1 Introduction

The cellular pattern of the quiet solar photosphere can be explained by an overshoot of convective motions into a stable atmosphere. This implies that the structure of solar granulation should be determined by the conditions in the convection zone beneath the photosphere, i.e., assuming Bénard-like convection, that the cell sizes should be comparable to the depth where the hydrogen ionization region is located (Simon & Leighton 1964). Contradictory to this view, numerical simulations of solar granulation (Stein & Nordlund 1989, 1998) show that the granular pattern is mainly a surface phenomenon driven by effective radiative cooling in the photosphere. The decrease of pressure in a cooled fluid parcel leads to a horizontal displacement away from warmer regions which triggers the spontaneous formation of localized downflowing plumes due to Rayleigh-Taylor instabilities. The granules (upflow regions) are, therefore, only a secondary phenomenon balancing the radiative loss in the photosphere and ensuring mass conservation in the convective layer (Rast 1995, 1999b). Observational studies of e.g. Rimmele et al. (1995) and Strous et al. (2000) show that the spontaneous formation of downflowing plumes does not only play a dominant role in the development of the convective motions close to the solar surface, but they are also expected to be efficient sources of acoustic waves on the Sun. This result is also supported by numerical models of e.g. Rast (1999a) or Stein & Nordlund (2001).

Since the Reynolds number in the lower solar photosphere is approximately 10¹², the granular flow field is expected to be highly turbulent. This is supported by the analysis of power spectra derived from data obtained with the Multichannel Subtractive Double Pass (MSDP) spectrometer by Espagnet et al. (1993) and from white light images obtained at the Swedish Vacuum Solar Telescope by Hirzberger et al. (1997). Opposite to this view, results from numerical granulation models (e.g. Steffen et al. 1989; Stein & Nordlund 1998; Gadun et al. 2000) show much less turbulence as expected or even almost laminar granular flow fields. Nordlund et al. (1997) interprete this lack of turbulence as produced by the small pressure scale height in the solar photosphere and, thus, by the rapid expansion of upflowing matter when it enters into higher photospheric levels. Vice versa, increased turbulence should be visible in the intergranular lanes. This idea is supported by observations of e.g. Nesis et al. (1993, 1999) and by the numerical models of Rast (1995).

Another approach to understand the phenomenon of solar granulation is the analysis of the behaviour of individual granules and the application of statistical methods to the resulting physical quantities such as lifetimes, sizes, and intensities of granules. Statistical studies of granular properties have been carried out, among others, by Mehltretter (1978), Dialetis et al. (1986), Roudier & Muller (1987), Title et al. (1989), and Hirzberger et al. (1997, 1999). In a recent study performed by Berrilli et al. (2002) spectral scans across several absorption lines have been used to obtain the (line-of-sight) flow velocity as an additional parameter for a statistical analysis.

Although the list of observational studies on solar granulation is quite long (for further references see the review articles of Spruit et al. 1990 and Muller 1999) our knowledge about the substructure of solar granules is still incomplete. Even in most of the statistical analyses individual granules are treated as objects with several physical properties but no internal variation of them. An attempt to analyze the intensity inhomogenities within a large granule using isophote contours was made by Bray & Loughhead (1984). However, the reason for this lack of knowledge is the disadvantageous ratio between granule sizes (about one arcsecond) and spatial resolution of present (and past) solar data (approximately $0\hbox{$.\!\!^{\prime\prime}$ }2$ in the very best cases).

More about the granular substructure can be stated from coherence analyses as carried out by e.g. Deubner (1988), Nesis et al. (1988), Komm et al. (1990) from slit spectrograms, Espagnet et al. (1995) from MSDP data, and by e.g. Salucci et al. (1994), Hirzberger et al. (2001) from two-dimensional spectrograms obtained with tunable narrow band filters based on Fabry-Perot interferometers. From these studies two main results can be extracted: (i) the velocity field retains its cellular character in the upper photosphere (although larger structures extend into higher levels than smaller ones, see Komm et al. 1991) whereas the intensity pattern vanishes rapidly and becomes even inverted in the uppermost photospherical levels higher than about 250 km; (ii) the coherence between temperature (intensity) fluctuations and vertical velocity fluctuations is high and the phase between them is zero only for structures larger than about $0\hbox{$.\!\!^{\prime\prime}$ }8$ up to $1^{\prime \prime }$. Most of the above mentioned studies show that the coherence drops for smaller structures which generally is explained by turbulent flows which become dominant at these structural scales. However, Hirzberger et al. (2001) have shown - using the same data as used in the present paper - that the coherence between photospheric intensity and velocity fluctuations remains high for structures with sizes far below one arcsecond, i.e. until the resolution limit of the data which is approximately $0\hbox{$.\!\!^{\prime\prime}$ }4$.

The aim of the present paper is to carry out a statistical analysis of the intensity and velocity patterns from different photospheric levels and to show some properties of the distribution of intensities and velocities within granular structures.