Characterizing the spatial pattern of solar supergranulation using the bispectrum
Vincent G. A. Böning1, Aaron C. Birch1, Laurent Gizon1,2,3, Thomas L. Duvall Jr.1, and Jesper Schou1
1 MPI für Sonnensystemforschung, 2 Institut für Astrophysik, Universität Göttingen
Our aim is to statistically characterize the spatial pattern of solar supergranulation beyond the power spectrum. The next-order statistic is the bispectrum. It measures correlations of three Fourier components and is related to the nonlinearities in the underlying physics. It also characterizes how a skewness in the dataset is generated by the coupling of three Fourier components. We estimated the bispectrum of supergranular horizontal surface divergence maps that were obtained using local correlation tracking (LCT) and time-distance helioseismology (TD) from one year of data from the helioseismic and magnetic imager on-board the solar dynamics observatory starting in May 2010. We find significantly nonzero and consistent estimates for the bispectrum using LCT and TD. The strongest nonlinearity is present when the three coupling wave vectors are at the supergranular scale. These are the same wave vectors that are present in regular hexagons, which have been used in analytical studies of solar convection. At these Fourier components, the bispectrum is positive, consistent with the positive skewness in the data and consistent with supergranules preferentially consisting of outflows surrounded by a network of inflows. We use the bispectral estimates to generate synthetic divergence maps that are very similar to the data. This is done by a model that consists of a Gaussian term and a weaker quadratic nonlinear component. Using this method, we estimate the fraction of the variance in the divergence maps from the nonlinear component to be of the order of 4–6%. We propose that bispectral analysis is useful for understanding the dynamics of solar turbulent convection, for example for comparing observations and numerical models of supergranular flows. This analysis may also be useful to generate synthetic flow fields.