2. Observations

We used the Hinode Operation Plan (HOP) 151 spectro-polarimetric dataset, Flux replacement in the photospheric solar network and internetwork (Gošić Reference Gošić2012), to detect and track SMEs on the solar photosphere. Hinode is a Japanese solar satellite that was launched in September 2006 by the Japan Aerospace Exploration Agency in collaboration with NASA and the Particle Physics and Astronomy Research Council. The dataset was acquired by the Solar Optical Telescope (SOT), a Gregorian telescope with a 50 cm diameter and a field of view (FOV) of about $360\times 200~\text{arcsec}^{2}$ , corresponding to about $252\times 146~\text{Mm}^{2}$ over the solar photosphere (Tsuneta et al. Reference Tsuneta, Ichimoto, Katsukawa, Nagata, Otsubo, Shimizu, Suematsu, Nakagiri, Noguchi, Tarbell, Title, Shine, Rosenberg, Hoffmann, Jurcevich, Kushner, Levay, Lites, Elmore, Matsushita, Kawaguchi, Saito, Mikami, Hill and Owens2008). Stokes V/I magnetograms, representing the longitudinal magnetic flux over the FOV, were acquired with the narrow-band filter imager (NFI) focal plane instrument (Kosugi et al. Reference Kosugi, Matsuzaki, Sakao, Shimizu, Sone, Tachikawa, Hashimoto, Minesugi, Ohnishi, Yamada, Tsuneta, Hara, Ichimoto, Suematsu, Shimojo, Watanabe, Shimada, Davis, Hill, Owens, Title, Culhane, Harra, Doschek and Golub2007). The dataset HOP 151 (acquired on 2 November 2010) contains 995 magnetograms with a cadence of 90 s corresponding to 25 hours of continuous observations of the quiet Sun at the disk centre. The spectral line used was $\text{Na}~\text{I}~\text{D}~589.6~\text{nm}$ , observed to two wavelengths ( $\pm 16~\text{nm}$ from the central line). The magnetograms, $2\times 2$ binned, have a final pixel area of $0.16\times 0.16~\text{arcsec}^{2}$ , and a spatial resolution of 0.3 arcsec. The pixel size corresponds to a distance of ${\sim}116~\text{km}$ on the solar photosphere. The individual filtergrams were aligned and trimmed to their common FOV. This step reduced the FOV to ${\sim}51\times 53~\text{Mm}^{2}$ . Solar rotation was compensated during the data acquisition, so as to monitor the evolution of the same quiet Sun region for as long as possible. The magnetogram noise level was evaluated via the standard deviation of the signal in a quiet region of the magnetogram. It corresponds to ${\it\sigma}_{B}\simeq 6~\text{G}$ for each magnetogram.

The large FOV allows investigation of magnetograms with a large number of SMEs. The magnetogram corresponding to the beginning of the time series and saturated at 25 G, to show the fine structure of the magnetic field, is reported in figure 1.

Figure 1.

Hinode magnetogram. The image corresponds to the beginning of the time series, and is saturated at $\pm 25~\text{G}$ to show the fine structure of the magnetic field.

For the preliminary detection and tracking of the magnetic elements we applied the tracking algorithm described in Del Moro (Reference Del Moro2004), which uses a segmentation technique via an iterative threshold procedure (Berrilli et al. Reference Berrilli, del Moro, Florio and Santillo2005) so as to resolve weak magnetic features and isolate the peaks of the largest magnetic features. The algorithm labels as magnetic every pixel whose associated magnetic field value satisfies $|B(x,y)|>T_{0}$ , where $T_{0}=3{\it\sigma}_{B}\times i$ ( $i$   = iteration number).

At the end of the identification and tracking procedure, a total of 20 145 magnetic elements were detected and tracked in 25 h. Their lifetimes range from 7.5 min (5 frames) to 11.1 h (444 frames).

