4 Results

4.1 Intensity and velocity distribution functions

Distribution functions of intensities and velocities in the data cubes are shown in Fig. 2. Intensity and velocity histograms of all pixels in the time series are plotted. At first glance the velocities seem to be symmetrically distributed around zero - this is partially caused by the definition of the zero points in the velocity maps (see Sect. 2) - but the curves do not have exact Gaussian shapes as found e.g. by Keil & Canfield (1978). The slopes towards negative velocities are significantly steeper than towards positive velocities. The maxima of the curves are at -125 m s^-1 (for v₀) and at -145 m s^-1 (for v₅₀), respectively. As already shown in Hirzberger et al. (2001) the velocity fluctuations exhibit only a small variation with photospheric height. The FWHM of the two curves are 1.125 km s^-1 (for v₀) and 1.380 km s^-1 (for v₅₀). A further conclusion extracted from the velocity distributions is that the upflow and downflow areas are almost equal. The fractional area of positive velocities (upflows) amounts to 48.6% (for v₀) and to 49.0% (for v₅₀). Both values are rather close to the fractional granular area detected by the applied image segmentation algorithm (see Sect. 3 above).

Figure 2: Distribution of velocities and intensities in the data cubes. Solid lines correspond to the maps v0 and I0, dotted curves correspond to v50 and I50, respectively. The dashed curve in the lower panel denotes the intensity distribution in the broad band images, $I_{\rm BB}$. All curves have been normalized by dividing by the total number of pixels in the time series.

The distribution functions of the intensities (lower panel of Fig. 2) show considerable variations with photospheric height. The curve corresponding to I₀ is almost symmetrical with a maximum at 0.998 and a FWHM of 0.098. In deeper photospheric levels the curves become broader - i.e. the rms contrasts are increasing - and the maxima of the curves are shifted to lower values. For I₅₀ the maximum of the distribution function is located at 0.978 ( FWHM=0.149) and that for $I_{\rm BB}$ is at 0.948 ( FWHM=0.232). The detected asymmetries of the distribution functions can also be seen in results from numerical simulations (Stein & Nordlund 1998) and do not imply that the ratios of areas brighter and fainter than the average values have to deviate significantly from one. The fractions of areas brighter than the mean intensities are 49.4% (for I₀), 47.1% (for I₅₀), and 47.0% (for $I_{\rm BB}$).

4.2 Structural parameters

An adequate method for describing shapes of structures is the calculation of structural parameters. The distributions of four such structural parameters are plotted in Fig. 3. They have been defined, according to Stoyan & Stoyan (1992), as follows:

The area-perimeter factor, $f_{\rm AP}$, denotes the ratio of a perimeter calculated from the area, A, of the structure assuming a circular shape and the actual perimeter, P.

$$f_{\rm AP}=4\pi A/P^2$$ (1)

$f_{\rm AP}$ is closely related to a fractal dimension, D, which can be derived from the area-perimeter relation of a sample of structures and which is frequently applied to solar granulation (see e.g. Roudier & Muller 1987; Hirzberger et al. 1997; Bovelet & Wiehr 2001).

The circularity factor, $f_{\rm C}$, denotes the ratio of a diameter calculated from the area of the structure assuming circularity and the so-called maximum Feret-diameter, $l_{\rm F}$, which corresponds to the maximum distance of two points on the boundary of the analyzed structure.

$$f_{\rm C}=\left(\frac{4A}{l_{\rm F}^2\pi}\right)^{1/2}\cdot$$ (2)

The elongation factor, $f_{\rm L}$, is defined by the ratio of the maximum Feret-diameter, $l_{\rm F}$, and the maximum extension, $w_{\rm F}$, of the structure perpendicular to $l_{\rm F}$.

$$f_{\rm L}=w_{\rm F}/l_{\rm F}.$$ (3)

The ellipticity factor, $f_{\rm E}$, denotes the ratio of the two semi-axes, a and b, of an ellipse approximating the shape of the structure.

$$f_{\rm E}=b/a.$$ (4)

The approximation of the ellipse can be carried out by the so-called area-perimeter method, i.e.:

$$a=\xi +\left[\xi^2-\frac{A}{\pi}\right]^{1/2}\quad\mbox{with}... ...{3}\left[\left(\frac{A}{\pi}\right)^{1/2}+\frac{P}{\pi}\right]$$

and

$$b=\frac{A}{\pi a}\cdot$$

Figure 3: Distribution functions of four structural parameters for granules (solid curves) and granular cells (dotted curves). The ordinates indicate the occurrence relative to the total number of granules in the analyzed sample.

