Omnigenous umbilic stellarators
To better understand the dependence of the magnetic field structure in the plasma edge on the plasma boundary shape, in the context of X-point and island divertor designs, we define and develop a class of stellarators called umbilic stellarators. These equilibria are characterized by a single continuous high-curvature edge on the plasma boundary that goes around multiple times toroidally before meeting itself. We develop a technique that allows us to simultaneously optimize the plasma boundary along with a curve lying on the boundary on which we impose a high curvature while imposing omnigenity -- a property of the magnetic field that ensures trapped particle confinement throughout the plasma volume. We find that umbilic stellarators naturally tend to favor piecewise omnigenity instead of omnigenity with a specific helicity. After generating omnigenous umbilic stellarators, we design coil sets for some of them and explore the field line structure in the edge and its sensitivity to small fluctuations in the plasma. Finally, using single-stage optimization, we simultaneously modify the plasma and coil shape and propose an experiment to modify an existing tokamak to a finite-beta stellarator using this technique and explore a potentially simpler way to convert a limited tokamak into a diverted stellarator.