Fractal and Spiral Organised Structures: Spectra and Diffusion

Abstract

Objects displaying a well-defined Kolmogorov capacity, like fractals or spirals, have attracted the interest of mathematicians and physicists in the last decades, mainly because of their occurence in a wide variety of situations. For example, scalar fields rolled-up in a spiral way (Gilbert [2]), or displaying a fractal interface (Nicolleau [6]), are very common features of fluid flows. It has been recently shown that they might have remarkable physical properties (Vassilicos [7], Flohr & Vassilicos [1], Gurbatov & Crighton [3]), due to the space-filling feature of the distribution of their gradients. One of the most remarkable features of these convoluted structures is their spectral signature. Moffatt [5] and Gilbert [2] showed that the energy spectrum of a rolled-up patch of scalar differs from the spectrum of an isolated patch of scalar, and that the exponent of their spectrum depends on the geometrical characteristics of the spiral. Vassilicos and Hunt[8] generalized these results to both fractals and spirals by showing that algebraic spirals have a well-defined Kolmogorov capacity (like fractals), and that a scalar field characterised by sharp interfaces with well-defined Kolmogorov capacity has an energy spectrum which depends on that capacity.