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The first part of the dissertation investigates the effects of multiple-ions on the propagation of shear Alfven waves. It is shown that the presence of a second ion-species allows for the formation of an ion-ion hybrid resonator in the presence of a magnetic well. A full- wave description is shown to explain the measured eigenfrequencies and spatial form of the resonator modes identified in experiments in the Large Plasma Device (LAPD) at UCLA. However, it is determined that neither electron collisions or radial convection of the mode due to coupling to either the compressional or ion-Bernstein wave can explain the observed dissipation.Ray tracing studies for shear Alfven waves are performed in various magnetic geometries of contemporary interest. In a tokamak, it is found that the hybrid resonator can exist in the cold-plasma regime, but that ion-temperature effects combined with curvature effects cause the wave reflection point to shift towards the cyclotron frequency of the heavier ion. A one-dimensional WKB model is applied to a tokamak geometry for conditions corresponding to a burning fusion plasma to characterize the resonator. Instability due to fusion-born alpha particles is assessed. An approximate form of the global eigenmode is considered. It is identified that magnetic field shear combined with large ion temperature can cause coupling to an ion-Bernstein wave, which can limit the instability. Finally, the radiation pattern of shear Alfven waves generated by a burst of charged particles in the presence of two-ion species is considered. The spectral content and spatial patterns of the radiated waves are determined.The second part of the dissertation considers the MHD stability of the plasma near a divertor in a tokamak. Two types of modes are considered: a ballooning mode and an axisymmetric, quasi-flute mode. Instability thresholds are derived for both modes and numerically evaluated for parameters relevant to recent experiments. This is done to determine whether these modes could be responsible for convection of the plasma in the vicinity of the poloidal null point. It is determined that MHD instability about a standard equilibrium is unlikely to explain the experimental results observed on the tokamak, TCV [Reimerdes et al., Plasma Phys. Contr. Fusion 55, 124027 (2013)]. From these results, it is concluded that the most likely explanation for the discrepancy is that the underlying equilibrium assumed in the calculation does not contain all the complexities present in the experiments.

Two Contemporary Problems in Magnetized Plasmas: the ion-ion hybrid resonator and MHD stability in a snowflake divertor