Determining the stability of steady inviscid flows through ``Imperfect Velocity-Impulse'' diagrams

More than a century ago, Lord Kelvin proposed a variational argument for determining the stability of steady inviscid flows; while the underpinnings of the method are well established, its application has been the subject of extensive debate. Considering, for example, a vortex configuration rotating at a rate $\Omega$ with impulse $J$ and energy $E$, Kelvin argued that an equilibrium corresponds to a stationary point of $H = E -\Omega J$. Since $H$ is conserved, the second variation $\delta^2 H$ constrains the dynamics and can be used to assess stability. Unfortunately, computation of $\delta^2 H$ is often impossible or impractical. Saffman \& Szeto (1980) suggested that extrema in a plot of $E$ vs $J$ could be used to identify changes in $\delta^2 H$. However, Dritschel (1985) later pointed out the lack of a firm link between $\delta^2 H$ and a plot of $E$ vs $J$. Furthermore, he stated that even if such link could be proven, changes of stability could also occur, at bifurcations, away from extrema in $E$ and $J$. We address both issues by proposing a new approach. We introduce a theorem from dynamical systems theory to prove that extrema in a plot of $J$ vs $\Omega$ (instead of $E$ vs $J$) are indeed related to the properties of $\delta^2 H$, while we use ideas from imperfection theory to ensure that bifurcations are detected by means of an ``imperfect velocity-impulse'' (IVI) diagram. By applying our approach to several classical flows, we obtain stability results in agreement with linear analysis, while additionally discovering new steady solutions.