Development of singular solutions to the axisymmetric Euler equations

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Abstract
A variety of initially smooth axisymmetric flows with swirl are simulated with a variable mesh, finite‐difference code with particular attention paid to the production of large (divergent) vorticity. Away from the symmetry axis, the evolution is entirely consistent with expectations based on the isomorphism with two‐dimensional convection. Vortex sheets form on the leading face of ‘‘plumes’’ and their trailing edges roll up. When a ‘‘plume’’ begins to fission, a cusp develops at the cleavage point via a Rayleigh–Taylor‐like instability and the maximum (three‐dimensional) vorticity diverges, approximately, as inverse time squared. For technical reasons, the Boussinesq approximation was employed for this part of the simulation which observed, overall, a 106 increase in vorticity. The diverging strain was generated progressively more locally, justifying the approximation. Analytic estimates are provided which significantly constrain the singular solutions.

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