It is shown how suitably scaled, order-$m$ moments, $D_m^{\pm }$, of the Elsässer vorticity fields in three-dimensional magnetohydrodynamics (MHD) can be used to identify three possible regimes for solutions of the MHD equations with magnetic Prandtl number $P_M=1$. These vorticity fields are defined by ${\mathbf{\omega }}^{\pm }=\text{curl}{z}^{\pm }=\mathbf{\omega }\pm \mathit{j}$, where $z^{\pm }$ are Elsässer variables, and where $\mathbf{\omega }$ and $\mathit{j}$ are, respectively, the fluid vorticity and current density. This study follows recent developments in the study of three-dimensional Navier-Stokes fluid turbulence [Gibbon et al., Nonlinearity 27, 2605 (2014)]. Our mathematical results are then compared with those from a variety of direct numerical simulations, which demonstrate that all solutions that have been investigated remain in only one of these regimes which has depleted nonlinearity. The exponents $q^{\pm }$ that characterize the inertial range power-law dependencies of the $z^{\pm }$ energy spectra, ${\mathcal{E}}^{\pm }\left(k\right)$, are then examined, and bounds are obtained. Comments are also made on  (a) the generalization of our results to the case $P_M\ne 1$ and (b) the relation between $D_m^{\pm }$ and the order-$m$ moments of gradients of magnetohydrodynamic fields, which are used to characterize intermittency in turbulent flows.

• Received 18 August 2015

DOI:https://doi.org/10.1103/PhysRevE.93.043104