Fractal dimensions of lines in chaotic advection

Field: Mathematics (Physical Sciences)

We release a line in a flow produced by a blinking vortex1 which is known to advect fluid elements ‘‘chaotically’’ for a certain range of parameters. This is a similar problem to that of a line evolving in phase space through the action of an area-preserving map, addressed a decade ago by Berry et al.2 These authors have classified the convolutions of such a line as being either ‘‘tendril shaped’’ or ‘‘whorl shaped.’’ Recent numerical simulations of lines released in 2-D turbulence3 have shown that they only develop ‘‘whorls,’’ i.e., spiral structures. These spiral structures are produced by the eddying regions of the flow4 and are responsible for the noninteger value of the ‘‘fractal’’ dimension 𝒟K of the line, as measured by the box-counting algorithm. This ‘‘fractal’’ dimension is actually a Kolmogorov capacity. It has also been shown recently3 that the Kolmogorov capacity is a measure of local self-similarity, whereas the Hausdorff dimension 𝒟H is a measure of global self-similarity. Spirals are a good example of locally self-similar objects, for which 𝒟K&gt;1 but 𝒟H=1. For conciseness we call a line H fractal when 𝒟H&gt;1, and K fractal when 𝒟K&gt;1. Most experimental and numerical evidence to date for ‘‘fractal’’ interfaces in turbulent flows is in fact evidence showing these interfaces to be K fractal. In fact, in the numerical simulations of Vassilicos,3 lines have been found not to be H fractal. Whether a line in chaotic advection becomes K fractal or H fractal is not a trivial question. If one neglects the effect of the unsteadiness of the flow, and thinks in terms of a single vortex at a fixed point in space wrapping the line around it, then it is easy to show that the spiral thus created has a 𝒟K&gt;1 but 𝒟H=1 (and, in particular, that 𝒟K&lt;2; the question of whether a line in chaotic advection is space filling is therefore not a trivial one either). But if one concentrates on the similarity between Aref’s blinking vortex and a two-dimensional map, then one may be reminded of the Hénon attractor5 which is known to have a transversal Cantor-like structure that is H fractal. In fact, pictures of the line in the blinking vortex flow show that line to have a comparable stretched and folded structure to that of the Hénon attractor. We measure 𝒟H by measuring the length of the line with various resolutions and find that 𝒟H grows with time above 1. By zooming into the pictures of our line we can see its self-similar structure, and are therefore inclined to conclude that lines in chaotic advection do become H fractal. We also measure 𝒟K by the box-counting algorithm, and find that it also grows with time above 1, but is not equal to 𝒟H. It is a known mathematical fact that in general 𝒟K≥𝒟H, and our findings are consistent with this requirement. But we do not yet understand what this nonvanishing difference between 𝒟K and 𝒟H means for a line in chaotic advection. Furthermore, we find that both 𝒟K and 𝒟H increase as the switching of the vortex from one location to the other becomes faster. It is not clear whether these two fractal dimensions tend, asymptotically with time, toward a value strictly smaller than 2 or not. The interest of this work is to show how efficient unsteadiness (which is the central component of 2-D chaotic advection) can be for creating H fractal structures through a process of folding that it adds to stretching of the flow.6 We compare with numerical simulations of 2-D turbulence3 where the simulated, self-similar cascade of eddies fails to produce H fractal structures, and only produces K fractals.