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5 Discussion

Using Method A (Table 2), we can conclude that the temporal window [2$^{\rm h}-3^{\rm h}$] seems to give the best results following the criterion of the greatest correlation coefficient ($r_{\rm c}= 0.29$) . One can observe also that, for this time interval, the probability P that two uncorrelated distributions give a correlation coefficient greater than that found with the actual measurements, is the smallest (P = 0.49). In Fig. 10 we show a graphic global analysis of these data.

Using Methods B(I)-B(II) (Tables 3-4), we observe that when computed without the first 106 m, a better correlation ($r_{\rm c} = 0.33$) is achieved than with the first 106 m ($r_{\rm c}= 0.20$). The criterium of the minimum standard deviation, leads to the opposite conclusion, i.e. $\sigma$ computed with the first 106 m ($\sigma$ = 0.50) is better than that obtained without the surface layer ($\sigma$ = 0.54). In Figs. 11-13 we show the data of Table 3 analyzed with the Method B(I) and in Figs. 14-16 the data of Table 4 analyzed with the Method B(II). The Meso-Nh $\varepsilon 
_{\rm FA}$values are different in the two Methods B(I) and B(II) because the calibration coefficient is computed using the seeing over the whole atmosphere $\varepsilon _{\rm TOT}$. We emphasize that method B only, based to the $C_{\rm N}^2 \ $ profiles, can analyze model performances on selected parts of the atmosphere. In particular, as we will see later,we can study separately the boundary layer and the free atmosphere.

\psfig {figure=ds7968f10.eps,angle=-90,width=8.8cm}
 \end{figure} Figure 10: Method A: statistical analysis summary for the [2$^{\rm h}-3^{\rm h}$] temporal window. The calibration coefficient is $\alpha=1.43$ (Table 2)  

\psfig {figure=ds7968f11.eps,angle=-90,width=8.8cm}
 \end{figure} Figure 11: Method B(I): statistical analysis summary for the total seeing $\varepsilon_{\rm Tot}$.The calibration coefficient is $\beta=4.48$ (Table 3)  

\psfig {figure=ds7968f12.eps,angle=-90,width=8.8cm}
 \end{figure} Figure 12: Method B(I): statistical analysis summary for the boundary layer seeing $\varepsilon _{\rm BL}$.The calibration coefficient is $\beta=4.48$ (Table 3)  

\psfig {figure=ds7968f13.eps,angle=-90,width=8.8cm}
 \end{figure} Figure 13: Method B(I): statistical analysis summary for the free atmosphere seeing $\varepsilon 
_{\rm FA}$. The calibration coefficient is $\beta=4.48$ (Table 3)  

\psfig {figure=ds7968f14.eps,angle=-90,width=8.8cm}
 \end{figure} Figure 14: Method B(II): statistical analysis summary for the total seeing $\varepsilon_{\rm Tot}$.The calibration coefficient is $\beta=2.86$ (Table 3)  

\psfig {figure=ds7968f15.eps,angle=-90,width=8.8cm}
 \end{figure} Figure 15: Method B(II): statistical analysis summary for the boundary layer seeing $\varepsilon _{\rm BL}$.The calibration coefficient is $\beta=2.86$ (Table 3)  

\psfig {figure=ds7968f16.eps,angle=-90,width=8.8cm}
 \end{figure} Figure 16: Method B(II): statistical analysis summary for the free atmosphere seeing $\varepsilon 
_{\rm FA}$. The calibration coefficient is $\beta=2.86$ (Table 3)  

We discussed, until now, the results from a global point of view. Can we have a finer analysis of the estimation of the energy in the low and in the high part of the atmosphere? Using both Methods A and B, we achieve correlation coefficients smaller than that characterizing the optical measurements (see $r_{\rm c} = 0.82$ in Sect. 2). Looking at the Figs. 12, 13 and Figs. 15, 16 it is obvious that the correlation level, especially in the high part of the atmosphere ($\varepsilon 
_{\rm FA}$$\!$) is weak. Analyzing the whole campaign, night by night, we remarked that this fact is due to a poor reliability of the Antofagasta radiosounding that often provides erroneous vertical (p, T and $\mathbf{ \vec{V}}$$\!\!$) profiles in the middle atmosphere. In this case, the model generates unrealistic $C_{\rm N}^2 \ $ layers which hamper the statistical analysis (Masciadri et al. 1997). We found that, frequently, the dynamic instability generated by the model is related to the radiosounding temperature profiles.

In the analyzed cases (with and without the first 106 m of atmosphere) the slopes of the regression line (after calibration) related to $\varepsilon _{\rm BL}$ and $\varepsilon 
_{\rm FA}$ are of the same order of magnitude. This means that the model has a comparable sensitivity in both the low and high parts of the atmosphere.

In the boundary layer, in both the Methods B(I) and B(II) there is a small systematic tendency of the model to underestimate the turbulence. In fact the slope is, in both cases, smaller than 1. It is possible that a better configuration of the model near the ground could give better results. In particular, as explained in the next section, the ground radiative contribution in the global energy budget, could be an important element. In the free atmosphere, the correlation coefficient is not good but the points are distributed in a more random way than in the boundary layer. We suppose that, in this case, no systematic error is made by the model.