Figure 3 shows the distribution of the four previously defined structural parameters calculated for granules and granular cells. The distributions corresponding to the granules are generally broader than those for the granular cells. Especially the distributions of $f_{\rm AP}$ exhibit completely different shapes. The curve for the granular cells shows almost a Gaussian distribution around 0.5 reflecting that the cells have mainly regular shapes. Their perimeters are by a factor $\sqrt{2}$ longer than expected for a circle. The distribution for the granules has its maximum at $f_{\rm AP}=0.15$ and a long tail up to values of $f_{\rm AP}=0.8$. Hence, most of the granules have extremely complex shapes but, however, a quite large number of them - mainly very small granules whose perimeters might be artificially shortened by the finite pixel size of the data - are rather regularly shaped.

For $f_{\rm C}$ the differences between granules and granular cells are not so large, yet still significant. Both distributions have a well-defined maximum and broad tails on both sides of it. The big difference between the distributions of $f_{\rm C}$ and $f_{\rm AP}$ for granules means that it is true that the granules have mainly very complex boundaries but their overall shapes do not deviate much from being circular.

The last statement has to be weakened when looking at the distribution functions of $f_{\rm L}$. Both curves cover almost the entire range between 0 and 1. Clearly defined maxima are not visible although the curves converge rather fast to zero below $f_{\rm L}=0.3$ (granules) and $f_{\rm L}=0.4$ (granular cells). Hence, most granules might have overall regular shapes (large $f_{\rm C}$) with irregular boundaries (leading to a low $f_{\rm AP}$) but their overall shapes are significantly elongated and not circular. The peak at $f_{\rm L}=0.5$ for granules is an artefact due to the finite pixel size yielding many structures whose lengths are exactly twice as long as their widths.

The ellipticity factor $f_{\rm E}$ is extremely low for all structures, granules and granular cells. Hence, the granules do have rather elongated shapes but the shapes are far from being elliptical.

Figure 4: Average broad band intensities ( $\left < I_{\rm BB}\right >$, upper panel) and velocities ( $\left < v_{50}\right >$, lower panel) for the 5509 detected granules vs. granular radius, $r_{\rm g}$. The abrupt edge for granular radii $r_{\rm g}<0\hbox{$.\!\!^{\prime\prime}$ }17$ appears since granules with areas $A_{\rm g}\le 9$ pixel have been removed from the sample.

4.3 Granular intensities and velocities

The broad band intensities and velocities (from level $i=50\%$) were averaged within each of the 5509 granules detected in the time series. Their dependence on the granular size is plotted in Fig. 4, in which the effective granular diameter, $r_{\rm g}$, is defined assuming circular shapes: $r_{\rm g}:=\sqrt{A_{\rm g}/\pi}$.

The average granular intensities, $\left < I_{\rm BB}\right >$, show an increase for small radii and remain nearly constant for granules larger than approximately $r_{\rm g}=0\hbox{$.\!\!^{\prime\prime}$ }5$. Qualitatively this is in good agreement with former results found in Hirzberger et al. (1997) and Berrilli et al. (2002) and does also agree with numerical simulations of Gadun et al. (2000). However, the position of the elbow is slightly shifted to smaller granules as found in the above cited references. This might be a consequence of the excellent spatial resolution of the data since the image segmentation algorithm used here is the same as applied in Hirzberger et al. (1997). In the lower panel of Fig. 4 the corresponding mean granular velocities, $\left < v_{50}\right >$, are shown. Their dependence on the granular radius agrees almost perfectly with that of the broad band intensities.

