Systematics of the magnetic-Prandtl-number dependence of homogeneous, isotropic magnetohydrodynamic turbulence

Abstract

We present the results of our detailed pseudospectral direct numerical simulation (DNS) studies, with up to 1024 3 collocation points, of incompressible, magnetohydrodynamic (MHD) turbulence in three dimensions, without a mean magnetic field. Our study concentrates on the dependence of various statistical properties of both decaying and statistically steady MHD turbulence on the magnetic Prandtl number Pr M over a large range, namely, 0.01 ≤ Pr M ≤ 10. We obtain data for a wide variety of statistical measures such as probability distribution functions (PDFs) of moduli of the vorticity and current density, the energy dissipation rates, and velocity and magnetic-field increments, energy and other spectra, velocity and magnetic-field structure functions, which we use to characterise intermittency, isosurfaces of quantities such as the moduli of the vorticity and current, and joint PDFs such as those of fluid and magnetic dissipation rates. Our systematic study uncovers interesting results that have not been noted hitherto. In particular, we find a crossover from larger intermittency in the magnetic field than in the velocity field, at large Pr M , to smaller intermittency in the magnetic field than in the velocity field, at low Pr M . Furthermore, a comparison of our results for decaying MHD turbulence and its forced, statistically steady analogue suggests that we have strong universality in the sense that, for a fixed value of Pr M , multiscaling exponent ratios agree, at least within our errorbars, for both decaying and statistically steady homogeneous, isotropic MHD turbulence.

Modeling Statistical Properties of Solar Active Regions Through Direct Numerical Simulations of 3d-MHD Turbulence

“…As can be observed from the data points in Figures 1(c) and (d), the information obtained from both ESS and LSA is extremely similar. However, LSA gives better error estimates than ESS, providing additional confidence on the data, as was observed by Sahoo et al (2011). Although the ζ p values up to an order of four for all the cases are very close to each other and do not yield much information on the nature of the structures, the higher-order structure function exponents do give useful information.…”

“…The standard deviation of the rest of the values from this average value is estimated from each of the curves constituting the error at each order (cf. Perlekar & Pandit 2009;Sahoo et al 2011). These estimated ζ p values from both the methods are shown in Tables 3-5 …”

“…For simplicity, these fluctuations are assumed to only depend on (as represented by Equation (2b)). Here ζ p is a p-dependent scaling exponent and b is the magnetic field, where p represents the "order" and δb p the longitudinal fluctuations in Sahoo et al 2011) and extended self-similarity (ESS) analysis (Benzi et al 1993;Biskamp & Müller 2000;Biskamp 2003) will be explored in order to determine the value of ζ p . To model ζ p Equation (3) is used, where the parameters x, g, related to the basic spatial scaling of the turbulent fields, ∼ 1/g , and C 0 , the codimension of the most singular dissipative structures in the flow, are determined on physical and phenomenological grounds.…”

Multiscaling in Hall-Magnetohydrodynamic Turbulence: Insights from a Shell Model

“…We carry out extensive numerical studies of this shell model, obtain the scaling exponents for its structure functions, in both the low-k and high-k power-law ranges of three-dimensional Hall-magnetohydrodynamic, and find that the extended-self-similarity procedure is helpful in extracting the multiscaling nature of structure functions in the high-k regime, which otherwise appears to display simple scaling. Our results shed light on intriguing solar-wind measurements.Turbulent plasmas abound in accretion disks, galaxies, stars, the solar wind, and laboratory experiments [1,2]; thus, the characterization of the statistical properties [1,3,4] of turbulence in such plasmas is a problem of central importance in astrophysics, plasma physics, fluid dynamics, and nonequilibrium statistical mechanics. Such a characterization begins with the energy spectra: e.g., in homogeneous and isotropic fluid turbulence, the energy spectrum EðkÞ assumes the scaling form EðkÞ $ k À if the Reynolds numbers Re is large and k is in the inertial range; the phenomenological theory of Kolmogorov (K41) yields [5,6] ¼ 5=3.…”

“…For k d > k > k I , E b ðkÞ / k À b;2 ; the value of b;2 depends on whether we consider the electron-MHD [13] limit or the ion-MHD (IMHD) limit. In the electron-MHD limit, where the induction term is subdominant to the Hall term, we obtain a single, characteristic scale and K41 phenomenology yields b;2 ¼ 7=3; in the IMHD case, a comparison of transfer times obtained from the Hall and the induction terms, and dimensional analysis yields b;2 ¼ 11=3 [10,12].Direct numerical simulations (DNSs) [9-11] have begun to resolve these two scaling ranges, but their spatial resolution is much more limited than has been achieved in direct numerical simulations of MHD turbulence [3,4]. Thus, they have not been used to study the multiscaling properties of order-p fluid and magnetic structure functions.…”