Discontinuous galerkin modeling of wave propagation, scattering, and nonlinear growth in inhomogeneous plasmas [electronic resource]
Keywords: Electrical Engineering

Modeling very low frequency (VLF, 3-30 kHz) wave propagation in the near-Earth space environment remains a significant computational challenge. Because of the strong inhomogeneity of the lower ionosphere, VLF waves propagating in these regions have an extremely wide range of wavelengths, varying from nearly zero to nearly infinite for some angles of propagation. In addition, the characteristic frequencies of a plasma may be an order of magnitude higher than the VLF frequencies of interest, presenting another significant problem for time-domain numerical solution methods because explicit time-stepping techniques are limited by the largest characteristic frequency present in the system. Further, the plasma environment of the Earth's magnetosphere is frequently unstable to VLF waves, leading to a wide variety of natural emissions and wave amplification phenomena, many of which are still poorly understood due to the relative paucity of high-resolution data and the difficulty in developing a complete theory of nonlinear wave growth in unstable plasmas. Early approaches to modeling wave propagation in this environment relied heavily on analytical approximations under some set of physical assumptions, e.g., by linearization or by assuming the solution is smooth or slowly varying. In recent years, supported by rapid increases in computer processing speed and memory capacity, so-called continuum time-domain methods such as the finite difference time-domain (FDTD) method have gained in popularity for solving such problems. FDTD works by discretizing a space into a finite number of points and then evolving the solution on these points forward in time, step by step. Although useful and simple, the FDTD method is not an ideal solution method for modeling wave propagation in the inhomogeneous ionosphere. FDTD is only accurate when both the wavelengths present in the system are adequately sampled, and when the FDTD grid is highly regular (plaid) and slowly varying. The consequence of these restrictions is that when using FDTD, ionospheric propagation problems of any reasonable size are either severely underresolved in the short-wavelength regions of the space (meaning the solution is inaccurate), or overresolved in the long-wavelength regions of the space (meaning the memory requirements are too large). Another continuum, time-domain solution technique, called the discontinuous Galerkin (DG) method, does not suffer from these restrictions. Due to its different formulation, the DG method is easily adapted for use on unstructured grids, allowing for increased spatial resolution only where it is required. The DG method is also highly accurate, yielding high-quality solutions with smaller memory and computational requirements than possible with low-order FDTD methods on structured grids. This dissertation describes our work our approach to adapting the DG method for modeling wave propagation in plasmas, specifically addressing the challenges that arise at VLF in the near-Earth plasma environment. We first present a procedure to incorporate any anisotropic linear dispersive material in the DG framework, with application to the perfectly matched layer (PML) and magnetized plasmas. As a semi-discrete formulation, we can exploit fully modern implicit-explicit Runge-Kutta time-stepping methods in order to circumvent the timestep restriction imposed by plasma frequency and gyrofrequency-induced stiffness, resulting in speedups of 5-10 times for typical mid-latitude ionospheres without the accuracy penalty typical of methods relying on low-order Strang splitting. We next describe the hybrid DG-PIC (discontinuous Galerkin particle-in-cell) method for studying nonlinear wave growth and damping in magnetized plasmas, with specific attention paid to the scheme's efficient parallelization in distributed computing environments. Our DG-PIC method is a hybrid technique, which works by splitting the plasma into a cold, dense background plasma (modeled as a fluid) and a hot, strongly nonlinear and highly interacting population of energetic particles (modeled with superparticles). This model is far more efficient than a direct PIC method and also closely matches the physics of the real magnetosphere, making it an ideal framework to investigate nonlinear wave phenomena there. We demonstrate our techniques on two difficult problems of interest at VLF in the near-Earth space environment. First, we simulate scattering of VLF signals from intense lightning-induced ionospheric disturbances. VLF signals propagating in the Earth-ionosphere waveguide can be strongly scattered via interaction with density or temperature perturbations in the lower ionosphere, producing measurable perturbations on these signals when observed from ground-based receivers. To date, all simulations of this scattering process have relied heavily on approximations (such as smoothness or weak scattering) that do not, in general, hold. Our simulations represent, to our knowledge, the first direct simulations of scattering from such disturbances over a large 3D volume. The results have revealed the full spatial structure of the scattered field from intense lightning-induced disturbances as observed by a ground-based receiver. Second, we use our hybrid DG-PIC technique to model the spontaneous generation of coherent rising emissions in an unstable, inhomogeneous magnetized plasma. The Earth's magnetosphere naturally and frequently produces a variety of spontaneous emissions, driven by instabilities in the relatively energetic population of radiation belt particles that are injected into the Earth's magnetosphere during active geomagnetic storms. One type of emission, termed chorus, is characterized by discrete, quasi-periodic, coherent rising tones. The nonlinear processes driving these emissions are, at present, still relatively poorly understood. Our simulations demonstrate that it is possible to consistently and spontaneously generate quasi-periodic, coherent rising emissions, provided that the hot electron distribution driving the growth process is sufficiently unstable. The emissions are generated in a region just upstream of the magnetic equator and are subsequently amplified as they travel through the equatorial region and away from the generation region. We additionally show that many features of these rising emissions are well-predicted by linearized growth rates alone, demonstrating that while the chorus generation process is indeed nonlinear, there is nonetheless a close relationship between chorus generation and the linear growth processes that produce the high wave amplitudes necessary to induce nonlinear effects in the medium.