Dynamics of Trees of Fragmenting Granules in the Quiet Sun: Hinode/SOT Observations Compared to Numerical Simulation

Abstract

We compare horizontal velocities, vertical magnetic fields, and the evolution of trees of fragmenting granules (TFG, also named families of granules) derived in the quiet Sun at disk center from observations at solar minimum and maximum of the Solar Optical Telescope (SOT on board Hinode) and results of a recent 3D numerical simulation of the magneto-convection. We used 24-hour sequences of a 2D field of view (FOV) with high spatial and temporal resolution recorded by the SOT Broad band Filter Imager (BFI) and Narrow band Filter Imager (NFI). TFG were evidenced by segmentation and labeling of continuum intensities. Horizontal velocities were obtained from local correlation tracking (LCT) of proper motions of granules. Stokes V provided a proxy of the line-of-sight magnetic field (BLOS). The MHD simulation (performed independently) produced granulation intensities, velocity, and magnetic field vectors. We discovered that TFG also form in the simulation and show that it is able to reproduce the main properties of solar TFG: lifetime and size, associated horizontal motions, corks, and diffusive index are close to observations. The largest (but not numerous) families are related in both cases to the strongest flows and could play a major role in supergranule and magnetic network formation. We found that observations do not reveal any significant variation in TFG between solar minimum and maximum.

1 Introduction

Roudier et al. (2016), referenced below as paper I, found from Hinode observations a relationship between the evolution of trees of fragmenting granules (TFG) and horizontal motions of magnetic features in the quiet Sun. They suggested that TFG could contribute to the formation of the magnetic network.

Reviews by Sheeley (2005) and Ossendrijver (2003) suggested that the quiet Sun contributes significantly to the solar magnetism. Understanding how magnetic fields form in the convective zone and are advected and diffused to the surface is a challenge that requires investigating flows at various length scales (granulation, meso- and super-granulation), both from observations and magnetohydrodynamic (MHD) simulation (see review by Stein 2012).

The solar granulation is organized into families of granules (TFG) that can live several hours. They are formed by successively exploding granules originating from a single parent (for the history of the discovery, we refer to paper I and Roudier et al. 2003). We assume that TFG are implied in network formation through mesoscale surface flows. Observations and simulations at similar space and time resolutions are necessary to characterize and understand their impact on magnetic elements.

In the present paper, the goal is to compare several Hinode observations at different dates in the cycle with an independent numerical simulation to examine how realistic it is in terms of TFG formation, evolution, and interaction with magnetic elements. For this purpose, we analyze results of several 24-hour sequences observed with Hinode (near solar minimum in 2007 and maximum in 2013 and 2014) and the recent (2014) numerical simulation of magneto-convection based on Stein and Nordlund (1998). We study the dynamics of the solar surface at the mesoscale, relationships between TFG and horizontal motions, spatial power spectra, TFG size and lifetime, corks, and the diffusive index. We also investigate the action of TFG in the formation of the magnetic network.

2 Description of Observations and Processing Methods

We used multiwavelength datasets of the Solar Optical Telescope (SOT) on board Hinode (for a description of the SOT, see Ichimoto et al. 2005; Suematsu et al. 2008). The 0.50 m aperture of the SOT provides a spatial resolution of about 0.25″ (180 km) at 450 nm. Observations were made in the frame of HOP217 (determination of the properties of granule families and formation of the photospheric network) and HOP295 (connection of granule families to the formation of the magnetic network). We observed at disk center, so that horizontal velocities can be derived from the proper motions of granules. Families can be detected from the time evolution of granules.

For horizontal velocity measurements and the TFG analysis, the Broad Band Filter Imager (BFI) of the SOT provided 24-hour sequences of the blue continuum at 450.4 nm. Observations were recorded continuously on 29 – 30 August 2007 from 10:48 to 10:40 UT and on 30 – 31 August 2007 from 10:50 to 10:18 UT (two consecutive sequences, 2007a and 2007b, which can be combined to form a long 48-hour sequence). It must be noted that the 2007 sequence is not far from a small remnant activity located southeast. We also selected the 24-hour sequences of 28 – 29 April 2013 from 06:13 to 06:13 UT and 28 – 29 December 2014 from 22:14 to 22:56 UT (see Table 1).

