The properties of the boundary plasma in a tokamak are now recognized to play a key role in determining the achievable fusion power and the lifetimes of plasma-facing components. Accurate quantitative modeling and improved qualitative understanding of the boundary plasma ultimately require five-dimensional gyrokinetic turbulence simulations, which have been successful in predicting turbulence and transport in the core. The additional challenges of boundary-plasma simulation necessitate the development of new gyrokinetic codes or major modifications to existing core gyrokinetic codes. In this thesis, we develop the first gyrokinetic continuum code capable of simulating plasma turbulence on open magnetic field lines, which is a key feature of a tokamak scrape-off layer. In contrast to prior attempts at this problem, we use an energy-conserving discontinuous Galerkin discretization in space. To model the interaction between the plasma and the wall, we design conducting-sheath boundary conditions that permit local currents into and out of the wall. We start by designing spatially one-dimensional kinetic models of parallel SOL dynamics and solve these systems using novel continuum algorithms. By generalizing these algorithms to higher dimensions and adding a model for collisions, we present results from the first gyrokinetic continuum simulations of turbulence on two types of open-field-line systems. The first simulation features uniform and straight field lines, such as found in some linear plasma devices. The second simulation is of a hypothetical model we developed of the NSTX scrape-off layer featuring helical field lines. These developments comprise a major step towards a gyrokinetic continuum code for quantitative predictions of turbulence and transport in the boundary plasma of magnetic fusion devices.
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