Abstract
We present the results of our detailed pseudospectral direct numerical simulation (DNS) studies, with up to 10243 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 PrM over a large range, namely 0.01⩽PrM⩽10. We obtain data for a wide variety of statistical measures, such as probability distribution functions (PDFs) of the 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 characterize intermittency, isosurfaces of quantities, such as the moduli of the vorticity and current density, 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 a larger intermittency in the magnetic field than in the velocity field, at large PrM, to a smaller intermittency in the magnetic field than in the velocity field, at low PrM. 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 PrM, multiscaling exponent ratios agree, at least within our error bars, for both decaying and statistically steady homogeneous, isotropic MHD turbulence.
Highlights
The hydrodynamics of conducting fluids is of great importance in many terrestrial and astrophysical phenomena
We present the results of our detailed pseudospectral direct numerical simulation (DNS) studies, with up to 10243 collocation points, of incompressible, magnetohydrodynamic (MHD) turbulence in three dimensions, without a mean magnetic field
We give representative probability distribution functions (PDFs) of the pressure p, modulus of vorticity ω = |ω|, and the local energy dissipation in Figs. 3.1(a), 3.1(b), and 3.1(c), respectively; note that the PDF of the pressure is negatively skewed. (iv) Inertialrange structure functions Spu(l) ∼ lζpu show significant deviations [24] from the K41 result ζpuK41 = p/3 especially for p > 3. From these structure functions we can obtain the hyperflatness F6u(l); this increases as the length scale l decreases, a clear signature of intermittency, as shown, e.g., in Refs. [48, 65]. This intermittency leads to non-Gaussian tails, especially for small l, in PDFs of velocity increments such as δu (l). (v) Small-scale structures in turbulent flows can be visualised via isosurfaces [71] of, say, ω, and p, illustrative plots of which are given in Figs. 3.1(a)-3.1(c); these show that regions of large ω are organised into slender tubes whereas isosurfaces of look like shredded sheets; pressure isosurfaces show tubes [36, 46] but some studies have suggested the term cloud-like for them [60]. (vi) Joint PDFs provide useful information about turbulent flows; in particular, contour plots of the joint PDF of Q and R, as in the representative Fig. 5, show a characteristic tear-drop structure
Summary
The hydrodynamics of conducting fluids is of great importance in many terrestrial and astrophysical phenomena. Examples include the generation of magnetic fields via dynamo action in the interstellar medium, stars, and planets [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], and liquid-metal systems [12, 13, 14, 15, 16, 17, 18] that are studied in laboratories The flows in such settings, which can be described at the simplest level by the equations of magnetohydrodynamics (MHD), are often turbulent [5]. Two dissipative scales play an important role in MHD; they are the Kolmogorov scale d
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