of these flexibilities, together with pre-processing facilities, PTAk constitutes a framework for devising extensions of multidimensional methods such ascorrespondence analysis, discriminant analysis, and multidimensional scaling, also enabling spatio-temporal constraints.