A conspicuous result to be noticed in Fig. 4 is that the distributions of both parameters, average granular intensities and velocities, do not increase monotonically with $r_{\rm g}$, i.e. the maximum values can be found in the range $0\hbox{$.\!\!^{\prime\prime}$ }4 < r_{\rm g} < 0\hbox{$.\!\!^{\prime\prime}$ }7$. This is in contrast to the distribution of the maximum granular intensities and velocities (not shown) which converge monotonically to asymptotic values (see also Figs. 12 and 13 in Berrilli et al. 2002). Therefore, granules larger than about $r_{\rm g}=0\hbox{$.\!\!^{\prime\prime}$ }7$ must have a rich internal structure containing very bright regions but also rather dark regions leading to a reduction of the average granular intensity. This conclusion is quite obvious when looking at a granular image, e.g. the one shown in Fig. 1. Also the development of dark centers in large granules is well known for many years. According to numerical simulations (e.g. Stein & Nordlund 1998; Gadun et al. 2000) this arises from a pressure excess developing above large granules which reduces the upflow velocity and, consequently, the transport of thermal energy from below. The evident close similarity of the two distributions shown in Fig. 4 is, thus, well supported by theoretical models.

The average granular intensities and velocities from higher photospheric levels, $\left < I_{0}\right >$ and $\left < v_{0}\right >$, are plotted in Fig. 5 vs. those from lower photospheric heights. The correlation between $\left < v_{0}\right >$ and $\left < v_{50}\right >$ is nearly perfect and independent on the granular radius. The slope of the distribution is only slightly smaller than unity which means that the average velocities do not decrease significantly in the probed height interval. The same is valid for the maximum velocities which are not shown here.

The correlation between $\left < I_{0}\right >$ and $\left < I_{50}\right >$ is not that good as found for the velocities and the correlation is considerably lower for small granules than for larger ones. This is due to several reasons: (i) The overall scatter is significantly larger as for the velocities; (ii) the slope of the distributions is significantly steeper for larger granules than for smaller ones. Thus, smaller granules exhibit in average a much faster reduction of their intensity with photospheric height than larger ones; (iii) some small structures exhibit an extremely high line center intensity. Possibly, these are magnetic structures which additionally bias the correlation between $\left < I_{0}\right >$ and $\left < I_{50}\right >$.

Figure 5: Upper panel: Average granular line center intensities, $\left < I_{0}\right >$, vs. those from level i=50% (line wings), $\left < I_{50}\right >$; lower panel: average granular velocities, $\left < v_{0}\right >$ vs. $\left < v_{50}\right >$. Blue dots denote granules smaller than $r_{\rm g}=0\hbox{$.\!\!^{\prime\prime}$ }7$, red dots denote granules with $r_{\rm g}\geq 0\hbox{$.\!\!^{\prime\prime}$ }7$. The overplotted solid lines represent linear fits to the granules with $r_{\rm g}\geq 0\hbox{$.\!\!^{\prime\prime}$ }7$, dotted lines are fits to granules with $r_{\rm g}< 0\hbox{$.\!\!^{\prime\prime}$ }7$. $\kappa$ and c are slopes of linear regression fits and their corresponding correlations. Dash-dotted curves represent the medians, $\kappa =1$.

It can be concluded from Fig. 5 that large granules exhibit an intensity excess also at higher photospheric levels whereas smaller granules dissolve at much lower heights. Hence, this result agrees partially with those found in Komm et al. (1991) that the typical length scale of intensity variations increases with photospheric height. For granular velocities this variation cannot be detected which contradicts the result of Komm et al. (1991).

4.4 Granular substructure

In Sect. 4.3 it was pointed out that (at least large) granules must have a rich internal intensity and velocity structure. Thus, an interesting question is how the intensities and velocities are distributed within the granules. As a first approximation to answer this question, the distances between the granular barycenters, $\vec x_{\rm bc}$, defined as

$$\vec x_{\rm bc}=\frac{\sum\limits_{j=1}^{N_{\rm p}}\vec x_jA_{\rm p}}{A_{\rm g}}$$ (5)

and the location of the maximum intensities and maximum velocities in the granules can be computed. The quantities $N_{\rm p}$ and $A_{\rm p}$denote the number of pixels belonging to a granule and the area of one pixel, respectively. In Fig. 6 the dependences of the relative - i.e. normalized to the granular radius, $r_{\rm g}$- distances of the maximum granular broad band intensities, $I_{\rm BB,max}$, and velocities, $v_{\rm 50,max}$, from the barycenters of the granules on the granular radius are shown.

