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

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 $|{\lambda }^{\max }|$ in a forced flow can be expressed using this single parameter, as a function of the Reynolds number as $\sqrt{|λ^{\rm max}|}/(2πk_f)\simeq 0.5\Rey^{1/2}$, 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 $|λ^{\rm max}|^{1/2}/(2πk_f)\simeq 0.47\Gr_f^{0.20}$. 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.

arXiv comment24 pages. Article accepted for publication in the Journal of Fluid Mechanics

References in corpus (4)

• Large scale flow effects, energy transfer, and self-similarity on turbulence
• An effective two-dimensional model for MHD flows with transverse magnetic field
• Small scales and anisotropy in low $Rm$ magnetohydrodynamic turbulence
• Bounds on the attractor dimension for low-R m wall-bound magnetohydrodynamic turbulence

Cited by in corpus (8)

• Strong anisotropy in quasi-static magnetohydrodynamic turbulence for high interaction parameters
• Why, how and when MHD turbulence at low $Rm$ becomes three-dimensional
• Anisotropic energy transfers in quasi-static magnetohydrodynamic turbulence
• Modeling quasi-static magnetohydrodynamic turbulence with variable energy flux
• Direct numerical simulation of quasi-two-dimensional MHD turbulent shear flows
• Stability of pulsatile quasi-two-dimensional duct flows under a transverse magnetic field
• The decay of wall-bounded MHD turbulence at low Rm
• Bounds on the attractor dimension for magnetohydrodynamic channel flow with parallel magnetic field at low magnetic Reynolds number