Table 1 List of observations

After data alignment (which slightly reduced the initial field of view; FOV), we applied a subsonic Fourier filter in the $k\text{--}\omega$ space, where $k$ and $\omega$ are the horizontal wave number and pulsation, respectively, in order to remove continuum intensity oscillations. All Fourier components for which $\omega \geq C_{\textrm{s}} k$ (where $C_{\textrm{s}}= 6~\text{km}\, \text{s}^{-1}$ is the sound speed and $k = \sqrt{k_{x}^{2}+k_{y}^{2}}$ is the horizontal wave number) were retained to keep only convective motions (Title et al. 1989). In order to analyze TFG, we detected exploding granules and labeled them in time after segmentation as described by Roudier et al. (2003). Families or their different branches correspond to the mesogranular scale.

Horizontal velocities ($v_{x}$ and $v_{y}$) were also derived from aligned and filtered data of the blue continuum using the local correlation tracking (LCT) technique (November and Simon 1988, Roudier et al. 1999) with a temporal window of 30 minutes and a spatial window of $3.5''$, providing mean motions at the mesoscale or larger. As such flows evolve slowly and families need at least 6 hours to develop, this explains why we worked with 24-hour sequences. LCT is based on the detection of spatially and time-averaged granular proper motions.

For vertical magnetic fields (BLOS), we used the Narrow Band Filter Imager (NFI) in shuttered Stokes $I$ and $V$ mode to record the $V$ signal (a proxy of the magnetic field) in the blue wing of the Fe i 630.2 nm line (exposure time 0.12 seconds). The observations were recorded together with those of the BFI, at the same time and cadence. The Lyot filter was tuned to a single wavelength 12 pm apart from line core. $I+V$ and $I-V$ images were taken, then added or subtracted (providing Stokes $I$ and $V$, respectively) on board the spacecraft. We also selected the long sequence of the very quiet Sun in the Na i D1 589.6 nm line (shuttered $V/I$ mode, exposure time 0.20 seconds) from 30 December 2008 (10:30 UT) to 5 January 2009 (05:37 UT) with a five-minute time step, which has a better sensitivity and a much lower JPEG-type compression noise (see Table 1).

3 Description of the MHD Simulation

We used the results of the 3D magneto-convection stagger code, which was run independently and is not designed to model solar TFG. It solves the equations of mass, momentum, and internal energy in conservative form plus the induction equation of the magnetic field for a compressible flow on a staggered mesh (Stein and Nordlund 1998; Stein et al. 2009; review by Stein 2012). Solar rotation is included. Scalar variables (density, internal energy, and temperature) are volume centered, momenta and magnetic field are face-centered, while currents and electric field are edge-centered. Time integration is performed by a third-order low-memory Runge–Kutta scheme. Parallelization is achieved with MPI, communicating the three overlap zones that are needed in the sixth- and fifth-order derivative and interpolation stencils.

Boundaries are periodic horizontally and open at the top and bottom. The code uses a tabular equation of state that includes local thermodynamic equilibrium (LTE) ionization of the abundant elements as well as hydrogen molecule formation to obtain the pressure and temperature as a function of log density and internal energy per unit mass. Radiative heating/cooling is calculated by explicitly solving the radiation transfer equation in both continua and lines assuming LTE. The number of wavelengths for which the transfer equation is solved is drastically reduced by using a multi-group method that accurately models the photospheric structure as revealed in line profiles and limb darkening.

Because the stagger code includes all the significant physical processes occurring near the solar surface and resolves the thin thermal boundary layer at the top of the convection zone, its results can be used to make predictions, and the results can be compared quantitatively with various solar observations, such as oscillations (Rosenthal et al. 1999; Stein and Nordlund 2001), Fe i line formation (Asplund et al. 2000), G-band magneto-convection (Carlsson et al. 2004), facular variability (De Pontieu et al. 2006), velocities of magnetic elements (Langangen et al. 2007), helioseismology (Zhao et al. 2007), or correlation tracking (Georgobiani et al. 2007).