Figure 6: Two-dimensional distributions of relative distances of maximum broad band intensities ( $\Delta d_{\rm BB}/r_{\rm g}$, upper panel) and velocities ( $\Delta d_{\rm v,50}/r_{\rm g}$, middle panel) from the granule barycenters. The distributions have been normalized to the total number of granules in the corresponding radius interval, $N_{\rm tot}$, taken from the histogram in the lower panel.

It is evident from the distributions plotted in Fig. 6 that for small granules the maximum intensities as well as the maximum velocities are located much closer to the granule barycenters as for larger ones. The dependences on the granular radius show nearly linear trends. This result confirms statistically the results of e.g. Nesis et al. (1992) or Krieg et al. (2000) who claimed that the maximum granular intensities and velocities tend to be located close to the granule boundaries although this claim seems to be valid only for very large granules. This result also agrees well with numerical models of Rast (1995) in which the fastest and brightest granular regions are locations of displaced material when fast downdrafts develop. Therefore, the fastest and brightest granular regions are located close to the intergranular lanes, i.e. at the granular boundaries.

The very large relative distances, $\Delta d_{\rm BB}/r_{\rm g}>1$ and $\Delta d_{\rm v,50}/r_{\rm g}>1$ which can be found in Fig. 6 do not mean that the brightest pixels are located outside the granule, i.e. that the segmentation algorithm is not working properly. This is rather due to the very complex granular shapes, e.g. corresponding to elongation factors far below $f_{\rm L}=0.5$ (see Fig. 3).

The location of pixels with maximum intensities and velocities has low statistical significance in cases where granules have areas of several hundred pixels. Moreover, the distribution shown in the middle panel of Fig. 6 seems to be also biased by a residual distortion between broad band images and velocity maps. This can be seen in the region $0\hbox{$.\!\!^{\prime\prime}$ }2<r_{\rm g}<0\hbox{$.\!\!^{\prime\prime}$ }4$ and $1<\Delta d_{\rm v,50}/r_{\rm g}<2$ where many displacements are found. For these tiny granules the detected shapes from the broad band images might be somewhat distorted with respect to the velocity maps. This is a consequence of the limited spatial resolution of the narrow band data which is in the range of $0\hbox{$.\!\!^{\prime\prime}$ }4$, and thus approximately 4 pixels (see Hirzberger et al. 2001).

Figure 7: Upper panel: relative area inertial radii of the granular cells vs. granular radius; lower panel: ratios between the inertial radii derived from the broad band intensity and the area inertial radii vs. granule radius.

To overcome this problem and to increase the statistical significance of the measured intensity and velocity structure within the granules, "inertial radii'', $r_{\rm\Theta}$, can be defined from the moment of inertia around a vertical axis through the granular barycenters. For convenience, i.e. to include also the structure of the intergranular lanes surrounding the granules, $r_{\rm\Theta}$ has been computed from the entire granular cells including granular and intergranular regions as defined in Sect. 3.

$$r_{\rm\Theta}=\left(\frac{\sum\limits_{j=1}^{N_{\rm p}} d_j^2\rho\left( d_j\right) A_{\rm p}}{M}\right)^{1/2}$$ (6)

where

$$M=\sum\limits_{j=1}^{N_{\rm p}}\rho\left( d_j\right) A_{\rm p}$$

denotes the granular "mass'' and the quantities d_j and $\rho\left( d_j\right)$ are the distance of the jth pixel from the granule barycenter and the local density (i.e. intensity, velocity, etc.) measured at the jth pixel, respectively. Here $N_{\rm p}$ denotes the number of pixels belonging to a granular cell.

