The Logarithmic Spiral: Mathematical Aspects and Modeling in Turbulence

We consider mathematical aspects of the logarithmic spiral and its utility in turbulence modeling. We consider mathematically the set of point crossings resulting from a linear intersection through the center of a logarithmic spiral. We derive analytically the fractal dimension as a function of scale for this set of crossings. We also derive analytically the power spectrum of the thresholded function corresponding to these point crossings. Our mathematical results have implications for turbulence modeling which are motivated by experimental observations of logarithmic spiral structures of scalar fields in turbulence.