Gradient enhancement and filament ejection for a non-uniform elliptic vortex in two-dimensional turbulence

Abstract

The axisymmetrization of a two-dimensional non-uniform elliptic vortex is studied in terms of the growth of palinstrophy, the squared vorticity gradient. First, it is pointed out that the equation for palinstrophy growth, if written in terms of the strain rate tensor, has a similar form to that of enstrophy growth in three-dimensions – the vortex-stretching equation. Then palinstrophy production is analysed, particularly for non-uniform elliptic vortices. It is shown analytically and verified numerically that a non-uniform elliptic vortex in general has a quadrupole structure for palinstrophy production, and that in the positive production regions, vortex filaments are ejected following the gradient enhancement process for vorticity. Numerical simulations are conducted for two different initial conditions, compact support and Gaussian vorticity distributions. These are characterized by distinctly different features of filament ejection and energy spectra. For both cases, the total palinstrophy production is a good indicator of the development of small-scale vorticity. In particular for the compact support case, a possible intermittency mechanism in the filament ejection process is proposed.

Topology of Two-Dimensional Turbulence

“…These axisymmetric centers are the likely predecessors of the coherent structures found in the latter stages of 2D turbulent decay [3,4,12,20,21]. They are stable [20,22,23], steady-state solutions of the Euler equation in which there is no nonlinear transfer of enstrophy and, hence, no enstrophy cascade [24].…”

2D vortex interaction in a non-uniform flow

“…If the vortices lie initially on the axis of the saddle points, the regime diagram is simpler, but again, evidences the competition between these two influences. An analysis of vortex deformation with the Okubo-Weiss criterion (see previous work by [8], [16], [3]), distinguishing vortexinduced strain and external strain, will be achieved to better understand the influence of vortex erosion during their interaction. A generalization of this study to stratified flows will be undertaken for oceanic applications.…”

The physics of vortex merger: Further insight

“…Fluids 18, 051701 ͑2006͒ dividing plane of the vortex pair. As discussed by Kimura and Herring, 14 in positive P s regions, isocontours are squeezed together while at the same time, they are extended in the orthogonal direction. The opposite is true for negative P s regions.…”

“…A positive P s corresponds to ٌ orienting towards the direction of the compressive strain, i.e., gradient amplification by compressive straining. Figure 5 shows the regions of positive/ negative P s , which exhibit the quadrupole structure associated with elliptic vortices, 14 and is asymmetric about the…”