Figure 7 (upper panel) shows area inertial radii, $r_{\rm A}$, i.e. assuming that the density $\rho$ is constant in the granular cell, for the 5509 detected granules vs. the granule radius. The absolute $r_{\rm A}$ have been normalized to the cell radii, $r_{\rm c}:=\sqrt{A_{\rm c}/\pi}$. The distribution shows no clear trend. For most of the small granules the relative area inertial radius, $r_{\rm A}/r_{\rm c}$, lies in the range between 0.8 and 1. For the sake of comparison, the inertial radius (assuming constant density) for a circle with radius R is $r_{\rm A}/R=1/\sqrt{2}$ and for a rectangle with a side ratio of 1/3 the relative area inertial radius is $\sqrt{10\pi /36}\approx 0.934$. For larger structures the $r_{\rm A}/r_{\rm c}$ scatter is in a range between 0.75 and 1.4 although a slight tendency for a general decrease of $r_{\rm A}/r_{\rm c}$ with $r_{\rm g}$ can be detected. This means that most of the large granular cells are more roundish than smaller ones - this might be slightly biased by the finite pixel size - but some of them (those with $r_{\rm A}/r_{\rm c}>1$) must have very elongated shapes, too.

Figure 8: Averaged ratios of the inertial radii derived from the intensity and velocity maps and the area inertial radii. The average has been carried out in overlapping bins of $0\hbox{$.\!\!^{\prime\prime}$ }1$ width. The dashed curves represent standard deviations (error bars) in each bin of the plot; the lower one has been calculated for $r_{\rm BB}/r_{\rm A}$ (squares), the upper one for $r_{\rm I,0}/r_{\rm A}$ (diamonds).

The lower panel of Fig. 7 shows ratios between the broad band inertial radii of the granular cells, $r_{\rm BB}$, and the area inertial radii. The $r_{\rm BB}$ have been calculated setting $\rho=I_{\rm BB}$ in Eq. (6). For very small granules with $r_{\rm g}<0\hbox{$.\!\!^{\prime\prime}$ }4$ the ratio increases with decreasing size, achieving values larger than 0.9 for the tiniest structures. Hence, these granules must have an almost homogeneous intensity structure ( $r_{\rm BB}=r_{\rm A}$ for constant $I_{\rm BB}$). Of course the ratio does not achieve unity because the intergranular lanes around these small granules reduces $r_{\rm BB}$ compared to $r_{\rm A}$. For granules in the range $0\hbox{$.\!\!^{\prime\prime}$ }4 < r_{\rm g} < 0\hbox{$.\!\!^{\prime\prime}$ }7$ the ratio of inertial radii exhibits a clear minimum. For these granules bright intensity maxima must exist close to their barycenters. For granules larger than $r_{\rm g}=0\hbox{$.\!\!^{\prime\prime}$ }7$ the ratio increases with the granule radius, i.e. the larger the granules are the closer the brightest regions are situated to the granule boundaries. This result is in good agreement with Fig. 6 but has a much higher statistical significance.

In Fig. 8 intensity inertial radii computed from the broad band images and the intensity maps together with velocity inertial radii computed from the velocity maps are plotted vs. $r_{\rm g}$. The absolute inertial radii, $r_{\rm BB}$, $r_{\rm I,0}$, $r_{\rm I,50}$, $r_{\rm v,0}$, and $r_{\rm v,50}$ have been normalized to the area inertial radius $r_{\rm A}$ of each cell. For the sake of an easier representation the resulting ratios have been averaged in overlapping bins of $0\hbox{$.\!\!^{\prime\prime}$ }1$ width (moving window method). This method using overlapping bins has the advantage that the averaged curves additionally become effectively smoothed. Qualitatively, all the curves plotted in Fig. 8 have the same shapes, i.e. they are close to one for the very smallest and very largest granules and have a minimum at intermediate granular radii. The two curves calculated from the velocity maps (v₀ and v₅₀) are almost identical, i.e. the velocity structure does not change in the probed photospheric height interval. The curve derived from I₀ deviates significantly from the others: (i) the minimum is shifted to larger granules ( $r_{\rm g}\approx 0\hbox{$.\!\!^{\prime\prime}$ }7$) and (ii) the minimum is much shallower than those of the other three curves. The shift of the minimum can be explained by the fact that the intensity excess of smaller granules dissolves at lower photospheric heights than that of larger ones (see also Fig. 5). The shallow minimum is due to the much lower rms intensity fluctuations measured in I₀ compared to those in I₅₀ and $I_{\rm BB}$ (see Hirzberger et al. 2001).

