Are We Seeing Turbulence?

The next question is, “Do the correlation function plots presented above imply that turbulence was being measured?” We can compare these plots for a day that was known to be turbulent, and one that was calm. Recall that January 15 and 17, 2008 were in the former category, and January 14, 2008 was in the latter. In the following, we present the correlation plots for the de-trended data covering two time periods on the 14^th (Figure 112 and Figure 113), four on the 15^th (Figure 114 through Figure 117), and two on the 17^th (Figure 118 and Figure 119). Some of these plots have already been shown above, but we repeat them here so all the relevant figures are in the same place. The first, and most obvious difference is that there is no correlation structure in the 900 cm^-1 region on January 14, whereas all the other cases do show a structure, except for January 15, 2008, Data Set 2, 1^st 5000 points (Figure 116). Second, the magnitude of the correlations seen in the 1500 and 2300 cm^-1 regions is larger on the turbulent days than on the calm day. However, the differences are not dramatic.

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Figure 112. Correlation plot for 14 Jan 2008, Data Set 1, second 5000 points.

Figure 113. Correlation plot for 14 Jan 2008, Data Set 2.

Figure 114. Correlation plot from 15 Jan 2008, Data Set 1, second 5000 points.

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Figure 115. Correlation plot from 15 Jan 2008, Data Set 1, third 5000 points

Figure 116. Correlation plot from 15 Jan 2008, Data Set 2, first 5000 points.

Figure 117. Correlation plot from 15 Jan 2008, Data Set 2, second 5000 points.

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Figure 118. Correlation plot from 17 Jan 2008, Data Set 1, first 5000 points.

Figure 119. Correlation plot from 17 Jan 2008, Data Set 3.

Next, consider power spectra and model fits for selected wavenumbers on the three days. (As with the correlation plots, some of the subsequent spectral plots have been shown above, but we include them here for ease of comparison.) We have chosen wavenumbers that correspond to the three regions of interest: around 900, 1500, and 2350 cm^-1. In each case, the von Karman model is fit from the 2^nd through last frequency points, and the quadratic model uses the 10^th through last.

First, consider spectra from January 14, 2008, Data Set 1. From the correlation plots (Figure 112 and Figure 113), it can be seen that the data around 900 cm^-1 is noise-dominated, and so there is no need to look at spectral plots in that range (except to verify that the data is noise-like) – there is no information contained in them. However, the spectra from wavenumbers 1504 and 2355, seen in Figure 120 and Figure 121, do show a good match to the von Karman model. The 1504 wavenumber power spectrum has an order of magnitude worth of data above the noise floor, whereas the 2355 wavenumber power spectrum is less pronounced. As might be expected from the correlation plot for January 14, 2008, Data Set 2 (Figure 113), there is not much signature above the noise in the power spectrum. (Note that the scales for these and the subsequent power spectral plots are not necessarily the same. It is best to consider the height of the low frequency

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portions above the noise floor seen in the higher frequencies.) The data from wavenumber 2355 is noise-dominated, whereas there is a very weak signature at wavenumber 1520 (Figure 122), although it could easily be argued that this is not statistically significant.

Figure 120. Power spectrum and fits for wavenumber 1504 from 14 Jan 2008, Data Set 1, 2^nd 5000 points.

Figure 121. Power spectrum and fits for wavenumber 2355 from 14 Jan 2008, Data Set 1, 2^nd 5000 points.

Figure 122. Power spectrum and fits for wavenumber 1520 from 14 Jan 2008, Data Set 2, 1^st 5000 points.

