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 a homogeneous magnetic field in the ez direction. The basis is then still a subset of the Fourier space, but ordered by growing linear decay rate |λ| (i.e. according to the least dissipative modes). We show that because the lines of constant energy tend to follow those of constant |λ| in the Fourier space, the scaling for the smallest scales |λmax| in a forced flow can be expressed, using this single parameter, as a function of the Reynolds number as $\(\sqrt{1\lambda^{max}|}/(2\upi k_f)\simeq 0.5\Rey^{1/2}\)$, where kf is the forcing wavelength, or as a function of the Grashof number Gf, which gives a non-dimensional measure of the forcing, as |λmax|1/2/(2πkf) ≃ 0.47Gf0.20. This scaling is also found to be consistent with heuristic scalings derived by Alemany et al. (J. Mec., vol. 18, 1979, pp. 277–313) and Pothérat & Alboussière (Phys. Fluids, vol. 15, 2003, pp. 3170–3180) for interaction parameter S ≳ 1, and which we are able to numerically quantify as k⊥max/kf ≃ 0.5Re1/2 and kzmax/kf ≃ 0.8kfRe/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 magnetohydrodynamic (MHD) flows.