For the very smallest granules the ratio of the inertial radii approaches almost one for the curves derived from I₀, I₅₀, v₀, and v₅₀, respectively, whereas the curve corresponding to $I_{\rm BB}$ reaches only 0.92. This might be explained by a residual distortion of the intensity and velocity maps compared to the broad band images. Hence, the granular cells detected in the broad band images are not exactly co-aligned with the corresponding granules in the intensity and velocity maps. However, this effect becomes crucial only for the very smallest structures with $r_{\rm g}<0\hbox{$.\!\!^{\prime\prime}$ }2$ because for these structures the expected residual distortion - which is in the range of the expected spatial resolution of the narrow band data, i.e., approximately $0\hbox{$.\!\!^{\prime\prime}$ }4$ - is in the range of the sizes of the granules.

Figure 9: Two-dimensional - azimuthally averaged along circles of constant wavenumbers $k=\sqrt {k_x^2+k_y^2}$ - power spectra of the broad band image (solid line) and the gradient image (dotted line) displayed in Fig. 1.

Figure 10: Upper panel: maximum gradient of the broad band intensity of the 5509 granules vs. maximum broad band intensities in the granules; lower panel: maximum gradient of the velocity (at i=50%) vs. maximum velocity.

4.5 Horizontal intensity and velocity gradients

The result from the previous section - highest intensities and velocities are located close to the granular boundaries - implies a sharp transition of both parameters from the granules to the intergranular lanes, i.e. high horizontal intensity and velocity gradients. In the lower panel of Fig. 1 an example of the broad band intensity gradient, $\left\vert\nabla I_{\rm BB}\right\vert =\left[\left(\partial I_{\rm BB}/\partial x\right)^2 + \left(\partial I_{\rm BB}/\partial y\right)^2\right]^{1/2}$, is displayed. The derivatives have been calculated using a three point Lagrangian interpolation algorithm. The structures visible in this image coincide, as expected, well with the contours of the granules. At first glance it seems that all visible structures have the same width of about $0\hbox{$.\!\!^{\prime\prime}$ }5$ which means that the width of the transition zone between granules and intergranular lanes is independent on the size and the intensity of the granules.

Figure 9 shows power spectra of the broad band image and of the gradient map displayed in Fig. 1. For structures with diameters, $d<2^{\prime\prime}$, the power spectrum of the broad band image falls nearly exponentially to zero (for a discussion of the power laws of granulation images see e.g. Espagnet et al. 1993; Hirzberger et al. 1997; Nordlund et al. 1997) whereas the power spectrum of the gradient image shows a linear decrease down to a wavenumber of approximately k=17 Mm^-1 ( $d=0\hbox{$.\!\!^{\prime\prime}$ }51$). Since the structures visible in the gradient image (Fig. 1) are mainly thin and elongated, this result can be interpreted such that it is dominated by structures with a maximum length of approximately $4^{\prime\prime}$ (k=2.17 Mm^-1) and with a minimum extension or typical width of $0\hbox{$.\!\!^{\prime\prime}$ }5$.

In the upper panel of Fig. 10 the maximum gradient of the broad band intensity in each of the 5509 granular cells (granules plus surrounding intergranular lanes) vs. the maximum broad band intensity in each granular cell is plotted. In the lower panel of Fig. 10 the maximum gradients of the velocities (i=50%) vs. the maximum velocities in the granules are shown. Both plots exhibit a clear and nearly linear trend although the lower one is tainted with somewhat higher scatter resulting in a flattening of the trend for structures with maximum velocities below zero. These latter structures do have maximum intensities below one, i.e. they are small structures which seem to be slightly affected by some residual noise in the velocity maps.

