A stable tripole vortex model in two-dimensional Euler flows. (10th November 2019) - Viúdez, A.
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 ([STIX]x1D707, [STIX]x1D708) by the even radial and angular order-0 Mathieu functions Je_0([STIX]x1D707)ce_0([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 [STIX]x1D701_0Je_0([STIX]x1D707_1)ce_0(0)/2, where [STIX]x1D707_1 is the first zero of Je_0^′ ([STIX]x1D707) and [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.