3. The motion of small magnetic elements

The motion of SMEs in the solar photosphere has been described in terms of an advection–diffusion process (see e.g. Wang Reference Wang1988; Berger et al. Reference Berger, Löfdahl, Shine and Title1998; Cadavid et al. Reference Cadavid, Lawrence, Ruzmaikin, Walton and Tarbell1998; Cadavid, Lawrence & Ruzmaikin Reference Cadavid, Lawrence and Ruzmaikin1999; Hagenaar et al. Reference Hagenaar, Schrijver, Title and Shine1999; Lawrence et al. Reference Lawrence, Cadavid, Ruzmaikin and Berger2001; Sánchez Almeida et al. Reference Sánchez Almeida, Bonet, Viticchié and Del Moro2010; Abramenko et al. Reference Abramenko, Carbone, Yurchyshyn, Goode, Stein, Lepreti, Capparelli and Vecchio2011; Manso Sainz, Martínez González & Asensio Ramos Reference Manso Sainz, Martínez González and Asensio Ramos2011; Giannattasio et al. Reference Giannattasio, Del Moro, Berrilli, Bellot Rubio, Gošić and Orozco Suárez2013, Reference Giannattasio, Stangalini, Berrilli, Del Moro and Bellot Rubio2014a , Reference Giannattasio, Berrilli, Biferale, Del Moro, Sbragaglia, Bellot Rubio, Gošić and Orozco Suárez2014b ; Del Moro et al. Reference Del Moro, Giannattasio, Berrilli, Consolini, Lepreti and Gošić2015), for which the mean-square displacement $\langle {\rm\Delta}l^{2}\rangle$ increases with time ${\it\tau}$ (measured from the first detection of each magnetic element) according to a power law

where $P(l,{\it\tau})$ is the PDF of displacements and ${\it\gamma}$ is the spectral index. The special case of a two-dimensional (2D) continuous Brownian motion with a normal distribution of displacements is represented by the so-called normal diffusion. In this case, the mean-square displacement (which corresponds to the variance of the PDF) reads $\langle {\rm\Delta}l^{2}\rangle =4K{\it\tau}$ , where the constant $K$ is the diffusion coefficient, which represents a measure of the efficiency of the transport regime. As a consequence of normal diffusion, the spectral index assumes the value ${\it\gamma}=1$ . However, in most cases an anomalous diffusion is observed, for which it is still possible to define a diffusion coefficient

which depends on both the probed spatial and temporal scales. In this case, the mean-square displacement may be modelled with a power law with spectral index ${\it\gamma}\neq 1$ , and superdiffusive ( ${\it\gamma}>1$ ) or subdiffusive ( ${\it\gamma}<1$ ) regimes can be observed. In the case of pure long-correlated advection the spectral index assumes the value ${\it\gamma}=2$ . Thus we expect a superdiffusive regime plasma motion to be a competition between (or a superposition of) pure advection and a normal diffusion process. This should constrain the spectral index to be $1\leqslant {\it\gamma}\leqslant 2$ .

Another way to explain the observed anomalous diffusion was suggested by Schrijver & Martin (Reference Schrijver and Martin1990), who report that SMEs are usually located at the vertices of a fractal lattice in convective structures, and constraints exist for their motion.

Figure 2.

The mean-square displacement $\langle {\rm\Delta}l^{2}\rangle$ as a function of time step  ${\it\tau}$ . Solid and dashed lines are nonlinear power-law best fits. The vertical dotted line indicates the cross-over time ${\it\tau}^{\ast }$ between the two different regimes.

The mean-square displacement of all the observed SMEs is computed for each time step  ${\it\tau}$ . A double-slope power law provided the best fit to the data, indicating two different regimes at small and large temporal scales. In particular, we found a spectral index ${\it\gamma}=[1.34\pm 0.02]$ for times up to ${\it\tau}^{\ast }\sim 2000$ s and ${\it\gamma}=[1.20\pm 0.05]$ for times up to ${\sim}10\,000~\text{s}$ . The uncertainties on ${\it\gamma}$ correspond to the standard deviation of the spectral index after random subsamplings of the magnetic elements. Figure 2 shows the mean-square displacement for tracked SMEs. The superdiffusive behaviour produces a diffusivity of $300{-}400~\text{km}^{2}~\text{s}^{-1}$ associated with photospheric convective flows.

An alternative way to investigate the advective–diffusive motion of the magnetic elements is to look at the evolution with time of the 2D probability distribution function $P({\it\delta}\boldsymbol{r},{\it\tau})$ of the displacements ${\it\delta}\boldsymbol{r}=\{{\it\delta}x,{\it\delta}y\}$ . To this end we consider all the magnetic elements whose lifetime is longer than ${\it\tau}=40$ time steps (i.e. more than 1 h). The total number of identified magnetic elements consists of ${\sim}2000$ elements. Figure 3 shows the overall diffusive motion of these magnetic elements when their initial positions are rescaled to the origin. The motion appears to be isotropic.

Figure 3.

The overall diffusive motion of the selected magnetic elements. Their initial positions are rescaled to the origin. Here, ${\it\delta}x$ and ${\it\delta}y$ indicate east–west and south–north shifts, respectively. The inset shows the trajectories (coded in different colours) for six selected long-living magnetic elements.