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The term “statistically significant” in this context is explained as such: As we have seen above, the distribution functions for the de-trended radiances are well-matched to a Gaussian one. If one takes the power spectrum of Gaussian-distributed data, it has an exponential distribution. (Do not confuse what we are discussing here with “Gaussian white noise.” In the latter case, the data are uncorrelated, whereas in this case they can be.) That is, at each frequency, the power spectral level is distributed exponentially. Specifically, if we had multiple independent radiance time series that came from the same Gaussian distribution, then the spectral level at each frequency would represent a sample from an exponential distribution with mean and standard deviation values equal to the spectral level (definition of an exponential distribution). Unfortunately, this means that the power spectrum at each frequency has 100% noise – the standard deviation is equal to the mean. Therefore, at the lower frequencies, where the spectral level is higher, the errors are larger in magnitude (though not in percentage). It is important to note that in this context, the term “noise” refers to atmospheric fluctuations, not instrument errors. The latter is typically uncorrelated, and gives rise to the white spectrum at the higher frequencies. (We are speaking theoretically here: we are not considering the case where there are correlated sources of error as we have seen above with trends.) In the power spectra presented here, spectral averaging was performed (i.e. 9 spectra were averaged for a 5000 point window), and so the standard deviation is less than the mean. Nevertheless, one must be cautious in the interpretation of power spectra, and the associated fits, given the previous discussion.

Consider now the data from January 15, 2008. As mentioned above, on this day there were numerous pilot and automated reports of turbulence over the mountains and on the lee-side of the mountains. We look at four time periods on this day: Data Set 1, 2^nd and 3^rd 5000 points (Figure 123 through Figure 128); and Data Set 2, 1^st and 2^nd 5000 points (Figure 129 through Figure 132).

As seen in the correlation plots, there is more “activity” in the wavenumber regions of interest.

Most of the spectra are similar in that there is approximately an order of magnitude of signal above the noise, and there are good fits to the von Karman model. Comparing similar plots from the 17^th (Figure 133 through Figure 136), we see consistent findings with the other days. A good comparison between January 14^th and January 15^th is seen in Figure 121 and Figure 128. Both are for wavenumber 2355 and are on the same scale. It can be seen that the data from the 15^th shows much more pronounced spectral amplitude at the lower frequencies. Both cases illustrate good fits to the von Karman model. Is this a sign of more turbulence on the 15^th? The evidence seems consistent with that hypothesis, but at it is not conclusive.

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Figure 123. Power spectrum and fits for wavenumber 961 from 15 Jan 2008, Data Set 1, 2^nd 5000 points.

Figure 124. Power spectrum and fits for wavenumber 1504 from 15 Jan 2008, Data Set 1, 2^nd 5000 points.

Figure 125. Power spectrum and fits for wavenumber 2355 from 15 Jan 2008, Data Set 1, 2^nd 5000 points.

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Figure 126. Power spectrum and fits for wavenumber 961 from 15 Jan 2008, Data Set 1, 3^rd 5000 points.

Figure 127. Power spectrum and fits for wavenumber 1504 from 15 Jan 2008, Data Set 1, 3^rd 5000 points.

Figure 128. Power spectrum and fits for wavenumber 2355 from 15 Jan 2008, Data Set 1, 3^rd 5000 points.

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Figure 129. Power spectrum and fits for wavenumber 1504 from 15 Jan 2008, Data Set 2, 1^st 5000 points.

Figure 130. Power spectrum and fits for wavenumber 2355 from 15 Jan 2008, Data Set 2, 1^st 5000 points.

Figure 131. Power spectrum and fits for wavenumber 1504 from 15 Jan 2008, Data Set 2, 2^nd 5000 points.

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Figure 132. Power spectrum and fits for wavenumber 2354 from 15 Jan 2008, Data Set 2, 2^nd 5000 points.

Figure 133. Power spectrum and fits for wavenumber 1504 from 17 Jan 2008, Data Set 1, 1^st 5000 points.

Figure 134. Power spectrum and fits for wavenumber 2354 from 17 Jan 2008, Data Set 1, 1^st 5000 points.

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Figure 135. Power spectrum and fits for wavenumber 1504 from 17 Jan 2008, Data Set 3.

Figure 136. Power spectrum and fits for wavenumber 2354 from 17 Jan 2008, Data Set 3.