The initial state was a snapshot from a non-magnetic convective simulation. This in turn was the result of a series of hydrodynamic convection runs. All had the same 20 Mm depth. A 12 Mm wide simulation was run for two turnover times; then it was doubled three times to 96 Mm wide before the magnetic field was introduced (after 37.65 hours of hydrodynamic convection) as a 100 G uniform horizontal field, advected by inflows into the computational domain at the bottom. It took about two days (one turnover time) for significant magnetic flux to reach the surface.

The dimensions of the runs were $2016 \times 2016 \times 500$ with a resolution of 48 km horizontally and 12 – 80 km vertically. The quantities at the visible surface were binned $2 \times 2$ from 48 to 96 km (0.13″ pixel size) to be comparable to observations (0.11″). The physical parameters issued from the simulation are the emergent intensity, the velocity field, and the magnetic field vectors (we used only horizontal velocities and the BLOS component). The FOV was $131'' \times 131''$ and the time step was 60 seconds. The run duration covered a long time (100 hours), but we extracted a 26-hour sequence (start time: 59.5 hours; end time: 85.5 hours) for comparison to observations.

Several MP4 movies are attached as Electronic Supplemental Material (see the Appendix for details).

4 Horizontal Velocities of the Simulation: Comparison Between Plasma, LCT, FLCT, and CST

The numerical simulation provides the horizontal velocity vector over the 2D FOV during 26 hours with time steps of 60 seconds. It is a good opportunity to compare the plasma velocity to the velocity derived from indirect methods, such as LCT, Fourier LCT (FLCT, Fisher and Welsch 2008), or the coherent structure tracking (CST, Rieutord et al. 2007), that are applied to intensity structures as granules. The FLCT compares two consecutive images, while LCT and CST are time averaged. We used 30 minutes and $3.5''$ averaging windows. Rieutord et al. (2001) indeed showed that granules are good tracers of solar surface velocities for space-scales and timescales not shorter than these typical values. In order to compare the quality of structure tracking to plasma motion, we applied the same windows to horizontal flows. Figure 1 shows that the correlation between plasma and LCT or FLCT velocities is morphologically good; this is not the case of CST (which gives, in contrast, much better results at $1''$ resolution, see Roudier et al. 2012). The next two figures provide quantitative results.

Figure 1

Comparison between plasma, LCT, FLCT, and CST horizontal velocity modulus using an averaging 30-minute temporal window and a $3.5''$ spatial window. The FOV is $131'' \times 131''$. Top left: FLCT velocity; top right: plasma velocity; bottom left: CST velocity; bottom right: LCT velocity.

Figure 2 allows us to compare the performances of LCT, FLCT, and CST for windows of 30 minutes/3.5″. We analyzed the departures between the direction of horizontal velocities given by these methods and the plasma velocity vector (PLA), filtered by the same windows. The mean angle between LCT, FLCT, CST, and PLA is almost null, but the dispersion is 3.75 times higher with CST (a standard deviation of 34 degrees instead of 9 degrees for LCT or FLCT); the mean ratio between the PLA and LCT velocity modulus is 2.0; while between PLA and CST or PLA and FLCT, we obtained 1.15 and 1.12, respectively. The distribution function is definitely asymmetric for the CST, with a long tail. The best method is FLCT, but as it is slow for long sequences and large images, we used LCT instead to determine mesoscale horizontal flows, even if it underestimates the velocity modulus.

Figure 2

Comparison of PLA, LCT, FLCT, and CST horizontal velocities of the simulation. Top: mean angle between velocity vectors (solid: PLA, LCT; dotted: PLA, FLCT; dashed: PLA, CST). Bottom: mean ratio between velocity modulus (solid: PLA/LCT; dotted: PLA/FLCT; dashed: PLA/CST). Average values are indicated for direction and ratio.

