Direct Numerical Simulations of Low-$Rm$ MHD turbulence based on the least dissipative modes

TL;DR

This paper introduces a spectral method for simulating low-Rm MHD turbulence using eigenmodes of the dissipation operator, enabling efficient analysis of flow scales and transition to three-dimensionality.

Contribution

The novel approach employs least dissipative eigenmodes as basis functions, improving simulation of low-Rm MHD turbulence over traditional methods.

Findings

Scaling laws for smallest flow scales as a function of Reynolds and Grashof numbers.

Validation of least dissipative modes predicting the onset of three-dimensional structures.

Numerical demonstration of the method's effectiveness for 2D and 3D low-Rm MHD flows.

Abstract

We present a new spectral method for the Direct Numerical Simulation of Magnetohydrodynamic turbulence at low Magnetic Reynolds number. The originality of our approach is that instead of using traditional bases of functions, it relies on the basis of eigenmodes of the dissipation operator, which represents viscous and Joule dissipation. We apply this idea to the simple case of a periodic domain in the three directions of space, with an homogeneous magnetic field in the $\mathbf e_z$ direction. The basis is then still as subset of the Fourier space, but ordered by growing linear decay rate $|\lambda|$ (\emph{i.e} according to the \emph{least dissipative modes}). We show that because the lines of constant energy tend to follow those of constant $|\lambda|$ in the Fourier space, the scaling for the the smallest scales $|\lambda^{\rm max}|$ in a forced flow can be expressed using this single parameter, as a function of the Reynolds number as $\sqrt{|\lambda^{\rm max}|}/(2\pi k_f)\simeq 0.5\Rey^{1/2}$, where $k_f$ is the forcing wavelength, or as a function of the Grashof number $\Gr_f$, which gives a non-dimensional measure of the forcing, as $|\lambda^{\rm max}|^{1/2}/(2\pi k_f)\simeq 0.47\Gr_f^{0.20}$. This scaling is also found consistent with heuristic scalings, and which we are able to numerically quantify as $k_\perp^{\rm max}/k_f\simeq 0.5 \Rey^{1/2}$ and $k_z^{\rm max}/k_f\simeq 0.8k_f \Rey/Ha$. Finally, we show that the set of least dissipative modes gives a relevant prediction for the scale of the first three-dimensional structure to appear in a forced, initially two-dimensional turbulent flow. This completes our numerical demonstration that the least dissipative modes can be used to simulate both two- and three-dimensional low-Rm MHD flows.