Numerical tests of cosmological Alfvén waves with Ohmic diffusion

Monthly Notices of the Royal Astronomical Society, vol. 541, pp. 1507–1519

Abstract

ABSTRACT Physical problems with a solution that can be expressed analytically are scarce; this holds even more true for problems set in a cosmological context. Such solutions are, however, invaluable tools for making comparisons between theory, numerical experimentation, and observations. In this work, we present what to our knowledge is the first set of non-trivial closed-form expressions describing the behaviour of a system governed by the equations of non-ideal magnetohydrodynamics (MHD), where the effects of Ohmic diffusion are considered, in a cosmologically expanding frame. We provide analytical solutions that describe the time evolution of linear perturbations to a homogeneous background in a radiation-dominated universe, yielding dissipative Alfvén waves. Although in our base framework solutions for any other cosmology of interest cannot be expressed in a closed form, they can still be obtained reliably through numerical integration of the coupled system of ordinary differential equations we provide. We compare our analytical solutions to numerical results obtained using our novel implementation of smoothed particle magnetohydrodynamics in the swift astrophysical simulation code, to find good agreement between the two. We find the code to display good convergence behaviour, its predictions agreeing with theory to within 0.1 per cent for a modest number of resolution elements and at a negligible computational cost. We aim this work as a companion and supplement to the cosmological ideal MHD wave tests recently presented in the literature, and suggest that it be adopted as part of standard testing of code implementations of MHD.

Authors 2

1. Orestis A. Karapiperis corresponding Aachen Leiden Observatory Lorentz Institute for Theoretical Physics

   Leiden University · Leiden Observatory

   Affiliation as printed

   Leiden Observatory, Leiden University , PO Box 9513, NL-2300 RA Leiden ,

   Lorentz Institute for Theoretical Physics, Leiden University , PO Box 9506, NL-2300 RA Leiden ,

   Lorentz Institute for Theoretical Physics, Leiden University, PO Box 9506, NL-2300 RA Leiden, the Netherlands

2. Matthieu Schaller Aachen Leiden Observatory Lorentz Institute for Theoretical Physics

   Leiden University · Leiden Observatory

   Affiliation as printed

   Leiden Observatory, Leiden University , PO Box 9513, NL-2300 RA Leiden ,

   Lorentz Institute for Theoretical Physics, Leiden University , PO Box 9506, NL-2300 RA Leiden ,

   Leiden Observatory, Leiden University, PO Box 9513, NL-2300 RA Leiden, the Netherlands