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

TL;DR

This work tests the von Kármán–Howarth similarity for photospheric magnetic turbulence by analyzing two-point spatial ACFs from a full-disk SDO/HMI magnetogram. After normalizing each ACF by its zero-lag energy and computing a domain-specific correlation length $\lambda$, the rescaled functions $\mathscr{R}(\ell/\lambda)$ collapse onto a quasi-universal exponential form, consistent with self-preserving turbulence. The correlation-length distribution is approximately lognormal, peaking near $\sim$1500 km, with active regions contributing longer lengths; a strong positive correlation between the mean magnetic field and $\lambda$ is quantified (Pearson $r$ ≈ 0.77). These results provide observational constraints for turbulence-transport models, offering data-driven boundary conditions for injecting and transporting turbulence from the photosphere into the corona and solar wind, and guiding future multi-height, vector-magnetogram studies.

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 Kármán 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 (\(λ\)) 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 \(λ\) over the photosphere is presented; the ``quiet Sun'' tends to have \(λ\sim 1500\) 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 \(λ\), 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.

Figures (8)

• Figure 1: Top: Line-of-sight magnetic field map, cropped at disk center from a full-disk 720-s averaged HMI magnetogram dated 2010.10.24 12:00:00 (TIA). Contours of Carrington longitude and latitude are shown. An upper limit of 1500 G has been imposed on the color map. Bottom: Mosaic of correlation lengths of magnetic fluctuations, corresponding to the magnetic field shown in top panel. See Sec. \ref{['sec:res']} and App. \ref{['sec:app0']} for details of computation and discussion. The shown latitude/longitude tick marks are approximate. Each element (or pixel) comprising the mosaic has an approximately 121 Mm long side.
• Figure 2: Top left: $R(\ell_x)$ are the two-point autocorrelation functions (ACFs) of magnetic fluctuations, with spatial lags $\ell_x$ in the horizontal ($\hat{\bm{x}}$) direction. Bottom left: Each ACF is normalized by its value at zero lag: $R'(\ell_x)=R(\ell_x)/R(0)$. Top right: The lags associated with each ACF $R'$ are rescaled by the respective ACF's correlation length $\lambda_x$. Resulting ACFs ($\mathscr{R}$) are plotted as grey curves. The mean across this ensemble of ACFs ($\langle\mathscr{R}\rangle$) is plotted as a cyan dashed curve and $1\sigma$ spread about the mean is indicated by the vertical bars. The quasi-universal form of the ACF ($\mathscr{R} = e^{-\ell_x/\lambda_x}$) is plotted as a deep-pink curve. Bottom right: "Box-and-whisker" plot where each element (horizontal line) of a box-and-whisker from bottom to top indicates minimum, $10^\text{th}$ percentile, median, $90^\text{th}$ percentile, and maximum, at the respective lag. Equivalently, the green-shaded region within each box indicates the middle 80% of the distribution. A reference exponential function is plotted in deep pink. See text for further details.
• Figure 3: Top: Histogram of correlation lengths $\lambda$ (that combines $\lambda_x$ and $\lambda_y$) of magnetic fluctuations, computed using Eq. \ref{['eq:lambda1']}. Bottom: "Stacked" bar plot showing the probability density function (PDF) of $\log \lambda$, with selected values of $\lambda$ indicated on top axis. Blue bars with thick outlines show the PDF with $\lambda$ values associated with ARs removed. The orange region above or below the top of each blue bar indicates the modification to the PDF when ARs are included in the ensemble. Blue and orange bars have stripes oriented at different angles. Dashed red curve shows a best-fit Gaussian to the blue PDF of $\log \lambda$. Green curve shows an "arithmetic" Gaussian , i.e., one with the same mean and standard deviation as the blue PDF. See text for more details.
• Figure 4: Scatterplot of average magnitude of magnetic field within each averaging domain versus correlation length computed within the respective domain. Averaging domains with active regions are identified by requiring the maximum value of the magnetic field within a domain to be greater than 500 G.
• Figure 5: Top: Example averaging domain in which the ACF $R(\ell_x\equiv\ell)$ is computed, with dimensions $200\times 40$ pixels, as indicated with tick marks. Averaging domains for $B_\text{left}$ and $B_\text{right}$ are indicated with cyan and magenta colored lines, respectively. See text for more details. Bottom: ACF computed in this domain.