Neutral modes of a two-dimensional vortex and their link to persistent cat's eyes
Turner, MR and Gilbert, AD and Bassom, AP, Neutral modes of a two-dimensional vortex and their link to persistent cat's eyes, Physics of Fluids, 20, (2) Article 027101. ISSN 1070-6631 (2008) [Refereed Article]
This paper considers the relaxation of a smooth two-dimensional vortex to axisymmetry after the application of an instantaneous, weak external strain field. In this limit the disturbance decays exponentially in time at a rate that is linked to a pole of the associated linear inviscid problem (known as a Landau pole). As a model of a typical vortex distribution that can give rise to catís eyes, here distributions are examined that have a basic Gaussian shape but whose profiles have been artificially flattened about some radius rc. A numerical study of the Landau poles for this family of vortices shows that as rc is varied so the decay rate of the disturbance moves smoothly between poles as the decay rates of two Landau poles cross. Catís eyes that occur in the nonlinear evolution of a vortex lead to an axisymmetric azimuthally averaged profile with an annulus of approximately uniform vorticity, rather like the artificially flattened profiles investigated. Based on the stability of such profiles it is found that finite thickness catís eyes can persist (i.e., the mean profile has a neutral mode) at two distinct radii, and in the limit of a thin flattened region the result that vanishingly thin catís eyes only persist at a single radius is recovered. The decay of nonaxisymmetric perturbations to these flattened profiles for larger times is investigated and a comparison made with the result for a Gaussian profile.