5 Horizontal Velocities from LCT: Observations and Simulation

We now compare the horizontal velocities provided by LCT applied to Hinode observations and simulation. We also add a new Interface Region Imaging Spectrograph observation (IRIS, HOP312), performed on 10 – 11 October 2015 (23:34 to 05:32 UT) at disk center with the slit jaw (ultraviolet continuum at 283.2 nm) with time steps of 13.8 seconds, a pixel size of 0.166″, and a duration sequence of 6 hours; the IRIS and Hinode and simulation datasets do not differ much in pixel size. We used the same temporal and spatial windows for all (30 minutes, 3.5″).

Figure 3 shows the behavior of the plasma velocity modulus and LCT proxy of the simulation as a function of time, together with LCT velocity modulus of Hinode (2007, 2013, and 2014) and IRIS (2015). Velocities are averaged on the FOV and remain almost constant in time. When we apply a factor 2.0 to LCT observations, we find a mean horizontal velocity of 0.8 km/s, which is close to the 0.65 – 0.70 km/s of the simulation.

Figure 3

Mean horizontal velocity over the FOV as a function of time. Solid line: plasma velocity (simulation); dotted line: LCT (simulation); dashed line: LCT Hinode 2007; dash-dotted line: LCT Hinode 2013; dash–dot-dotted line: LCT Hinode 2014; thick solid line: LCT IRIS 2015.

6 Magnetic Fields of the Quiet Sun

Figure 4 shows typical magnetic fields (BLOS) of Hinode 2007 and 2008 at disk center (0.16″ pixel size), together with the beginning and the end of the 26-hour numerical sequence (0.13″ pixel) extracted from the full run. The 2008 FOV shows several supergranular cells, but also that many spatially concentrated and mixed polarities exist inside the cells, as noted by Dominguez Cerdeña, Sanchez Almeida, and Kneer (2006). The 2007 FOV is more noisy and contains a well-structured supergranule with BLOS mainly to the south. The numerical simulation is not magnetically steady state (there is a constant input of horizontal field at the bottom of the box); at the surface, there is more field at the end of the sequence than at the beginning. Polarities are much more mixed than in observations, with narrow concentrated flux tubes, so that it is difficult to distinguish the network. As the spatial resolution of BLOS is much better in the simulation, we filtered it by the point-spread function (PSF) of the SOT (including the 50 cm aperture, central occultation, spider, and CCD, assuming no defocus). Orozco Suárez, Bellot Rubio, and Del Toro Iniesta (2007) pointed out the role of instrumental degradation for imaging magnetic fields with Hinode. The 2008 sequence (with the best magnetic sensitivity) exhibits a spatial distribution that is fairly close to the simulation, with well-formed network and mixed polarities inside.

Figure 4

Vertical magnetic field (blue/yellow for north/south polarities); top left: Hinode 2008; top right: Hinode 2007; bottom left: simulation start at 59.5 hours; bottom right: simulation end at 85.5 hours, filtered by the SOT PSF.

The distribution functions of the vertical magnetic field (Figure 5) exhibit an asymmetry between north and south polarities for Hinode 2007 (probably due to the vicinity of a weak active region). Conversely, there is almost no asymmetry in 2008 (very quiet-Sun region); polarities of the numerical simulation are well balanced but produce concentrated fields that are much more intense. In particular, BLOS is smaller than 200 gauss for Hinode 2007 (350 gauss for 2008), while kilogauss (kG) fields are produced by the MHD code. However, Figure 5 shows that kG fields vanish when the SOT PSF is applied to the simulation. BLOS is likely underestimated by the Stokes sensitivity and the spatial resolution of the SOT and NFI filter.

Figure 5

Distribution functions of magnetic polarities (blue/red for north/south polarities); solid line: Hinode 2008, Na i D1; dotted line: Hinode 2007, Fe i; dashed line: simulation. The green/yellow curves (north/south) are for the simulation filtered by the SOT PSF.

