Magnetic confinement fusion is a technique in which a strong magnetic field is used tocontain a hot plasma, which enables nuclear fusion. In terms of overall energy efficiency, the two most promising magnetic confinement concepts are tokamaks (axisymmetric devices) and stel- larators (nonaxisymmetric devices). The power P produced by a magnetically confined nuclear fusion device is proportional to Vβ2B4, where V is the volume of the device, β is the plasma pressure - magnetic pressure ratio, and B is the magnetic field strength. Most tokamaks and stellarators currently in operation are low-β devices. In general, there are three ways to increase P , one may increase the operating β, the magnetic field or the volume of the device. The cost of these devices is proportional to V , making large enough devices expensive. Similarly, a large magnetic field (>10T) requires superconducting magnets that, even after the recent innovations in HTS (High-Temperature Superconductors), are expensive to manufacture. High-β devices are an attractive idea to efficiently produce fusion energy. However, a high-β generally also implies a large gradient in plasma pressure that can be a source of numerous instabilities. If fusion devices could be optimized against such instabilities, high-β operation would become an attractive approach compared to high field or large-volume reactors. Therefore, this thesis explores the optimization of high-β tokamak and stellarator equilibrium equilibria against linear instabilities. We will start by investigating the stability of high-β tokamaks and stellarator equilibria against the infinite-n ideal ballooning mode, an important pressure-driven MHD instability. We stabilize these equilibria against the ideal ballooning mode. To achieve this, we formulate a gradient-based adjoint technique and demonstrate its speed and effectiveness by stabilizing these equilibria. We also explain how this technique can be easily extended to low-n ideal-MHD modes in both tokamaks and stellarators.

After demonstrating the adjoint technique for stabilizing against ideal MHD modes, wefirst analyze the kinetic stability of a sequence of axisymmetric equilibria. We study this by nu- merically solving the δf gyrokinetic model, a simplified version of the Vlasov-Maxwell model. Since these kinetic instabilities are driven by temperature and density gradients, we explore them by scanning multiple values of the plasma β, temperature and density gradients, and plasma boundary shapes, discovering interesting relationships between equilibrium-dependent quantities and growth rates of these instabilities. We then repeat the same process for two recently pub- lished stellarator equilibria with quasisymmetry — a favorable hidden symmetry in stellarators. With this study, we verify that our observations from high-β tokamaks can be generalized to quasisymmetric stellarators.

From our microstability study, we find that electromagnetic effects are important for high-βdevices. Hence, using the numerical tools and knowledge derived from the previous chapters we build an optimization framework that searches for stable equilibria. Due to the similarity between axisymmetry and quasisymmetry, we then use the microstability optimizer to search for ideally and kinetically-stable, quasisymmetric, high-β stellarators.