von Karman - Howarth Similarity of Spatial Correlations and the Distribution of Correlation Lengths in Solar Photospheric Turbulence

Abstract

Fluctuations in the Sun's photospheric magnetic field are the primary source of the turbulence that can heat and accelerate the solar atmosphere, and thus play an important role in the production and evolution of the solar wind that permeates the heliosphere. A key parameter that characterizes this turbulence is the correlation scale of fluctuations, which determines the injection of turbulent energy into the plasma and the diffusive transport of solar energetic particles. This study employs magnetogram data from the Helioseismic and Magnetic Imager on the Solar Dynamics Observatory to characterize an ensemble of spatial autocorrelation functions (ACFs) of turbulence in the photosphere. It is shown that the two-point ACFs satisfy the similarity-decay hypothesis of von K\'arm\'an and Howarth, a fundamental property of turbulent systems: rescaling the ACFs by their respective energies and correlation lengths yields a quasi-universal exponential form. The probability distribution function of transverse correlation lengths (\(\lambda\)) is shown to be approximately log-normal, which is consistent with observations of turbulence in the solar wind. A ``mosaic'' of the spatial distribution of \(\lambda\) over the photosphere is presented; the ``quiet Sun'' tends to have \(\lambda\sim 1500\) km (albeit with a wide distribution), which is close to the scale of solar granulation; systematically longer lengths are associated with active regions. A positive correlation is observed between mean magnetic field magnitude and \(\lambda\), and empirical fits quantify this relationship. These results improve our understanding of solar turbulence while providing observational constraints for models that describe turbulence transport from solar and stellar photospheres into their atmospheres.