In particular, the simulation reveals mainly vertical kG flux tubes only in intergranular lanes (median angle with the vertical 9 degrees, 90% of the angles are smaller than 17 degrees). The vertical velocity in flux tubes is downward (−2.23 km/s in average). The downflow increases with magnetic field strength (−1.92, −2.67, and −3.43 km/s for fields above 1.0, 1.5, and 2.0 kG, respectively). Ninety-seven percent of the flux ropes produce downdrafts. Unfortunately, the spatial resolution of Hinode does not allow us to check these predictions.

7 Spatial Power Spectra of Horizontal Velocities

Figure 6 provides the power spectra of horizontal velocities of Hinode/BFI observations as well as the simulation. The power spectrum is defined as the squared modulus of the 2D spatial Fourier transform averaged over the sequence duration and integrated over circular coronas. Power spectra of horizontal velocities computed with LCT are in full agreement at the supergranular (30 Mm) and mesogranular (5 Mm) scales for observations and simulation. The power spectrum of the plasma velocity of the simulation (convolved by the temporal and spatial LCT windows) also agrees with LCT results. The filtering windows cause the power spectra of the horizontal velocities to vanish at the granular scale (1 Mm); of course, this is not the case for plasma velocities of the simulation.

Figure 6

Power spectra of horizontal velocities of observations and simulation. LCT velocities of observations: 2007 (red), 2013 (green), and 2014 (dark green). Horizontal velocities of the simulation (blue): plasma velocity (thick), LCT velocity (thin), and plasma velocity after convolution by LCT windows (dashed). SG, MG, and G designate typical supergranular, mesogranular, and granular length scales.

8 Granule Merging and Splitting Rates

Splitting or exploding granules produce mesoscale horizontal flows and form families. We determined the splitting rate (fraction of exploding granules) and merging rate (fraction of collapsing granules) in BFI observations and simulation as a function of time. We did not find any significant time variation during the 24-hour sequences: the merging rate is 0.065 for the simulation (0.11 for Hinode 2007). The splitting rates are always higher: 0.115 for the simulation (0.14 for Hinode 2007). Splitting rates are always higher than merging rates; this explains the formation of TFG that appear in both observation and simulation. They are analyzed in the next section.

9 Dynamics of Trees of Fragmenting Granules

In paper I, we described the main properties of TFG. Their areas grow by a succession of granule explosions (some can occur simultaneously). TFG are able to push magnetic elements of the internetwork (IN) toward the frontiers of supergranules to form the network (NE). Explosions of granules at the TFG birth generate flows at the mesogranular scale in the range 0.5 to 0.8 km/s, which propagate outward when there is no counterpart flow of adjacent families. We also showed the existence of large velocity fronts affecting the NE shape. Magnetic features are often located at the edge of such fronts.

We discovered that TFG also form in the MHD simulation. Figure 7 displays TFG at the development time of the largest family (30″ or supergranular size), with the absolute value of BLOS superimposed. Magnetic fields are slowly advected to the borders of families in observations and simulation, while new magnetic elements (located in intergranules) appear in the IN.

Figure 7

TGF at maximum extension (top: simulation; bottom: Hinode 2007). Granules belonging to the same family have the same color. Magnetic fields (absolute value) are superimposed in white.

We now compare the main characteristics of families using observations (2007, 2013, and 2014) and numerical results. Figure 8 shows that the distribution function of family lifetimes is typically a power law of the form $t^{-1.88}$, where $t$ is the time. Most families develop to the mesoscale and have short lifetime (a few hours), while some rare TFG may last 24 hours or more and grow almost to the supergranular scale. The simulation appears in remarkable agreement with Hinode.

Figure 8

Distribution of TFG lifetimes: Hinode observations of 2007, 2013, and 2014: dashed and dotted lines; simulation: solid line.

