Fine-scale turbulence measurements with a pulsed lidar

The observations in Pauscher et al. (2016, Appendix C) motivate a more detailed inves-tigation of fine-scale turbulence measurements using lidars. This analysis is presented in Pauscher et al. (2017b, Appendix D) and Pauscher et al. (2016, Appendix C). As outlined in Section 2.1.3, knowledge of the spatial averaging is key to a derivation of small-scale turbulence from lidar measurements. Figure 3.6 compares the spectral transfer function ϕ²(k) derived from theoretical consideration to the observations ob-tained from measurements.

The theoretical averaging functions in Figure 2.2a show a good agreement with the experimentally determined ϕ²(k) from the cross-spectral method (for details on the method see Pauscher et al. (2017b, Appendix D)). The difference between the measured pulse and the assumption of a Gaussian pulse shape is rather small, if σ_l = 15.8 m is chosen. In contrast, ϕ²(k) obtained from the spectral method shows a much slower drop-off. It roughly corresponds to σl = 9 m, which is significantly smaller than the theoretical considerations. An observation which was also made by Angelou et al.

(2012) for a continuous wave lidar and is consistent with the observations in Pauscher et al. (2016, Appendix C) (see Erratum).

Due to its relation to the spectral density in the inertial sub-range, the dissipation rate of turbulent kinetic energy is well suited to model and characterise small scale turbulence. Pauscher et al. (2017b, Appendix D), therefore, focus on the estimation of from a pulsed lidar. Three methods are investigated in detail. First and as a baseline scenario, the ’classical’ spectral method exploiting the -5/3-slope of the spectrum in the inertial sub-range is applied. For the second method an approach originally proposed by Bouniol et al. (2003) and O’Connor et al. (2010) based on short term variances was improved and corrected for the contribution of larger turbulence scales. Compared to the original formulation a significant bias towards an overestimation of could be removed. The third method is based on the spatial structure function (Kristensen et al., 2011) and is the first experimental evaluation of this method. For details of the individual approaches the reader is referred to Pauscher et al. (2017b, Appendix D).

Figure 3.6. Comparison of ϕ²(k) obtained from theory in comparison to the experi-mentally determined values during the -Experiment; only wind directions ±5^◦ the beam direction of the lidar were used; the theoretical functions (dashed blue and orange lines) correspond to the spatial averaging functions in Figure 2.2.

In general, all methods produce acceptable results and_lidar shows a good correlation to the measurements from the reference sonic (Figure 3.7). The analysis of the statis-tical error in Pauscher et al. (2017b, Appendix D) showed that the statisstatis-tical random errors in the different methods are also similar and is between approx. 10-30 %. The majority of the scatter in the comparison in Figure 3.7 is thus likely to stem from the noise in the measurements. For all methods an average underestimation of the _sonic by_lidar which is in the order of magnitude of the random error can be observed. For the spectral method this underestimation a first order correction using|ϕ(k)|³ (see also Equation 2.9) can be applied. For the wave-number interval k = 0.0454 - 0.1 m⁻¹ the median underestimation can be reduced from 39 % to 2 % applying this correction.

For the short term variance method the underestimation is likely to be caused by the fact that scales outside the inertial sub-range contribute to the short term variances.

In case of the structure function method the underestimation (19 %) might be related to uncertainties in ϕ(k) and correlated noise along the measurement beam. For the structure-function and the short-term variance method a removal of the noise for small , which are connected to week turbulence, is important.

From Figure 3.7 it is difficult to make a judgement, which of the methods is the best to estimatefrom lidar measurements. The differences between the individual methods rather lie in their applicability to different measurement scenarios. The spectral as well as the short term variance method require a fast sampling rate at the same point to allow for the calculation of the temporal statistics at wave number intervals which lie within the inertial sub-range. Currently available pulsed lidar technology requires

3.2. Turbulence measurements using lidars

(a) (b)

(c)

Figure 3.7. Comparison of as estimated from the reference sonic and the lidar mea-surements for(a)the spectral method; the colors indicate two different wave number intervals (given in the legend) which were used for the estimation of _lidar; for the red line_lidar is corrected using|ϕ(k)|³ (b)_lidar based on short term variances; blue:

noise was removed using the auto covariance method (Frehlich, 2001); red: no noise removal;(c)_lidar based on the structure function method; color code see(b); darker markers and dashed line indicate the binned median values.

(a) (b)

Figure 3.8. (a) median derived from the reference sonic and the lidar during the -Experiment using the spectral method for different wave-number intervals; (b) influence of the choice of the separation distances (r) on the median lidar derived from the structure-function method for the theoretically derived ϕ(k) (σ_l = 15.8 m) and the experimentally determined ϕ(k) (σ_l = 9 m); the black line indicates the median value derived from the sonic measurements using the spectral method; figures taken from Pauscher et al. (2017b, Appendix D).

accumulation times of at least in the order of 0.5 - 1 s. Moreover, currently available scanner heads usually have a maximum speed of approx 30^◦s⁻¹. The spectral and short-term variance method are mainly suitable for staring configurations and, thus, primarily interesting for research applications. The short-term variance method should only be applied in the corrected version presented in Pauscher et al. (2017b, Appendix D). The downside of the corrected method is, that the equations become more complex and it looses some of the previous appeal, which was its simplicity.

One of the advantages of the spectral method is, that the spectral analysis of the measurement data allows a better judgement, whether the scales used to derive are contained within the inertial sub-range and at which scale noise will become dominant.

An example of such an analysis is shown in Figure 3.8a. The first minimum in lidar measurements (black crosses) is reassuring that the signal at high wave numbers is not dominated by noise.

The structure function method exploits the capability of a pulsed lidar to sample multiple points of the atmosphere along the measurement beam quasi-simultaneously.

This reduces the need for a high sampling rate at the same point. It is, thus, more suitable for more complex scan patterns like DBS, plane-position indicator or range-height indicator scans and presents the most flexible option. One thing to keep in mind when applying the structure function method is the increased sensitivity of the estimate of to uncertainties in ϕ(k) with decreasing r (Figure 3.8b). This consideration needs