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4. RESULTS AND DISCUSSIONFigure 2 shows an example of C_{n}^{2} values derived for a single launch in Camborne on the 1^{st} January 2002 at 5AM local time (0600 UTC). What appears to be a vertical line on the left hand side of the figure shows the data points that were identified as stable layers (C_{n}^{2} set to 10^{21}). The dots scattered across the figure show the C_{n}^{2} values for all those data points considered to be in turbulent layers. The figure shows, as expected Bufton [1973], Barat [1982], that turbulence, in the free atmosphere is confined to thin layers separated by nonturbulent regions . For altitudes above around 17 km there are turbulent layers that have C_{n}^{2} values that are smaller than that assigned to stable layers. This shows that the value of C_{n}^{2} chosen for stable layers is too high for high altitudes and this may lead to a slight overestimation in the statistical results of C_{n}^{2} for these height ranges. Even though this figure depicts only results for a single radiosonde launch, the values observed are consistent with those observed by radar (e.g. Rao, VanZandt) Figure 3 shows the histogram of C_{n}^{2} for Camborne at a height of 6000 meters. The data shown covers a period of 4 years with 4 radiosonde launches per day. The histogram class for stable layers (C_{n}^{2}=10^{21}) is outside of the figure and contains around 45% of the total number of samples (55% of the samples at this height are turbulent and shown in the histogram). The median value of the histogram is around 10^{14}. It should be noted that this histogram shows only the turbulent samples and, when normalized to the total number of turbulent samples, would represent the probability density function conditioned to the presence of turbulence. The lognormal behaviour of the histogram is evident and agrees with previous observations regarding the statistical behaviour of C_{n}^{2} as discussed in section 1.2.1. This result seems to indicate that the approach taken here is in principle correct. Using statistical descriptors such as the mean and standard deviation care has to be taken when either describing the full process, turbulent and stable, or only the turbulent part. When only considering the lognormal turbulent part, the mean and standard deviation have to be derived using the stochastic variable log C_{n}^{2}. Since the data ranges over several orders of magnitude, a single high value would dominate a linear mean which would subsequently misrepresent the turbulent part. The radar community also uses as in radar measurements the reflectivity is measured in logarithmic units. Hence also for the purpose of comparison using is a good practice. However, if all measurements are taken into account (turbulent and stable), would lead to an underestimation of the mean (or median) of C_{n}^{2} when there is turbulence. In this case, for all measurements (turbulent and stable), the linear mean will be closer to when only turbulent measurements are considered even if is, in the opinion of the authors, a poor descriptor for the probability of C_{n}^{2} conditioned to the occurrence of turbulence. It should be noted that some authors have presented mean results using the linear mean, see Vasseur [1999]. Figure 4 shows the overall statistical results for Camborne (4 years and 4 launches per day). The results shown were derived from the cumulative distributions of C_{n}^{2} for each height. Only the 10, 50 and 90 percentiles are shown, the 25 and 75 percentiles were omitted so that the figure is not overcrowded. It can be seen, for instance, that at 10 km the structure constant in the 90 percentile is approximately 10^{16}, indicating that only 10 % of the measurements would exceed this value. In order to distinguish between the various lines in the figure but preserve the visibility of the fast variation of the structure constant with height, every thirtieth data point has been replotted using the various plot symbols. The mean shown in the figure is . For applications in the troposphere where it is sufficient to use a simplified exponential model the median is fitted to yield a single expression for the C_{n}^{2} with height. The dashed line in the figure is the medianfit and was constructed between 2.1 and 8 km: C_{n}^{2}= 1.1×10^{15} exp(h/2014). Note that this value is very similar to that derived, also for the median case, for a Belgian site with a similar latitude and climate (Uccle) by Vasseur [1999]. The probability of occurrence of turbulence as a function of height is presented in Figure 5, this probability was derived by normalising the number turbulent samples at a given height to the total number of samples. Figure 6 shows the percentiles of the cumulative distribution conditioned to the occurrence of turbulence as a function of height. The data in Figure 4 can be derived from these two figures using:
Where z is the height, x is a given value of C_{n}^{2}, and P_{Cn2>x}(z), P_{Turb}(z) and P_{Cn2>x Turb}(z) are represented in Figures 4, 5 and 6 respectively. From figure 5, the probability of having a turbulent layer between 2 and 8 kilometres is always higher than 50%. This explains why the median values derived from figure 4 are always smaller than those in figure 5, i.e. the median of all samples underestimates the most probable case when there is turbulence. Thus the median of the histogram in Figure 3 is C_{n}^{2}=10^{14}, whereas the median for the overall histogram for the same height is C_{n}^{2}=10^{16}. The presence of the boundary layer (where the atmosphere interfaces with the surface of the Earth) can be clearly identified below 2 km where there is an increased probability of turbulence. Note that in figure 6 below around 12 km the percentiles are almost symmetrical around the median. This illustrates again the lognormal behaviour of C_{n}^{2} as already discussed for figure 3. Figure 7 shows the percentiles as a function of height derived from the cumulative distribution of the turbulent layer thickness for each height. The thickness of the turbulent layers was derived first from the Richardson analysis (see 3.2) and then by creating a histogram from a thickness classification for each height. The figure shows for all percentiles above 25%, as expected, thicker turbulent layers in the boundary layer (below 2 km), an almost constant thickness up to the tropopause and then a decrease above it. In the stratosphere (up to 20km) the figure shows again a slowly decreasing value. The values shown here are consistent with those for the outer scale of turbulence derived for different seasons in Eaton & Nastrom [1998]. The trend however is different. This may be due to the different techniques used, different climatology and orography (Eaton & Nastrom measurements were carried out close to a mountain range, 2700 m) and especially due to the different variables that are being compared (turbulent layer thickness and outer scale of turbulence). Figures 8, 9 and 10 show the results for Lerwick, Gibraltar and St. Helena in the form of percentiles derived from the cumulative distributions of C_{n}^{2} for each height. Only the 10, 50 and 90 percentiles are shown, the 25 and 75 percentiles were omitted so that the figures are not overcrowded. For Lerwick the median value is very low as may be expected from a northern site. The dashed line in the figure is the medianfit and was constructed between 2 and 8 km: C_{n}^{2 }= 8.9×10^{16} exp(h/1054). With h the height in meters. Gibraltar shows levels of C_{n}^{2} that are much higher than expected for a site at this latitude. This may be due to orographic effects and the proximity of the radiosonde launches to the Rock. The median was fitted between 3 and 10 km giving C_{n}^{2} = 2.37×10^{13} exp(h/991). St Helena shows values of C_{n}^{2} that higher than either Camborne or Lerwick as may be expected for a site closer to the Equator. The median was fitted between 2 and 12 km: C_{n}^{2} = 6.8 ×10^{15} e(h/1284). The data shown in this figure is noisier than that for all other sites, this is due to the smaller number of samples available (only one launch per day). For the lowest altitudes a discontinuity in the data can be seen, this is due to the altitude of the site from where the launches are performed (400 m). Some influence of the local orography can also be observed in the median and 90% percentile. The island has a peak at 832 meters. 