We also examined the distribution functions of the surface of families and found a typical power law of the form $t^{-1.74}$ (Figure 9). The maximum size corresponds to the supergranular scale: 28 – 29″ for the simulation and Hinode 2007, 18 – 20″ for 2013 and 2014. However, large families are not numerous: only ten families have areas in the range 500 – 900 arcsec² in the simulation; for the 2007 observations, the ten largest families are in the range 300 – 900 arcsec². Most supergranules instead appear formed of several families at the mesoscale (8″ typical size): in a unit area of $1'\times 1'$, we found 113 families in the simulation above 8″ and 80 in observations (regardless of the date).

Figure 9

Distribution of TFG sizes: Hinode observations of 2007, 2013, and 2014: dashed/dotted lines; simulation: solid line.

Figure 10 shows the contribution of families in the FOV for two different lifetime thresholds as a function of time. Families lasting at least 3 hours represent 75 to 85% of the FOV, while long-life families (lasting more than 12 hours) cover 20 to 45% of the solar surface (the dispersion is higher because the number of long-duration families is small).

Figure 10

Fraction of the FOV covered by families lasting more than 3 hours (solid lines) or more than 12 hours (dashed lines): simulation (blue), 2007 (red), 2013 (cyan), and 2014 observations (magenta).

Hence, we conclude that the spatio-temporal properties of TFG as seen by Hinode or issued from the simulation fit well.

We note that the largest TFG are the most dynamic. Mean and maximum horizontal velocities of families are reported in Figure 11 as a function of their maximum area. For observations, velocities are derived from LCT; for the simulation, we used both LCT and plasma velocities (filtered by LCT windows). While average velocities do not vary much with family area, velocity maxima increase with size. We conclude that strongest flows are generated by largest families, which are composed of many simultaneously exploding granules. The results are similar for Hinode and the MHD code.

Figure 11

Mean (bottom) and maximum (top) horizontal velocities. Simulation: velocity plasma filtered by the SOT PSF (thick line) and LCT (thin line); observations: 2007 (2 sequences, dashed and dotted), 2013 (dash-dotted), and 2014 (dash–dot-dotted).

Movies 1 and 2 display interactions between horizontal flows and BLOS for the 2007 observations and simulation. Velocity fronts contribute to transport magnetic fields toward the boundaries of supergranules that are delineated by the NE. The strongest fronts occur at the border of large families, as shown in movies 3 and 4, suggesting that the dynamics of largest (but not numerous) TFG could play an important role in the NE buildup.

The NE formation at supergranular scale is illustrated (Figure 12) by the displacements of free corks for Hinode 2007 as well as for the simulation. At time $t=0$, corks were uniformly distributed on a regular grid. After the initial time, corks moved at the horizontal velocity of the LCT, and their trajectories are drawn. The final positions coincide with the magnetic network and delineate the boundaries of supergranules. Here again, we found an impressive agreement between observations and simulation. Movies 5 and 6 display the density of corks as a function of time, showing that corks are pushed toward the magnetic NE (which also correspond to bright He ii 30.4 nm hot structures of Hinode/EIS 2007).

Figure 12

Cork trajectories. Initial positions are indicated by green crosses. Magnetic polarities (green/blue and yellow/red) are superimposed. Top: simulation; bottom: Hinode 2007.

Cork motions can be characterized by a coefficient $\gamma$ assuming that the square of the distance $d^{2}$ from the initial position to the final position has the form of a power law $t^{\gamma}$. For pure diffusion or random motion, $\gamma$ is about 1, but for advective motions, $\gamma$ is higher. The LCT systematically suppresses Brownian motion, so that it does not appear in observations. Conversely, this component is obvious in the simulation and is superimposed on the shift toward the NE. We computed the mean $\gamma$ coefficient of observations and simulation. We found the following results.

• i) Simulation: plasma velocity at 0.13″, time step 60 seconds: $\gamma$ = 1.02 (Brownian motion dominates).

• ii) Simulation: plasma velocity at 3.5″, time step 30 minutes: $\gamma$ = 1.59 (advection dominates).

