Topological and geometrical properties of plasma waves

Abstract

Geometry and topology are two different but closely related concepts in mathematical physics. Geometry is concerned with local differential properties, such as curvatures, while topology is often used to study global properties, such as the number of holes in an object. This thesis demonstrates how geometry and topology manifest themselves in the study of linear waves in plasma. The fundamental objects are the waves' polarization vector and associated Berry curvatures. The integral of Berry curvatures over a closed parameter space yields a topological index called the Chern number. Topological surface plasma waves can be found when two sides of the surface have different Chern numbers, which can be rigorously proved from an index theorem connecting the Chern numbers and the number of spectral flows. The topological surface waves in cold plasma and Hall magnetohydrodynamics are investigated. As topological waves, they possess the so-called topological robustness, i.e., they are unidirectional and free of scattering and reflection. The dispersion and propagation of the topological surface waves are analyzed and benchmarked numerically. When a quasi-monochromatic electromagnetic wave packet propagates in non-uniform plasmas, the wave's polarization changes over time and, therefore, picks up a geometric phase. As the gradient of the wave phase, the local wave vector is modified accordingly, and as a result, so is the group velocity. This causes additional bending of the ray trajectory. An electron-cyclotron wave beam in a typical toroidal fusion plasma can deviate by as much as ten wavelengths (0.1 m) in the poloidal direction relative to the lowest-order ray trajectory. This is a geometrical effect called the spin Hall effect of waves.