Using the spatial distribution of the selected magnetic elements, reported in figure 3, we construct the 2D-PDFs $P({\it\delta}\boldsymbol{r},{\it\tau})$ at different times ${\it\tau}$ , which will allow us to study the features of the diffusive–advective motion in terms of temporal evolution of the PDFs.

Computation of PDFs is done using the kernel method described in Kaiser & Schreiber (Reference Kaiser and Schreiber2002). In particular, this approach permits investigation of the Markovian nature of this motion. Figure 4 shows the evolution of the 2D-PDFs with time. Although at very short time scales (typically up to 360 s) the PDFs are affected by pixel resolution, for times shorter than ${\sim}2000$  s the PDFs seem to broaden symmetrically with time. In contrast, at longer times a multi-peaked feature of the PDFs emerges, which suggests the non-trivial nature of the diffusion–advection process of magnetic elements. In particular, the distance between the peaks at ${\it\tau}=2880~\text{s}$ is ${\sim}870~\text{km}$ , which is consistent with the typical spatial scale of solar granulation, i.e. of solar convection. Furthermore, we observe that magnetic element motion at time scales of the order of 3000 s shows strong anisotropy with preferential directions.

Figure 4.

Evolution of the 2D-PDFs $P({\it\delta}\boldsymbol{r};{\it\tau})$ with ${\it\tau}$ .

To evaluate the Markovian character of the observed evolution of PDFs we can apply the following approach. Indeed, in general the PDF at time $t$ , when it is known at time $t_{0}$ , can be written as

where $P({\it\delta}\boldsymbol{r},t\mid {\it\delta}\boldsymbol{r}_{0},t_{0})$ is the probability of transition from ${\it\delta}\boldsymbol{r}_{0}\rightarrow {\it\delta}\boldsymbol{r}$ in the time interval $[t_{0},t]$ . For a Markov process without memory, the previous expression can be written as

where ${\rm\Delta}{\it\delta}\boldsymbol{r}={\it\delta}\boldsymbol{r}-{\it\delta}\boldsymbol{r}_{0}$ and ${\it\tau}=t-t_{0}$ .

For Markov processes, such as random motions described by a simple stochastic Langevin equation,

where ${\it\sigma}$ is a constant indicating the noise amplitude, and ${\it\eta}(t)$ is a unit variance stochastic ${\it\delta}$ -correlated noise ( $\langle {\it\eta}(t^{\prime }){\it\eta}(t)\rangle ={\it\delta}(t^{\prime }-t)$ ), equation (3.6) implies that the 2D-PDF at time $t+{\it\tau}$ can be written in terms of a simple convolution of the 2D-PDFs at time ${\it\tau}$ , i.e.

Using this expression and moving from the 2D-PDF $P({\it\delta}\boldsymbol{r}_{0};t)$ for $t=180~\text{s}$ , we can compute the evolution of the 2D-PDFs for a simple Markov process without memory effects. Figure 5 shows the comparison between the measured 2D-PDFs and the corresponding one obtained by (3.6) at ${\it\tau}=3600~\text{s}$ . Large differences (approximately 20 %) are observed in the central part of the 2D-PDF. In particular, the actual 2D-PDF shows more prominent tails. In general, the probability of finding a magnetic element in the core of the distribution is reduced, while it is enhanced in the tail. This is the counterpart of the superdiffusive nature of magnetic element motion. Furthermore, we note that the 2D-PDFs at this time show a drift in the direction of  $-{\it\delta}x$ .

Figure 5.

(a) Comparison between the actual 2D-PDFs $P({\it\delta}\boldsymbol{r};{\it\tau})$ and the corresponding $P_{T}({\it\delta}\boldsymbol{r};{\it\tau})$ (dashed lines) obtained using (3.6) at ${\it\tau}=3600~\text{s}$ . (b) Difference ${\it\delta}P=P-P_{T}$ between the two distributions. Red and blue indicate negative and positive differences respectively.

The superdiffusive feature of the magnetic element motion could be due to different physical processes, from fractional Brownian motion to Lévy walks and/or flights, etc. In order to get more information on the nature of this motion, we can explore the evolution of the shape of the 2D-PDFs and their convergence to an asymptotic one. Because we deal with discretized steps, the 2D-PDFs expected at the shortest time scale should display a shape consistent with a 2D-binomial distribution, while for increasing time it should converge to a Gaussian one. In figure 6 we show the comparison between the measured 2D-PDFs for ${\it\tau}=90~\text{s}$ and ${\it\tau}=3600~\text{s}$ and the corresponding 2D-binomial and Gaussian PDFs. Good agreement is found in the case of the PDF at ${\it\tau}=3600~\text{s}$ , while the distribution of the initial step does not seem to be very well correlated with a 2D-binomial one.