• iii) LCT (3.5″, time step 30 minutes):

  • Simulation: $\gamma$ = 1.82

  • BFI 2007: $\gamma$ = 1.66

  • BFI 2013: $\gamma$ = 1.67

  • BFI 2014: $\gamma$ = 1.64

The $\gamma$ values provided by the LCT for various datasets are in good agreement. After a few hours, $\gamma$ decreases because corks have formed the NE, as shown by movies 5 and 6, where the surface density of corks is plotted. When corks reach the cell boundaries, they slowly drift along them and tend to collapse at specific points that correspond to the intersections of several supergranules. This phenomenon is more clearly visible in observations (movie 5) than in the simulation (movie 6), where the magnetic NE is more diffuse. Our results can be related to those of Giannattasio et al. (2014a, 2014b), which were based on NFI magnetograms in the Na i D1 line: they found 1.44 for magnetic element displacements and 1.55 for magnetic pairs. However, corks are not magnetic elements; they move freely at the speed of the plasma flow.

10 Discussion

The surface of the Sun is covered by solar granules that are grouped in TFG or families. We used continuum intensities to show and label exploding granules that form families. The BFI provided intensities at 450 nm, but synthetic emerging intensities are at 500 nm. Danilovic et al. (2008) compared the granulation contrast seen by the Hinode spectropolarimeter (SP) with an MHD simulation and found that at 630 nm, the simulated contrast decreased from 0.14 to 0.07 after the SOT PSF was applied (this is close to the observed value). Wedemeyer-Böhm and Rouppe van der Voort (2009) compared BFI images in the blue (450 nm), green (555 nm), and red (668 nm) continua to synthetic images degraded by the PSF and found 0.11 in the blue. In the present study, the BFI contrast at 450 nm is 0.13. Granulation images of the simulation have a contrast of 0.155 at 500 nm, which reduces to 0.105 after filtering by the SOT PSF. We found that the recognition of TFG is little affected by the PSF because granule evolution between two consecutive times relies on the detection of a common surface. The PSF removes the small granules or details, but does not affect the formation of TFG, which are built essentially by the medium- and large-sized granules (although some branches may be cut). This is the reason why we recently discovered (work still in progress) that TFG are still detected at the SDO HMI resolution.

Averaged horizontal velocities were computed by LCT through windows of 30 minutes and 3.5″; using the simulation, we found a good agreement between LCT applied to continuum images and plasma motion, except that the velocity modulus was underestimated by a factor two. Using another simulation, Verma, Steffen, and Denker (2013) showed that the LCT recovers the main features of the granulation dynamics, but proper motions may be underestimated by a factor of three. Louis et al. (2015) have also compared this technique to simulated flow fields and concluded that LCT is a viable and fast tool to retrieve velocities for large datasets, with a good correlation between vectors and an underestimation of the magnitude. Alternative methods are discussed by Welsch et al. (2007). Recently, Asensio Ramos, Requerey, and Vitas (2017) developed a new algorithm (DeepVel) working on consecutive images to show smallscale instantaneous motions, but providing similar results to LCT for averaged velocity fields.

TFG sizes are distributed continuously with a decreasing slope from small (mesoscale) to large (supergranule); this is also the case of their lifetimes. TFG compete with each other, but the largest, which are not numerous (power law), generate the strongest horizontal flows that push the IN magnetic field to the border of supergranules to form the quiet NE. Thus, TFG appear to be an essential part of the supergranulation. This schematic view of TFG dynamics is compatible with the evolution of the IN and maintenance of the NE described by Gošic̀ et al. (2014). Using NFI observations in Na i D1, they showed a flux transfer from the IN to the NE, supplying as much flux as is present in the NE in 24 hours. We report here large-scale horizontal flows generated by the TFG regardless of the date in the solar cycle (2007, 2013, and 2014) and suggest that their associated flows could contribute to the transport of magnetic elements from the IN to the NE.

