A stable tripole vortex model in two-dimensional Euler flows

Abstract

An exact solution of a stable vortex tripole in two-dimensional (2-D) Euler flows is provided. The stable tripole is composed of an inner elliptical vortex and two small-amplitude lateral vortices. The non-vanishing vorticity field of this tripole, referred to as here as an embedded tripole because of the closeness of its vortices, is given in elliptical coordinates $(\unicode[STIX]{x1D707},\unicode[STIX]{x1D708})$ by the even radial and angular order-0 Mathieu functions $\text{Je}_{0}(\unicode[STIX]{x1D707})\text{ce}_{0}(\unicode[STIX]{x1D708})$ truncated at the external branch of the vorticity isoline passing through the two critical points closest to the vortex centre. This tripole mode has a rigid vorticity field which rotates with constant angular velocity equal to $\unicode[STIX]{x1D701}_{0}\text{Je}_{0}(\unicode[STIX]{x1D707}_{1})\text{ce}_{0}(0)/2$, where $\unicode[STIX]{x1D707}_{1}$ is the first zero of $\text{Je}_{0}^{\prime }(\unicode[STIX]{x1D707})$ and $\unicode[STIX]{x1D701}_{0}$ is a constant modal amplitude. It is argued that embedded 2-D tripoles may be conceptually regarded as the superposition of two asymmetric Chaplygin–Lamb dipoles, separated a distance equal to $2R$, as long as their individual trajectory curvature radius $R$ is much shorter than their dipole extent radius.