Astron. Astrophys. 363, 289-294 (2000)

## 3. Theoretical correlative relationships

In this section we present height-dependent correlative relationships derived from 2-D models. In particular, we shall deal with two different sorts of correlations. The first of them is called a two-component representation, when thermodynamic quantities are averaged over up- and downflows, separately. Such relationships can be directly compared with the results, produced by inversion codes (Bellot Rubio et al. 1999, Frutiger et al. 1999, for instance). The second kind of correlations are linear correlation coefficients which reflect the linear dependence of model quantities between each other. These correlative dependencies were studied in a large number of papers using observations of spectral lines formed at various heights in the photosphere.

### 3.1. Two-component representation

Fig. 1 shows temperature, gas pressure and density fluctuations averaged over up- and downflows, separately, and over the total evolution time. Hereafter, the surface level or h = 0 in the geometrical height scale corresponds to log = 0, where is the Rosseland optical depth averaged over time and space in our sequence of 2-D models.

 Fig. 1a and b. Relative temperature a , pressure and density b fluctuations in two-component representation: the quantities were averaged over up- and downflows, separately. The temperature and gas pressure in upflows are shown by solid lines. Dotted lines denote temperature and gas pressure in downflows. The dashed line marks the averaged density in upflows and the dotted-dashed line is mean density in downflows.

From Fig. 1a we can clearly see two crucial height levels where temperature fluctuations reverse their sign. The first reversal (Nelson & Musman 1977) occurs at  km and is caused by a cooling of thermal convective upflows which overshoot into stable photospheric layers and a heating of downflows due to their compression. Near the traditional temperature minimum there exists the second reversal of temperature fluctuations - oscillating upflows are seen to be hotter again due to compression of the medium as they move into the upper atmospheric layers. Such a behaviour of temperature fluctuations is very similar in 2-D models and in the output of inversion codes (Bellot Rubio et al. 1999, Frutiger et al. 1999).

Fig. 1b exhibits the pressure and density averaged inside up- and downflows and shown as relative variations around their mean values at certain height levels. They outline three regions:

• convectively unstable subphotospheric layers (negative correlation between density and vertical velocity),

• a region of overshooting convection (correlation between density and vertical velocity becomes positive but is significantly greater than and is positive (negative) in upflows (downflows),

• and oscillating photospheric layers being in radiative equilibrium ().

Based on these models we may conclude that overshooting convection extends to about 150-170 km in the photosphere. However, this extension of the overshooting convection region, varies in dependence on the size of the convective cell: for instance, above cells with horizontal sizes of about 180 km it only extends to below 70-75 km (Gadun et al. 2000).

### 3.2. Correlative relationships

We analyse two kinds of correlations: one- and two-point correlations. The one-point (local) correlations correspond to correlation coefficients calculated between spatial variations of model quantities at the same model (horizontal) layer. Two-point correlations were found between spatial fluctuations of model quantities when one point is fixed around the surface level and the other point will be taken at various heights. These correlations reflect changes in the columnar structure of the inhomogeneous atmosphere.

We have determined the correlations between model quantities or between selected line parameters for each model. The mean correlation coefficients are obtained by averaging over the modeling time interval or over the time interval of observations.

#### 3.2.1. Two-point correlations

We start our analysis with correlations between (spatial variations of emergent monochromatic intensity at 500 nm) and spatial variations of temperature fluctuations at each horizontal level i in the photosphere (). They again demonstrate a high correlation (Fig. 2) in the low photosphere, dropping rapidly with height and becoming even negative at h larger than 120-130 km. This occurs due to the first reversal of temperature fluctuations inside the photospheric columnar structure (Fig. 1a): overcooling of the matter above the central part of the convective cells and heating of gas above intercellular lanes. The largest anticorrelation is found in the middle photosphere at heights between 250 and 350 km, where the reversal of temperature fluctuations is most pronounced (Fig. 1a). At these heights the temperature fluctuations are almost a mirror image of the granular brightness field. In the upper photosphere this anticorrelation decreases but is still significant in spite of the second reversal of temperature fluctuations because oscillations and shearing flows break down the quasi columnar structure there. In Figs. 2-4 we have used models.

 Fig. 2. Correlations between monochromatic emergent intensity at 500 nm and selected model quantities derived at various height levels in the model photosphere. Error bars given as indicate the scatter in values.

 Fig. 3. Correlations between vertical velocity fluctuations at a depth 60 km below the surface and selected model quantities as well as correlations between horizontal velocities (dashed line). The error bars are standard deviations shown as .

 Fig. 4. Local correlations between spatial fluctuations of selected model quantities. Error bars are estimates of standard deviations which are given as .

The correlations between and spatial variations of vertical velocities () demonstrate another dependence on height: they decrease slowly from a high level of correlation in the low photosphere to almost zero level at or near the traditional temperature minimum, there is no reversal field of vertical velicities.The high correlation between and becomes smaller than 0.5 at a height of about 250 km.

and (correlations between and fluctuations of gas pressure and density, respectively) correspond to those as to be expected in the transition region from thermal convection to layers with radiative equilibrium. For instance, are in anticorrelation in subphotospheric layers - less dense matter is hotter and brighter as well, but in optically thin layers they become positive due to the buoyancy breaking effect.

The peak in height dependence of is located around the surface, i.e. deeper than for the stratification, and positive over almost the whole photosphere.

Let us comment on of the correlations between and (spatial variations of monochromatic opacity at 500 nm). In the low photosphere, where temperature fluctuations are larger, they follow the but in high photospheric layers almost coincides with spatial fluctuations of gas pressure. This is explained by the sensitivity of H- ions to electron concentrations; H- ions constitute the main absorber in the solar atmosphere. In the low photosphere, the electron concentration depends mainly on hydrogen ionization which is strongly temperature-dependent. However, in higher layers the metals are the main contributor. Since the metals are basically ionized due to the still high temperatures, the electron concentration in these layers is not very sensitive to temperature fluctuations.

In Fig. 3 we present a series of two-point correlations in which we use a profile of vertical () and horizontal () velocities at a depth of 60 km below the surface. They can serve as better indicators of the columnar structure of the model photosphere than correlations with .

Correlations with () show almost the same behaviour as the correlation coefficients with previously discussed. , however, exhibit positive correlation over a larger geometrical height range than . It is important that horizontal velocities are highly correlated in these models over almost the whole photosphere. We note that from obvious reasons the correlation of spatial fluctuations of horizontal velocities with or is absent (is close or equal to zero).

#### 3.2.2. One-point correlations

The one-point (local) correlations are given in Fig. 4. They show the correlation coefficients found between selected model quantities for each horizontal level.

The correlation demonstrates, as mentioned above, two reversals of temperature fluctuations in the model atmosphere and a large anticorrelation due to overcooling of thermal convective flows in optically thin layers. We may also note the relatively large positive correlations between vertical velocities and and : on the average, ascending flows produce denser atmospheric inhomogeneities which have higher pressure than downflows. and are highly correlated over the whole model atmosphere.

and are negatively correlated up to a height of about 30 km. This may serve as an argument that the top of thermal convection reaches the low photosphere. The same conclusion is followed from Figs. 1-3.

The positive correlation in the upper photosphere does not change significantly the negative values of and because the photosphere does not have a columnar structure in the upper layers.

© European Southern Observatory (ESO) 2000

Online publication: December 5, 2000