Magnetic field evolution in simulations with Euler potentials

TL;DR

This paper demonstrates that using Euler potentials with artificial diffusion in hydromagnetic simulations leads to incorrect magnetic field evolution, especially in turbulent and three-dimensional flows, due to fundamental incompatibilities.

Contribution

It reveals the limitations of Euler potentials in simulating magnetic fields with diffusion, contrasting them with standard vector potential methods.

Findings

Euler potentials produce incorrect decay in 3D simulations.

Artificial diffusion causes non-convergence of Euler potentials.

Standard methods permit dynamo action where Euler potentials do not.

Abstract

Using two- and three-dimensional hydromagnetic simulations for a range of different flows, including laminar and turbulent ones, it is shown that solutions expressing the field in terms of Euler potentials (EP) are in general incorrect if the EP are evolved with an artificial diffusion term. In three dimensions, standard methods using the magnetic vector potential are found to permit dynamo action when the EP give decaying solutions. With an imposed field, the EP method yields excessive power at small scales. This effect is more exaggerated in the dynamic case, suggesting an unrealistically reduced feedback from the Lorentz force. The EP approach agrees with standard methods only at early times when magnetic diffusivity did not have time to act. It is demonstrated that the usage of EP with even a small artificial magnetic diffusivity does not converge to a proper solution of hydromagnetic turbulence. The source of this disagreement is not connected with magnetic helicity or the three-dimensionality of the magnetic field, but is simply due to the fact that the nonlinear representation of the magnetic field in terms of EP that depend on the same coordinates is incompatible with the linear diffusion operator in the induction equation.