Variational Formulation of Vortex Dynamics

Vortex dynamics for a system of constant strength point vortices is described by the Kirchhoff-Routh (KR) function, where the resulting first-order differential equations have a special non-canonical Hamiltonian structure. Although this Hamiltonian is not derived from basic definitions in classical mechanics, it was widely adopted in Literature. In contrast, a new formulation of vortex dynamics is developed here using the principle of least action and formal variational principles for fluid mechanics. A system of non-deforming free vortex patches of constant strength is considered as a case study, where, for the first time, the resulting ODEs are second order in nature, defining acceleration. As a consequence, the dynamics is richer than that derived from the KR where vortices can attract and repel each other (i.e., there is a radial acceleration component along the line connecting each pair of vortices). Interestingly, the new formulation can recover the KR solution in the limit of vanishing core size (i.e., point vortices). Finally, the fact that the resulting model is derived from first principles (not devised to match certain dynamics) makes it applicable to arbitrary time-varying vortices.