Linear theory of plasma waves and instabilities: from ion-electron to MHD descriptions

Abstract. As the most common, fundamental state of matter in the universe, plasmas are observed in a variety of phenomena, ranging from small-sized fusion experiments to gas clouds on stellar and interstellar scales. As a consequence, an assortment of models exists to describe plasma behaviour on different scales. Within each model the natural oscillations and instabilities supported by a particular plasma configuration can be studied to interpret the plasma’s behaviour and evolution. In this thesis, we employ three models to investigate the behaviour of both mechanical and electromagnetic waves: the two-fluid, ion-electron model, Hall-magnetohydrodynamics (HMHD), and regular magnetohydrodynamics (MHD).

Firstly, we consider waves in a homogeneous, ideal ion-electron plasma at rest to address the literature’s inconsistent wave labelling schemes that rely on wave behaviour at parallel and perpendicular propagation to the magnetic field. Using a polynomial form of the dispersion relation, which describes six wave types, it is shown that at oblique propagation angles the well-known MHD frequency ordering extends to the ion-electron model, including its additional three wave types compared to MHD, which only has three wave types. In this respect, parallel and perpendicular propagation are unique because the frequency ordering is violated there, thus making these angles unfit for classifying wave types at all angles. This violation of the frequency ordering is characterised by waves crossing in the frequency-wave number diagram. These crossing are then replaced by avoided crossings at oblique angles. In addition, which wave types cross or avoid crossing depends on the specific parameters of the plasma environment. As it turns out, the parameter space contains six distinct regimes of crossing behaviour. Finally, the highly anisotropic behaviour of all wave types is highlighted by their phase and group speed diagrams.

Subsequently, this ion-electron model is applied to a couple of relevant use cases. Whistler waves, which travel in Earth’s magnetosphere and are characterised by rapid variations in group speed for small changes in frequency, were recently witnessed to travel at oblique angles to the magnetic field, unlike previous surveys indicating their propagation along the magnetic field lines. With the use of the polynomial ion-electron dispersion relation, known whistler approximations at parallel propagation are meaningfully extended to oblique angles, though any damping effects are inherently absent in this description. Furthermore, considering the avoided crossings at oblique angles, their influence on the whistling behaviour is charted, exposing the split of whistling behaviour across two different wave types in select cases. In a second application, the widely-used, magneto-ionic Appleton-Hartree equation is extended to include the effect of a non-zero, thermal electron velocity. A final application shows that in a warm plasma the emission of laser-induced Cherenkov radiation is restricted to a cone centered around the laser beam.

In the second half of the thesis, the methodology changes to a spectroscopic approach, computing all natural oscillations and instabilities of a specific configuration numerically with the Legolas code, in the HMHD and MHD models. Here, we present how the HMHD spectrum of a homogeneous slab captures the analytic wave solutions, and how the inclusion of electron inertia modifies the short wavelength MHD behaviour to be in line with the ion-electron model.

With the shift to numerical spectroscopy, Legolas’s low numerical cost is exploited to examine the resistive tearing instability parametrically. This instability is a form of spontaneous magnetic reconnection, which is a necessary process for many eruptive and disruptive events in the solar corona and Earth’s magnetosphere. One of the outstanding questions relating to magnetic reconnection, however, is the fact that observed reconnection rates do not match theoretical predictions. In this regard, it is recognised that the tearing growth rate is affected by Hall physics, ambient flow, and viscosity. The significance of all three factors is scrutinised in compressible conditions, contrary to many reviews limiting themselves to the incompressible case.

In this analysis of the tearing growth rate, the popular Harris current sheet, which features a magnetic field reversal across the sheet, is adopted as the equilibrium configuration. Unlike the incompressible case, the introduction of a “guide field”, i.e. a constant magnetic field component perpendicular to the reversing component, causes the Hall current to suppress the tearing growth rate in the compressible case. This is not the only difference between the compressible and incompressible configurations, though, since the incompressible, analytic scaling laws as a function of the resistivity also require adjustment to account for compressibility. This effect is especially pronounced in the presence of ambient flow, where the flow completely eliminates the instability in certain regions of the parameter space. In this regard, the effect of viscosity is similar to that of background flow.

Finally, we offer perspectives for future research. There, we outline the extension of the ion-electron study to multifluid models, in which a numerical approach similar to the \Legolas{} code may prove useful to quantify waves and instabilities. Ultimately, the linear analysis presented here should be compared to non-linear simulations, to identify when and where non-linear effects take over. At the same time, the linear results may aid in the interpretation of non-linear simulations.

Errata.

• Chapter 3: All calculated frequencies (\(\omega\)) are angular frequencies. The listed values are thus in rad/s and not in Hz. To obtain values in Hz, divide by \(2\pi\).
• Section 5.1.2, Harris current sheet, Linear stage of tearing reconnection: Due to a bug in the script to generate Fig. 5.5(c,d), the wrong mode was selected and this figure does not show the tearing mode. Consequently, the discussion in this section is incorrect. For an updated discussion, please see De Jonghe & Keppens (2024).

Recommended citation: De Jonghe, J. (2023). Linear theory of plasma waves and instabilities: from ion-electron to MHD descriptions. PhD thesis. http://jordidj.github.io/files/thesis.pdf