A vortex dynamics approach is used to study the underlying mechanisms leading to polar vortex breakdown during stratospheric sudden warmings (SSWs). Observational data are used in chapter 2 to construct climatologies of the Arctic polar vortex structure during vortex-splitting and vortex-displacement SSWs occurring between 1958 and 2002. During vortex-splitting SSWs, polar vortex breakdown is shown to be typically independent of height (barotropic), whereas breakdown during vortex-displacement SSWs is shown to be strongly height dependent (baroclinic). In the remainder of the thesis (chapters 3-7), a hierarchy of models approach is used to investigate a possible resonant excitation mechanism which is responsible for the vortex breakdown seen in our observational study. A single layer topographically forced vortex model is shown to exhibit vortex-splitting behaviour similar to that observed during SSWs. Two analytical reductions, the first a fully nonlinear analytical model of an elliptical vortex in strain and rotation velocity fields, the second a weakly nonlinear asymptotic theory applied to a topographically forced vortex, show that vortex-splitting in the model occurs due to a self-tuning resonance of the vortex with the underlying topography. Resonant excitation of an idealized polar vortex by topographic forcing is then investigated in a three-dimensional quasi-geostrophic model, with emphasis on the vertical structure of the vortex during breakdown. It is shown that vortex breakdown similar to that observed during displacement SSWs occurs due to a linear resonance of a baroclinic mode of the vortex, whereas breakdown similar to that observed during splitting SSWs occurs due to a resonance of the barotropic mode. The role of self-tuning in these resonant behaviours is then discussed in relation to the analytic reductions of the single layer model.