11 Conclusion

We have studied the dynamics of the quiet Sun from disk center observations of the Hinode SOT in terms of mesoscale horizontal flows, evolution of TFG, and LOS magnetic fields, throughout the cycle (2007 at solar minimum, 2013 and 2014 near solar maximum) using 24-hour sequences; the results were compared to those issued from the magneto-convection code at similar space and time resolutions after filtering by the SOT PSF. Horizontal flows in the observations were computed using the LCT applied to intensities; for the simulation, we used both LCT and plasma velocity and found good agreement with Hinode. In all cases, TFG appear after a few hours, most at the mesoscale, but some (composed of several branches) reach the supergranular scale. Families are associated with velocity fronts that advect magnetic fields and contribute to form the network. The largest TFG are not numerous but are the most dynamic; their development could be an efficient mechanism to build the network. The simulation provides realistic results for the properties, dynamics of TFG, and network interaction. However, small discrepancies do exist, such as granule merging/splitting rates or magnetic field polarities, which appear to be more mixed in the simulation than in observations (although this may come from instrumental effects and remnant activity).

We do not see any striking variation of the dynamics between solar minimum and maximum, confirming a previous study by Roudier, Malherbe, and Mirouh (2017) based on SDO/HMI observations. However, we need to analyze more data between 2007 and 2013 in the ascending phase, but unfortunately, the number of exploitable sequences at disk center in the blue continuum is rather limited. G-band sequences are more frequent but also more difficult to analyze with the LCT (bright points). We checked that IRIS observations (slit jaw continuum at 283 nm) are LCT compatible, so that new data could be used in the future to cover longer periods in the cycle. For this purpose, HOP312 with IRIS has been set up.

Appendix

The Electronic Supplemental Material movies are available in MP4 format.

• Movie 1: horizontal velocities ($\sqrt{v_{x}^{2}+v_{y}^{2}}$) of Hinode/BFI from LCT of the blue continuum, 29 – 31 August 2007, sequence duration 48 hours, FOV $65'' \times 75''$. Velocities in gray levels (LCT windows of 30 minutes/$3.5''$). The LOS magnetic field (Stokes V as a proxy of BLOS from Hinode/NFI blue wing of Fei 630.2 nm, pixel size $0.16''$, five-minute-averaged) is shown in blue/orange for north/south polarities.

• Movie 2: horizontal plasma velocities averaged through 30-minute/$3.5''$ filters (LCT windows for comparison with movie 1) of the numerical simulation, sequence duration 26 hours, FOV $131'' \times 131''$. Velocities in gray levels. The vertical component of the magnetic field (pixel size $0.13''$, five-minute-averaged) is superimposed in blue/orange for north/south polarities.

• Movie 3: Families of granules (TFG in various colors) derived from the Hinode/BFI blue continuum at 450.4 nm, 29 – 31 August 2007, sequence duration 48 hours, FOV $90'' \times 105''$, pixel size $0.11''$, together with horizontal velocities from LCT technique (30-minute/$3.5''$ windows, gray levels).

• Movie 4: Families of granules (TFG in various colors) provided by the numerical simulation, sequence duration 25 hours, FOV $131'' \times 131''$, pixel size $0.13''$, together with horizontal plasma velocities (gray levels, 30-minute and $3.5''$ filtered through LCT windows for comparison with movie 3).

• Movie 5: cork density (initially uniformly distributed over the FOV) represented by disks (size proportional to corks number, 10, 30, 100, 300, 1000, 3000, and 10000 or more). The corks are driven by horizontal LCT velocities (30-minute/$3.5''$ windows) from the Hinode/BFI blue continuum at 450.4 nm, 29 – 31 August 2007, sequence duration 48 hours, FOV $90'' \times 105''$. BLOS (Hinode/NFI blue wing of Fei 630.2 nm) is superimposed in blue/red for north/south polarities. In the background, Heii 30.4 nm intensities from Hinode/EIS are displayed in green.

• Movie 6: cork density of the simulation represented by disks (size proportional to the number of corks). The corks are driven by horizontal plasma velocities, sequence duration 26 hours, FOV $131'' \times 131''$. The vertical magnetic field is superimposed in blue/orange for north/south polarities. The time step is ten minutes and the averaging window $3.5''$ (to allow comparison with movie 5).