Following recent work, we discuss waves in a warm ideal two-fluid plasma consisting of electrons and ions starting from a completely general, ideal two-fluid dispersion relation. The plasma is characterised by five variables: the electron and ion magnetisations, the squared electron and ion sound speeds, and a parameter describing the angle between the propagation vector and the magnetic field. The dispersion relation describes 6 pairs of waves which we label S, A, F, M, O, and X. Varying the angle, it is argued that parallel and perpendicular propagation (with respect to the magnetic field) exhibit unique behaviour. This behaviour is characterised by the crossing of wave modes which is prohibited at oblique angles. We identify up to 6 different parameter regimes where a varying number of exact mode crossings in the special parallel or perpendicular orientations can occur. We point out how any ion-electron plasma has a critical magnetisation (or electron cyclotron frequency) at which the cutoff ordering changes, leading to different crossing behaviour. These are relevant for exotic plasma conditions found in pulsar and magnetar environments. Our discussion is fully consistent with ideal relativistic MHD and contains light waves. Additionally, exploiting the general nature of the dispersion relation, phase and group speed diagrams can be computed at arbitrary wavelengths for any parameter regime. Finally, we recover earlier approximate dispersion relations that focus on low-frequency limits and make direct correspondences with some selected kinetic theory results.

Magnetohydrodynamic (MHD) spectroscopy is central to many astrophysical disciplines, ranging from helio-to asteroseismology, over solar coronal (loop) seismology, to the study of waves and instabilities in jets, accretion disks, or solar/stellar atmospheres. MHD spectroscopy quantifies all linear (standing or travelling) wave modes, including overstable (i.e. growing) or damped modes, for a given configuration that achieves force and thermodynamic balance. Here, we present Legolas a) , a novel, open-source numerical code to calculate the full MHD spectrum of one-dimensional equilibria with flow, that balance pressure gradients, Lorentz forces, centrifugal effects and gravity, enriched with non-adiabatic aspects like radiative losses, thermal conduction and resistivity. The governing equations use Fourier representations in the ignorable coordinates, and the set of linearised equations are discretised using Finite Elements in the important height or radial variation, handling Cartesian and cylindrical geometries using the same implementation. A weak Galerkin formulation results in a generalised (non-Hermitian) matrix eigenvalue problem, and linear algebraic algorithms calculate all eigenvalues and corresponding eigenvectors. We showcase a plethora of well-established results, ranging from p-and g-modes in magnetised, stratified atmospheres, over modes relevant for coronal loop seismology, thermal instabilities and discrete overstable Alfvén modes related to solar prominences, to stability studies for astrophysical jet flows. We encounter (quasi-)Parker, (quasi-)interchange, currentdriven and Kelvin-Helmholtz instabilities, as well as non-ideal quasi-modes, resistive tearing modes, up to magneto-thermal instabilities. The use of high resolution sheds new light on previously calculated spectra, revealing interesting spectral regions that have yet to be investigated.

Many linear stability aspects in plasmas are heavily influenced by non-ideal effects beyond the basic ideal magnetohydrodynamics (MHD) description. Here, the extension of the modern open-source MHD spectroscopy code Legolas with viscosity and the Hall current is highlighted and benchmarked on a stringent set of historic and recent findings. The viscosity extension is demonstrated in a cylindrical set-up featuring Taylor–Couette flow and in a viscoresistive plasma slab with a tearing mode. For the Hall extension, we show how the full eigenmode spectrum relates to the analytic dispersion relation in an infinite homogeneous medium. We quantify the Hall term influence on the resistive tearing mode in a Harris current sheet, including the effect of compressibility, which is absent in earlier studies. Furthermore, we illustrate how Legolas mimics the incompressible limit easily to compare with literature results. Going beyond published findings, we emphasise the importance of computing the full eigenmode spectrum, and how elements of the spectrum are modified by compressibility. These extensions allow for future stability studies with Legolas that are relevant to ongoing dynamo experiments, protoplanetary disks or magnetic reconnection.

The detection of whistler waves was first recorded in 1918 by radio operators who observed audio signals with a rapidly changing pitch (Barkhausen, 1919). Presently, they are covered in various plasma physics

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