Physics of Fluids, Volume 33, Issue 5, May 2021. Solutions of robust axisymmetric neutral vortices, that is, vortices with zero amount of vorticity, in two-dimensional (2D) Euler flows with distributed vorticity are obtained. These solutions are particular linear combinations of vorticity layer-modes, which are defined as truncated, shifted, and conveniently normalized Bessel functions of order-0, each one occupying a circular layer defined by a zero of the Bessel function of order-1. It is found that some linear combinations of these modes have a vanishing net amount of vorticity and remain axysimmetrically robust to small amplitude vorticity perturbations. These neutral vortices are quiescent and remain steady in the presence of similar vortices. Other linear combinations of these vorticity layer-modes give rise to unstable neutral vortices that develop into neutral tripoles, pentapoles, etc. It is found numerically that the robustness of these neutral vortices is related to the spiralization and axisymmetrization of the initially growing vorticity disturbances as are advected by a convex azimuthal velocity distribution beyond its first inflection point. In particular, it is found that two co-rotating neutral tripoles attract due to the phase synchronization of their respective octupolar potential flow but repel when touched due to vorticity exchange. This interaction mechanism makes possible equilibrium states for sets of a large number of neutral tripoles. Other linear combinations of these vorticity layer-modes give rise to non-neutral shielded vortices which interact and may form coherent vortex structures as pairs of co-rotating shielded vortices sharing their outermost vorticity layer or counter-rotating shielded vortices translating with uniform speed as vortex dipoles.

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