Emergence and evolution of tripole vortices from net-circulation initial conditions

Abstract

The emergence of coherent vortical structures is a hallmark of the evolution of two-dimensional turbulence. Two fundamental processes of this evolution have been identified in vortex merging and vortex axisymmetrization. The question of whether axisymmetrization is a universal process has recently been answered in the negative. In the linear approximation, vortices indeed become axisymmetric, due to shear-enhanced diffusion. In the case of nonlinear interactions, other outcomes are possible; in the present work, we discuss a situation in which the flow reorganizes into a tripolar vortex. By performing an extensive numerical study, spanning the parameter space, we pursue the questions of what dictates if the flow will become axisymmetric or will develop into a quasisteady tripolar vortex, and what are the stages and the time scales of the flow evolution. The initial condition in this study consists of a Gaussian monopole with a quadrupolar perturbation. The amplitude of the perturbation and the Reynolds number determine the evolution. A tripole emerges for sufficiently large amplitude of the perturbation, and we seek to find a critical amplitude that varies with Reynolds number. We make several physical observations derived from visualizing and postprocessing numerous flow simulations: looking at the decay of the perturbation with respect to viscous or shear diffusion time scales; applying mixing theory; obtaining the first few azimuthal modes of the vorticity field; and describing the long-time evolution.

Linear and nonlinear decay of cat's eyes in two-dimensional vortices, and the link to Landau poles

“…These can take the form of multipoles; for example the addition of a pure azimuthal wave number m = 2 vorticity distribution can cause the vortex to evolve into a tripole consisting of a vortex core and two opposite-signed satellite vortices (Rossi et al 1997;Barba & Leonard 2007). Another possibility is to take a stable axisymmetric vortex and expose it to an external irrotational strain field, another m = 2 disturbance.…”

Thresholds for the formation of satellites in two-dimensional vortices

“…In the case when these two parameters are real we were able to construct a contour in this parameter space, within which the stream function decays in time and the vortex returns to an axisymmetric state and outside which the vortex remains elliptical and a satellite structure persists in the vortex. The results from this paper were all constructed with Z 2 (r) real, and hence Γ and ∆ real, to agree with the studies of Rossi et al (1997) and Barba & Leonard (2007), but could be extended to cover the full three-dimensional parameter space. Also it should be noted that these results are universal in that they apply to any family of vortices for which the BLSY theory becomes valid, in other words which asymptotically have the structure of a compact vortex supporting a normal mode, plus a weak skirt of vorticity.…”

“…used by the Navier-Stokes simulations of Rossi et al (1997) and Barba & Leonard (2007), we have a contour in the (Γ, ∆) plane given by…”

“…The threshold parameter δ shows a dependence on the Reynolds number: does δ(R) have a limiting value in the inviscid limit as R → ∞? Barba & Leonard (2007) have run simulations showing that the threshold value δ(R) decreases as R is increased, but the large Reynolds number limit has not been extensively researched.…”

“…The vortex evolves into a tripole which consists of a vortex core with two opposite-signed satellites of vorticity rotating around it, if δ is above some threshold value (Rossi et al 1997;Barba 2006;Barba & Leonard 2007). This tripole structure can be seen in the top row of panels in figure 1, which show the evolution of (1.1) and (1.2) with R = 10 4 and δ = 0.5 at t = 0, t = 300 and t = 700.…”