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

TL;DR

This paper analyzes the nonlinear behavior of 3D magnetohydrodynamic turbulence using mathematical analysis and simulations, revealing a depletion of nonlinearity and identifying regimes of solution behavior.

Contribution

It introduces scaled moments to identify solution regimes in MHD turbulence and demonstrates that turbulence remains in a regime with depleted nonlinearity, supported by numerical simulations.

Findings

Turbulence solutions stay in the regime with depleted nonlinearity.

Mathematical analysis links moments to solution regimes.

Simulations confirm the theoretical regime classification.

Abstract

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.