Numerical tests of cosmological Alfv\'en waves with Ohmic diffusion

TL;DR

This paper derives analytical solutions for dissipative Alfvén waves in a cosmological setting with Ohmic diffusion, and validates these solutions through numerical simulations using a novel SPMHD implementation, enhancing MHD code testing standards.

Contribution

It provides the first non-trivial closed-form solutions for non-ideal MHD Alfvén waves in an expanding universe, and demonstrates their accuracy with a new numerical approach.

Findings

Analytical solutions accurately describe dissipative Alfvén waves in a cosmological context.

Numerical SPMHD simulations agree with analytical solutions within 0.1%.

The method offers a reliable and efficient way to test MHD codes.

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 Alfv\'en 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 (SPMHD) 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% 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.