The appearance of the linear trends in Fig. 10 means that brighter granules and granules containing faster upflows show a steeper drop of intensity and velocity towards the intergranular lanes than fainter ones. This follows from the fact that (i) the brighter the granules are the closer the location of the maximum intensities and velocities is situated to the granular boundary and (ii) that the width of the structures found in the gradient images (e.g. the one in Fig. 1) is independent on the granular size or intensity. However, this result does not mean that the brightest granules must be surrounded by the darkest intergranular lanes. Plotting maximum granular broad band intensities vs. minimum intergranular broad band intensities (not shown) exhibits a slight negative trend but with a correlation of only -0.21. The correlation between maximum granular velocities and minimum intergranular velocities is, with -0.18 (for i=0%) and with -0.16 (for i=50%), even weaker.

4.6 Coherence analysis

In the present data the time interval between two images is 70 s, which is in the range of the lifetimes of small granules (see Hirzberger et al. 1999). Hence, for studying the temporal evolution of the granulation pattern a direct tracking of individual structures is not possible. An alternative attempt is shown in Fig. 11. In this figure, the temporal variation of the coherence spectra of broad band images and line center intensity maps are plotted. The coherence spectrum, C(k), of two images, F(x,y) and G(x,y), is defined as

$$C(k)=\frac{\left\vert\left< \tilde{F}(k_x,k_y)\cdot\tilde{G}^... ...F}(k_x,k_y)\cdot\tilde{G}^{\ast}(k_x,k_y) \right\vert\right>},$$ (7)

where the $\tilde .$ denotes Fourier transformed quantities depending on the wavenumbers k_x and k_y. The brackets denote averages over a suitable region, e.g. in circles of constant $k=\sqrt {k_x^2+k_y^2}$ (azimuthal average).

Figure 11: Averaged coherence spectra of broad band images (left panel) and of line core intensity maps (right panel). The parameter $d=2\pi /k$ denotes a spatial wavelength of structures.

The columns in the images shown in Fig. 11 denote averages of coherence spectra, C(t_m,k), from images, F_l and F_l+m, separated by a constant time interval $t_m=m\cdot\Delta t$:

$$C(t_m,k)=\frac{\sum\limits_{l=0}^{N-m-1}C\left( F_l, F_{l+m}\right)} {N-m}~,\quad m=0,\dots,N-1$$ (8)

where $\Delta t=70$ s is the time interval between two images and N=36 is the number of images in the time series. The columns corresponding to t_m=0 (of the plots in Fig. 11) denote averaged coherence spectra when correlating images with themselves which is equal to unity for all wavenumbers, k. The second columns (t_m=70 s) represent coherence spectra when correlating images with the subsequent ones and so on.

Figure 11, hence, shows the temporal variation of the coherence of the granulation pattern in dependence on the structural sizes. In the left panel (broad band data) the coherence is high in the region around k=5 Mm^-1 ( $1\hbox{$.\!\!^{\prime\prime}$ }73$) and for $t_m\lesssim 10$ min which denotes typical sizes and lifetimes of large granules. For smaller structures (5 Mm ^-1<k<20 Mm^-1) the time interval of high coherences drops quickly to zero. For smaller wavenumbers the same is valid but the coherence shows additional peaks at $t_m\approx 23$ min and at $t_m\approx 36$ min for k=1.6 Mm^-1 and at $t_m\approx 40$ min for k=0.7 Mm^-1. A slight increase of the coherence is also visible at k=3.8 Mm^-1 and 30 min <t_m<40 min. These secondary peaks might be resulting from large and recurrent exploding granules first detected by Carlier et al. (1968) or from strong positive divergences (SPDs) found by Rieutord et al. (2000). Meso- and supergranular structures are expected to have longer lifetimes than the time intervals between these secondary peaks. Moreover, they should not be visible that clearly in the broad band data although it has to be assumed that the positions where recurrent granules are situated are somehow related to those large-scale flow fields (see e.g. Oda 1984; Title et al. 1989). Yet, a detailed study of these structures visible in the very low k-range is not the aim of the present work because of the limited field of view yielding a rather low wavenumber resolution in the range below k=3 Mm^-1.