Figure 6.

Comparison between the actual 2D-PDFs $P({\it\delta}\boldsymbol{r};{\it\tau})$ at ${\it\tau}=90~\text{s}$ and ${\it\tau}=3600~\text{s}$ and 2D-binomial and Gaussian PDFs (dashed white curves) at the two different times, respectively.

In order to quantify the consistency between the actual and expected PDFs we compute the Kullback–Leibler distance $D_{KL}$ (Kullback & Leibler Reference Kullback and Leibler1951). The Kullback–Leibler distance is an entropic measure of the distance between two distributions, defined by

where $P({\it\delta}\boldsymbol{r};{\it\tau})$ is the actual PDF and ${\it\Pi}({\it\delta}\boldsymbol{r};{\it\tau})$ is the reference one.

The Kullback–Leibler distance $D_{KL}$ is $D_{KL}=0.21$ and $D_{KL}=0.066$ for the two 2D-PDFs at times ${\it\tau}=90~\text{s}$ and ${\it\tau}=3600~\text{s}$ , respectively. In order to quantify the similarity/difference between the observed and theoretical distributions, we applied a hypothesis test evaluating a 5 % confidence limit via a Monte Carlo method. In detail, the evaluation of this confidence limit was achieved by: (i) computing more than $2\times 10^{4}$ distributions of a number of random variables, equal to the number of our magnetic elements, sampled from the same expected theoretical distributions, (ii) successively evaluating the corresponding statistics of the Kullback–Leibler distance for these ensembles of distribution functions, and (iii) setting the value of $D_{KL}$ corresponding to 95 % of the cumulative distribution function. This value is the 5 % confidence limit used to establish whether or not the PDFs agree with the theoretical PDFs. These numbers, when compared with the 5 % confidence limit, confirm that the 2D-PDF converges to a Gaussian PDF in the limit of very long times but is not consistent with a 2D-binomial PDF, which is that expected for uncorrelated discrete motion as a consequence of the inherent image resolution. In particular, the 2D-PDF at ${\it\tau}=90~\text{s}$ shows a more peaked shape than the binomial one.

4. Voronoi analysis

So far we have studied how convective photospheric plasma flows influence the motion of SMEs. However, in order to quantitatively investigate the nature of turbulent convective flows within the solar photosphere we analysed the development and dynamics of convective structures using an automated polygon subdivision based on SME localization, i.e. the Voronoi tessellation (Weaire, Kermode & Wejchert Reference Weaire, Kermode and Wejchert1986; Tanemura Reference Tanemura2003; Rapaport Reference Rapaport2006; Chiu et al. Reference Chiu, Stoyan, Kendall and Mecke2013). Indeed, the shape of cells in a Voronoi tessellation gives us information about the configuration of points, i.e. of SMEs detected in the magnetograms. It is known that a random distribution of points on a plane gives rise to a Voronoi diagram $V$ whose tile sizes are distributed according to Euler’s Gamma distribution (Tanemura Reference Tanemura2003):

where ${\it\alpha}={\it\beta}\in [3.5,3.6]$ (Weaire et al. Reference Weaire, Kermode and Wejchert1986). The stochastic variable $u$ is defined as $u\doteq A/\langle A\rangle$ , where $A$ is the size of the generic tile of $V$ and the Euler distribution function satisfies the following relations:

The parameters ${\it\alpha}$ and ${\it\beta}$ are extremely sensitive to the degree of correlation between points. In particular, their values promptly increase even at small spatial correlation values and eventually become very different when the correlation grows further.

Figure 7.

The probability density function of the normalized areas $u$ (black dots), as obtained from the complete image dataset, fitted to (4.1) (dotted line). The independent variable is $u=A/\langle A\rangle$ and the order of magnitude of the average area is $\langle A\rangle =8~\text{Mm}^{2}$ .

Figure 8.

Voronoi diagram formed using SMEs detected in the first image of the dataset.

In our case, every magnetogram is characterized by the coordinates of the SMEs $\{\boldsymbol{r}_{k}(t_{i})\}$ . We fit the data taken at different times with (4.1) and we study the distribution function of the $u$ variable and the evolution with time of the fitting parameters ${\it\alpha}$ and  ${\it\beta}$ .

For each set $\{\boldsymbol{r}_{k}(t_{i})\}$ , the corresponding Voronoi diagram $V_{i}$ is constructed and the overall probability density function of the normalized tile areas $u$ fitted to (4.1). Figure 7 shows the probability density function of the normalized tile area  $u$ , as obtained from the complete image dataset. Figure 8 shows a simple Voronoi diagram calculated using the first magnetogram of the dataset.

