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Continuum kinetic simulations are increasingly capable of resolving high-dimensional phase space with advances in computing. These capabilities can be more fully explored by using linear kinetic theory to initialize the self-consistent field and phase space perturbations of kinetic instabilities. The phase space perturbation of a kinetic eigenfunction in unmagnetized plasma has a simple analytic form, and in magnetized plasma may be well approximated by truncation of a cyclotron-harmonic expansion. We catalogue the most common use cases with a historical discussion of kinetic eigenfunctions and by conducting nonlinear Vlasov–Poisson and Vlasov–Maxwell simulations of singlemode and multimode two-stream, loss-cone and Weibel instabilities in unmagnetized and magnetized plasmas with one- and two-dimensional geometries. Applications to quasilinear kinetic theory are discussed and applied to the bump-on-tail instability. In order to compute eigenvalues we present novel representations of the dielectric function for ring distributions in magnetized plasmas with power series, hypergeometric and trigonometric integral forms. Eigenfunction phase space fluctuations are visualized for prototypical cases such as the Bernstein modes to build intuition. In addition, phase portraits are presented for the magnetic well associated with nonlinear saturation of the Weibel instability, distinguishing current-density-generating trapping structures from charge-density-generating ones. 

Plasma photonic crystals (PPCs) have the potential to significantly expand the capabilities of current millimeter wave technologies by providing high speed (microsecond time scale) control of energy transmission characteristics in the GHz through low THz range. Furthermore, plasma-based devices can be used in higher power applications than their solid-state counterparts without experiencing significant changes in function or incurring damage. Plasmas with periodic variations in density can be created externally, or result naturally from instabilities or self-organization. Due to plasma's diffuse nature, PPCs cannot support rapid changes in density. Despite this fact, most theoretical work in PPCs is based on solid-state photonic crystal methods and assumes constant material properties with abrupt changes at material interfaces. In this work, a linear model is derived for a one-dimensional cold-plasma photonic crystal with an arbitrary density profile. The model is validated against a discontinuous Galerkin method numerical solution of the same device configuration. Bandgap maps are then created from derived group velocity data to elucidate the operating regime of a theoretical PPC device. The bandgap maps are compared for one-dimensional PPCs with both smooth and discontinuous density profiles. This study finds that bandgap behavior is strongly correlated with the density profile Fourier content and that density profile shapes can be engineered to produce specific transmission characteristics. 

The accuracy of quasilinear theory applied to the electron bump-on-tail instability, a classic model problem, is explored with conservative high-order discontinuous Galerkin methods applied to both the quasilinear equations and to a direct simulation of the Vlasov–Poisson equations. The initial condition is chosen in the regime of beam parameters for which quasilinear theory should be applicable. Quasilinear diffusion is initially in good agreement with the direct simulation but later underestimates the turbulent momentum flux. The greater turbulent flux of the direct simulation leads to a correction from quasilinear evolution by quenching the instability in a finite time. Flux enhancement above quasilinear levels occurs as the phase space eddy turnover time in the largest amplitude wavepackets becomes comparable to the transit time of resonant phase fluid through wavepacket potentials. In this regime, eddies effectively turn over during wavepacket transit so that phase fluid predominantly disperses by eddy phase mixing rather than by randomly phased waves. The enhanced turbulent flux of resonant phase fluid leads, in turn, through energy conservation to an increase in non-resonant turbulent flux and, thus, to an enhanced heating of the main thermal body above quasilinear predictions. These findings shed light on the kinetic turbulence fluctuation spectrum and support the theory that collisionless momentum diffusion beyond the quasilinear approximation can be understood through the dynamics of phase space eddies (or clumps and granulations).