Turbulence, Fractals, and the Solar Granulation

Abstract

We give a brief, mostly qualitative introduction into the topics of convection and turbulence, and their description, mainly referring to laboratory experiments. Following Mandelbrot (1967), the concept of the fractal dimension is introduced and some earlier results of measurements of the fractal dimension in laboratory turbulence are discussed. Next, we address the question whether hints of turbulence have been observed in the solar photosphere, and describe three independent methods of determining the fractal dimension of the solar granulation: the area-perimeter relation, the line and the plane intersection method. An analysis of a set of high resolution granulation photographs taken with the balloon-borne’ spektro-Stratoskop’ telescope yields a fractal dimension of d ≈ 1.9 for granules of diameters > 1.32 arcsec and of d ≈ 1.3 for smaller granules, analysing the area-perimeter relation of approx. 40 000 granules. At first sight we seem to confirm the results obtained by Roudier and Muller (1986), Darvann and Kusoffsky (1989), and by Karpinsky (1990), who claim to see a splitting of the granulation into two regimes of different fractal dimension. However, a more detailed investigation of the analysis technique reveals that: i) there exists a smooth transition of the fractal dimension from small to large granules, and ii) the fractal dimension of small granules seems to be dominated by the finite resolution, and therefore no positive statement concerning the turbulent origin of small granules seems possible with the present technique. The fractal dimensions determined from the same material with the other two methods (i.e. the line and the plane intersection method), applied to both the intensity pattern itself and the lane map, all range between 1.88 and 1.97.