Low-Dimensional Invariant-Manifold Models of Vortex Instability

We analyze barotropic and baroclinic instabilities of axisymmetric vortices in a hierarchy of quasigeostrophic models. Revisiting the classical vortex profiles of Carton and McWilliams (1989), we show that these simple vortex profiles possess very low-dimensional unstable manifolds in the nondissipative limit.  Using numerical simulations of the potential vorticity together with analytical calculations, we construct systematic approximations of these unstable manifolds. We then derive reduced-order models using the theory of extended normal forms on the low-dimensional reduced dynamics on these manifolds. The resulting Stuart–Landau–type amplitude equations, obtained in both data-driven and equation-driven settings, capture growth rates, frequencies, and the early nonlinear evolution leading to vortex deformation and breakup. This yields an interpretable and low-dimensional predictive description of the dynamics of vortex instabilities.