\psfig {figure=ds7968f17.eps,angle=-90,width=8.8cm}
 \end{figure} Figure 17: $C_{\rm N}^2 \ $ profiles comparison from balloons (full line) and Meso-Nh simulations after a 3 - hour simulation (thin line) on 25 Mars 1992. Calibration coefficient $\beta=1.66$  
In any case, we can show that in particular favorable conditions, the model can simulate a good optical turbulence distribution in the free atmosphere as shown in Fig. 17. The measured $C_{\rm N}^2 \ $ profile is obtained by a balloon launched during the PARSCA92 campaign. A favorable condition means that a deep region in the troposphere has a weak stability associated to a strong gradient of the wind. A good $C_{\rm N}^2 \ $ profile reconstruction at these high altitudes is very interesting for the scintillation rate, the isoplanatic angle and the wavefront coherence time forecasts. All these parameters are very sensitive to this kind of turbulence. In particular, the predictive performance of the wavefront coherence time could be extremely good. We know, that its sensitivity increases with the 5/3 power of wind intensity. At these latitudes the jet stream wind speed can reach 60 m/s and it becomes an important parameter. Knowing that the wind prediction at the synoptic scales is generally reliable, this numerical technique could give good forecasts of the wavefront coherence time.

In order to estimate the statistical reliability of our technique, we compared the statistical estimators presented in the previous paragraph with those obtained by a simple forecast-by-persistence method. The forecast-by-persistence principle is based on the assumption that the seeing of the next night is the same as that of the present night. It gives good results especially during periods characterized by a stable weather. It fails when an abrupt modification of the atmospheric state occurs. In other words, the numerical technique could be useful when the persistence method fails. We constructed a sequence of seeing values following the principle just described: the seeing of the (J+1)-th night is equal to the seeing of the J-th night. We computed all the statistical estimators between Scidar measurements and this sequence of values. Figure 18 shows the results of this analysis.

\psfig {figure=ds7968f18.eps,angle=-90,width=8.8cm}
 \end{figure} Figure 18: Statistical analysis comparison between the forecast by persistence and Scidar measurements  

This result is compared to those obtained using Methods A and B. Following the criterion of the maximum correlation coefficient, the numerical Methods A, B(I) and B(II) give better results: $r_{\rm c,MNH}\gt r_{\rm c,persistence}$. Conversely, following the criterion of the minimum standard deviation, the forecast-by-persistence method gives better results: $\sigma 
_{\rm persistence}<\sigma _{\rm MNH}$. No conclusions can be drawn because of the small number of statistical points.

We recall that during PARSCA93 the weather was particularly favorable to the forecasting-by-persistence because we identified two periods characterized by a worse ($\overline{\varepsilon }$ = 0.94 for the 13-19 May nights) and a better ($\overline{\varepsilon }$ = 0.57 for the 20-26 May nights) seeing with respect to a mean value over the whole campaign $\overline{\varepsilon }$ = 0.75. During each period, the statistical fluctuations around these mean values are not great. Under these conditions the forecast by persistence method is particularly efficient. Hence, we can expect that under a realistic conditions (for example a longer campaign or over many campaigns in different part of the year), the forecast by persistence method might give worse results.

The present study confirms the good geographic location of the Paranal site. Most of the simulations (Figs. 35-42 in Masciadri et al. 1997) show that Paranal mountain is sufficiently far from the Chilean coast which is characterized, on the contrary, by a high turbulence production rate. To give an example of the different turbulence production over the coast and the Paranal mountain, we report, in Fig. 19, the temporal seeing evolution over a 4 hours simulation related to the 16 May night.

\psfig {figure=ds7968f19.eps,angle=-90,width=8.8cm}
 \end{figure} Figure 19: Temporal evolution of the seeing over a 4 hour simulation time above the Paranal mountain and the chilean cost on 16 May 1993 night  

Finally, important elements concerning the sensitivity and precision of the model appeared at the end of this study. We considered the worst ($\varepsilon = 1.38)$ and the best ($\varepsilon = 0.38$) seeing night of the PARSCA93 campaign (Table 1). We compared the seeing simulated by Meso-Nh after 4 hours simulation time and the seeing measured by Scidar during these two nights over the whole atmosphere. The ratio between the worst and the best seeing is $\varepsilon_{16}/\varepsilon_{25} = 3.6$ for the Scidar and $\varepsilon_{16}/\varepsilon_{25}$ = 3.8 for Meso-Nh. This means that the model is capable of discriminating seeing values in a range [0.38 -1.38 arcsec] typical for good sites. In the previously study (Bougeault et al. 1995) the model has shown important limitations for simulations of good seeing nights. Low seeing values were analyzed but the best spatial correlation between measurements and simulations was found at 9 km from the peak.

The ability to simulate the favorable geographic position of Paranal mountain and to discriminate between such low seeing values are positive elements for a future implementation of such a model for flexible scheduling of telescope usage.

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