The probability density function is fitted using (4.1). The agreement between the calculated distribution and the theoretical Euler Gamma distribution is remarkable: we have obtained an adjusted coefficient of determination $R^{2}=0.976$ . The maximum of the Gamma distribution fit corresponds to $u=0.5$ , i.e. a spatial scale of 2 Mm corresponding to granulation convective cells. The measured parameters ${\it\alpha}$ and ${\it\beta}$ range between [3.6, 4.2] and [5.2, 7.5], respectively – a result suggesting that there is a certain degree of correlation between SMEs (e.g. Berrilli et al. Reference Berrilli, Del Moro, Consolini, Pietropaolo, Duvall and Kosovichev2004) and that their spatial distribution is not purely random. In fact, the analysis of figure 8 reveals the existence of several clumps of SMEs embedded in regions of magnetic voids. Such organization is related to the presence of network and intranetwork magnetic fields and ephemeral active regions in the photospheric plasma. These organized structures convey a tessellation which is made up of two or more populations of magnetic structures. There is a strong contribution from clumped SMEs of small area, corresponding to the peak around $u=0.5$ , and a weak contribution at intermediate area values. This inhomogeneous spatial arrangement modifies the tail of the probability density function of the normalized areas (figure 7) in such a way that a two-parameter Gamma distribution function (4.1) is necessary for an appropriate fit. The observed spatial distribution of SMEs involves the complex details of highly nonlinear small-scale interactions of turbulent convective flows detected in solar photospheric plasma.

5. Summary

The turbulent convection regime and diffusion properties of SMEs in a quiet region of the solar photosphere are studied using the mean-square displacement, the evolution with time of the 2D probability distribution function for the displacements, and their spatial distribution. SMEs are detected and tracked from magnetograms observed by Hinode/SOT using the $\text{Na}~\text{I}~\text{D}~589.6~\text{nm}$ line. In this paper we investigate the following.

1. (i) The diffusion properties of magnetic elements in a supergranular convective cell. A double-slope power law provided the best fit to the data, indicating two different regimes at small and large temporal scales. In particular, we found a spectral index ${\it\gamma}=[1.34\pm 0.02]$ for times up to ${\it\tau}^{\ast }\sim 2000~\text{s}$ and ${\it\gamma}=[1.20\pm 0.05]$ for times up to ${\sim}10\,000~\text{s}$ . The uncertainties on ${\it\gamma}$ correspond to the standard deviation of the spectral index after random subsamplings of the magnetic elements. As a consequence, the typical diffusivity ranges between $\simeq 400~\text{km}^{2}~\text{s}^{-1}$ and $\simeq 300~\text{km}^{2}~\text{s}^{-1}$ .

2. (ii) The advective–diffusive motion of the magnetic elements is to study the evolution with time of the 2D probability distribution function $P({\it\delta}\boldsymbol{r},{\it\tau})$ of the displacements ${\it\delta}\boldsymbol{r}=\{{\it\delta}x,{\it\delta}y\}$ . We found that the probability of finding a magnetic element in the core of the distribution is reduced, while it is enhanced in the tail. This is the counterpart of the superdiffusive nature of magnetic element motion. Furthermore, we find evidence of the existence of a spatial scale of approximately 900 km, which emerges at time scales of the order of 45 min. This could be the fingerprint of the complex dynamics of solar convection, which manifests in spatio-temporal scales (Consolini et al. Reference Consolini, Berrilli, Florio, Pietropaolo and Smaldone2003).

3. (iii) Euler’s Gamma distribution displays a maximum that corresponds to $u=0.5$ , i.e. a spatial scale of 2 Mm corresponding to granulation convective cells. The measured parameters ${\it\alpha}$ and ${\it\beta}$ range between [3.6, 4.2] and [5.2, 7.5], respectively, a result suggesting that there is a certain degree of correlation between SMEs and that their spatial distribution is not random. Specifically, the inhomogeneous spatial arrangement of SMEs on the solar photosphere modifies the tail of the probability density function of the normalized areas (figure 7) in such a way that a two-parameter Gamma distribution function is necessary for an appropriate fit. The observed distribution of SMEs involves the complex details of highly nonlinear small-scale interactions of turbulent convective flows detected in solar photospheric plasma.

In conclusion, we have shown how it is possible to gain information on solar convection by investigating the motion of magnetic elements. In particular, the motion of these objects reveals the complex nature of solar convection, which manifests in competing spatio-temporal structures and characteristic scales. Further analyses are necessary to place the emerging scenario within a more theoretical framework.