Spectral Direct Numerical Simulations of low Rm MHD channel flows based on the least dissipative modes

Abstract

We put forward a new type of spectral method for the direct numerical simulation of flows where anisotropy or very fine boundary layers are present. The mean idea is to take advantage of the fact that such structures are dissipative and that their presence should reduce the number of degrees of freedom of the flow, when paradoxically, their fine resolution incurs extra computational cost in most current methods. The principle of this method is to use a functional basis with elements that already include these fine structure so as to avoid these extra costs. This leads us to develop an algorithm to implement a spectral method for arbitrary functional bases, and in particular, non-orthogonal ones. We construct a basic implementation of this algorithm to simulate Magnetohydrodynamic (MHD) channel flows with an externally imposed, transverse magnetic field, where very thin boundary layers are known to develop along the channel walls. In this case, the sought functional basis can be built out of the eigenfunctions of the dissipation operator, which incorporate these boundary layers, and it turns out to be non-orthogonal. We validate this new scheme against numerical simulations of freely decaying MHD turbulence based on a Finite Volume code and it is found to provide accurate results. Its ability to fully resolve wall-bounded turbulence with a number of modes close to that required by the dynamics is demonstrated on a simple example. This opens the way to full blown simulations of MHD turbulence under very high magnetic fields, which were until now too computationally expensive, as the computational cost of the proposed method, in contrast to traditional methods, does not depend on the intensity of the magnetic field.