Previous studies have shown that observed vortex-splitting SSWs are predominantly barotropic in nature, i.e. the vortex split occurs almost simultaneously with height, and that they occur at a fixed orientation to the Earth's surface. This suggests that better understanding of a very simple model, in which an idealized vortex in a single layer quasi-geostrophic model is subject to a topographic forcing, may yield significant insight into the underlying dynamics. The model flow is governed by two parameters; a topographic height h, which controls the excitation of planetary Rossby waves from the troposphere, and a background rotation parameter Ω, which determines the stratospheric `climate'.
For relatively low h, vortex splits similar to observed SSWs occur only for a narrow range of values of Ω. Further, an abrupt transition in parameter space can be observed - a small change in Ω (or h) past a critical value can lead to an abrupt transition between a low amplitude oscillatory state and a sudden vortex split. The model behaviour can be fully understood using two separate (but essentially equivalent) analytical reductions, one of which results in the Kida model of elliptical vortex motion in a strain flow, and the other to a forced nonlinear oscillator equation similar to that studied by Plumb. The abrupt transition in behaviour is a feature of both reductions, and corresponds to the onset of a nonlinear (self-tuning) resonance. Similar behaviour, strongly resembling observed SSWs, can also be seen in a three-dimensional model of a columnar vortex. The results add an important new aspect to the `resonant excitation' theory of SSWs; if the stratospheric climate and planetary wave forcing evolve only very slowly, for example due to the seasonal cycle, an abrupt transition leading to a vortex split can nevertheless occur when a critical threshold is crossed. Under this paradigm, it is not necessary to invoke an anomalous tropospheric planetary wave source, or unusually favourable conditions for upward wave propagation, in order to explain an SSW.