Linear and non-linear eccentric mode evolution in unstratified MHD discs

In this paper, we develop a framework for studying unstratified, magnetized eccentric discs and compute uniformly precessing eccentric modes in a cylindrical annulus which provide convenient initial conditions for numerical simulations. The presence of a magnetic field in an eccentric disc can be described by an effective gas with a modified equation of state. At magnetic field strengths relevant to the magneto-rotational instability the magnetic field has negligible influence on the evolution of the eccentric disc, however, the eccentric disc can significantly enhance the magnetic field strength over that in the a circular disc. We verify the suitability of these eccentric disc solutions by carrying out 2D simulations in RAMSES. Our simulated modes (in 2D) follow a similar evolution to the purely hydrodynamical modes, matching theoretical expectations, provided they are adequately resolved. Such solutions will provide equilibrium states for studies of the eccentric magneto-rotational instability and magnetized parametric instability in unstratified discs and are useful for exploring the response of disc turbulence on top of a fluid flow varying on the orbital time-scale.