Depletion of Nonlinearity in Magnetohydrodynamic Turbulence: Insights from Analysis and Simulations

• Published in: arXiv.org 2016-05
• Main Authors: Gibbon, J D, Gupta, A, Krstulovic, G, Pandit, R, Politano, H, Ponty, Y, Pouquet, A, Sahoo, G, Stawarz, J
• Format: Article
• Language: English
• Subjects: Computational fluid dynamics; Computer simulation; Depletion; Energy spectra; Fluid flow; Magnetohydrodynamic turbulence; Mathematical models; Nonlinearity; Prandtl number; Vorticity
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.1508.03756

We build on recent developments in the study of fluid turbulence [Gibbon \textit{et al.} Nonlinearity 27, 2605 (2014)] to define suitably scaled, order-\(m\) moments, \(D_m^{\pm}\), of \(\omega^\pm= \omega \pm j\), where \(\omega\) and \(j\) are, respectively, the vorticity and current density in three-dimensional magnetohydrodynamics (MHD). We show by mathematical analysis, for unit magnetic Prandtl number \(P_M\), how these moments can be used to identify three possible regimes for solutions of the MHD equations; these regimes are specified by inequalities for \(D_m^{\pm}\) and \(D_1^{\pm}\). We then compare our mathematical results with those from our direct numerical simulations (DNSs) and thus demonstrate that 3D MHD turbulence is like its fluid-turbulence counterpart insofar as all solutions, which we have investigated, remain in \textit{only one of these regimes}; this regime has depleted nonlinearity. We examine the implications of our results for the exponents \(q^{\pm}\) that characterize the power-law dependences of the energy spectra \(\mathcal{E}^{\pm}(k)\) on the wave number \(k\), in the inertial range of scales. We also comment on (a) the generalization of our results to the case \(P_M \neq 1\) and (b) the relation between \(D_m^{\pm}\) and the order-\(m\) moments of gradients of hydrodynamic fields, which are used in characterizing intermittency in turbulent flows.