Vortex-splitting stratospheric sudden warmings are dramatic events in the stratosphere where the polar vortex abruptly (over a few days) breaks up into two comparable pieces before reforming. In this talk, the dynamics of the stratospheric sudden warming is considered from the point of view of transitions between quasi-stable and unstable vortex states. We will focus on the situation in which the vortex adjusts adiabatically in response to a slowly varying wavenumber-two topographic forcing, tracing out a family of approximately elliptical quasi-steady states, which lose stability when the aspect ratio of the vortex exceeds a certain threshold. For this purpose, we use a single-layer shallow water model as an appropriate representation of the predominantly barotropic dynamics involved in wavenumber-two vortex-splitting sudden warmings. In addition to the freely-forced problem, we also consider the more realistic situation where the vortex is relaxed back to the equilibrium profile using a simple Newtonian relaxation on the height field.