The coherence spectra of the line core intensity maps (right panel of Fig. 11) show a gap in the region 4 min <t_m<10 min and 1 Mm^-1<k<3 Mm^-1. It can be concluded from this result that granules are almost completely dissolved at this photospheric height (320 km). Smaller structures are clearly visible and seem to have quite long lifetimes of more than 10 min. These structures cannot be granules because small ones should dissolve at much lower photospheric heights than larger ones. Possibly, the origin of the high coherence in that region are magnetic structures which become bright in high photospheric levels. In the line core maps a few of these structures are visible but their number is very small (see also Fig. 5) so that they do not considerably bias the statistics carried out in the previous sections.

The secondary peaks at k=1.6 Mm^-1 ( $t_m\approx 23$  min and $t_m\approx 36$ min) and k=0.7 Mm^-1 ( $t_m\approx 40$ min) are also visible in the line core coherence spectra but the increase of coherence at k=3.8 Mm^-1 has almost disappeared. Since broad band and narrow band data are obtained from independent observations, the secondary peaks should represent real physical phenomena. The very largest and brightest exploding granules which might be responsible for these secondary peaks are expected to produce positive temperature and brightness excesses also at high photospheric levels (see e.g. Roudier et al. 2001). Thus, it is not surprising that they are also visible in the line core data.

Figure 12: Granular coherence e-folding times, $t_{\rm e}$, for broad band images (solid line), line core intensity maps (dotted line), intensity maps corresponding to i=50% (dashed line), velocity maps from i=0% (dash-dotted line) and from i=50% (dash-dot-dot-dotted line). The inset shows the e-folding times for the broad band images in the range 3.2 Mm -1<k<20.5 Mm-1 overplotted with a linear fit to them.

It is possible to obtain lifetimes of granular features from the coherence spectra shown in Fig. 11. In Fig. 12 e-folding times, $t_{\rm e}(k)$, i.e. the times, t_m, in which the coherences fall below 1/e, vs. k are plotted. To overcome the relatively low temporal resolution of the analyzed data (70 s cadence between two images) the coherences have been interpolated using cubic splines before calculating the e-folding times. The curves are nearly identical for all the used data, except the curve corresponding to the line core intensity maps deviates slightly from the others. The e-folding times of the remaining parameters show well defined maxima at $k\approx 3.5$ Mm^-1 and a quite noisy behaviour for smaller k. This noisy character is probably caused by a combination of two effects: (i) the poor resolution in the low k-range and (ii) due to the large variety of phenomena in this range, e.g. large exploding granules (recurrent and non-recurring), SPDs, meso- and supergranules, etc.

In the range k>3.5 Mm^-1 the e-folding times exhibit an almost perfect linear (in this log-log representation) decrease from $t_{\rm e}=11$ min down to the time interval between two images in the time series is reached, which takes place at $k\approx 20$ Mm^-1. The linearity continues slightly below $t_{\rm e}=70$ s which is caused by the spline interpolation of the measured coherences. The inset in Fig. 12 shows once more the e-folding times for the broad band images (in the range 3.2 Mm ^-1<k<20.5 Mm^-1). The overplotted linear fit represents a power law of the form

$$t_{\rm e}\sim k^{-\beta}\quad\mbox{with}\quad\beta =1.512\approx 3/2.$$ (9)

The correlation of the fit amounts to c=0.998. Hence, the existence of a power law for granular lifetimes with increasing wavenumber has to be concluded from this result. This holds for both the granular velocity pattern and for the intensity pattern at low photospheric heights. The dissolution of the intensity pattern in the upper photosphere causes the deviation of the curve corresponding to the line core intensity maps. The peak at k=5 Mm^-1 might be produced by long-lasting magnetic structures which become rather large in these high photospheric levels since the equilibrium of the (reduced) gas pressure and the magnetic pressure can be maintained only if the magnetic structures expand.