Atmospheric Small-Scale Turbulence from Three-Dimensional Hot-film Data

Abstract

The behavior of small-scale atmospheric turbulence is investigated using the three-dimensional Canopy Horizontal Array Turbulence Study hot-film data. The analysis relies on an in situ calibration versus simultaneous sonic anemometer measurements. The calibration is based on King’s law and geometric relationships between the individual hot-film sensors, and is able to account for the errors associated with sensors’ misalignment and the high turbulence intensity. The details of the calibration are provided, and its performance is validated by comparing results of spectra and structure functions with standard wind-tunnel data and model spectra. A single 3 h block of data was selected, containing 33 subblocks of 2 min data without error gaps, whose statistics were averaged to provide smooth results. These data were measured above canopy under stable conditions, and correspond to a Taylor Reynolds number $Re_\lambda \approx 1550$. The agreement with wind tunnel results for a similar $Re_\lambda$ and with model predictions provides a validation for the in situ calibration method applied. Furthermore, the results indicate a presence of the bottleneck effect in the lateral and vertical spectra, in addition to a lack of inertial range in the second-order structure function due to the low Reynolds number. An additional analysis of the effect of Reynolds number on the inertial range is provided using atmospheric data from the literature.

Appendix: Illustration of the Calibration Procedure and Additional Statistics

The data processing required two steps, an initial data selection and a final data selection. The initial data selection consisted of going through all available data, and selecting the 30-min blocks that passed two quality criteria: mean wind direction relative to the hot-film’s orientation smaller than $10^\circ$ and a final number of 30-sec subblocks of at least 25. This stage was repeated correcting for angles $80^\circ \le \theta _z \le 90^\circ$, when the value of $\theta _z = 85^\circ$ was selected.

The final data selection consisted of a single 3-hour block, from which 33 subblocks of 2-min data without gaps were identified. These data were selected in order to increase the subblock length and the statistical convergence in the average between subblocks. It was also the only long period of several consecutive blocks that passed the initial data screening. See a summary in Fig. 12.

Figure 12 also illustrates the concept of blocks and subblocks. While the original data (hot-film voltage and sonic velocity) was separated in blocks (30-min and 3 hours long for the initial and final data selection, respectively), the final hot-film velocity presented gaps in the time series. Consecutive periods of data without gaps (30-sec and 2-min long for the initial and final data selection, respectively) were then selected as subblocks, which can start at the beginning of a block, immediately after a gap or after another subblock.

Fig. 12

Summary of data selection and illustration of the blocks and subblocks used in this study. While the block consisted of a fixed period in the original data (hotfilm voltage and sonic velocity, 30-min and 3-hours long), the subblocks were formed in the final hot-film velocity series by selecting consecutive periods without gaps

Figure 13 shows the mean and standard deviation of the three velocity components for each subblock, compared to the 3-hour value and comparing between sonic and hot-film values. Results show that the flow presented a slight increase in mean velocity and standard deviation over time, but it can be considered approximately steady-state, justifying the average over subblocks of all statistics presented in this study.

Fig. 13

Block average (3 hours, solid lines) versus subblocks statistics (2-min, circles). Filled (open) circles are sonic (hot-film) data. Mean (top left), standard deviation (top right) and delta parameter (bottom) of the streamwise (black), spanwise (red) and vertical (blue) velocities

In order to compare sonic and velocity data directly, it is important to filter both data at the frequencies in which they are comparable. As discussed in Sec. 2.5, ideally, at most a 0.3 Hz cut-off frequency should be used (see Fig. 3). However, a 2-min time series at 0.3 Hz of frequency has only 36 data points, which are not statistically meaninful. Instead, we filtered the two datasets at 2 Hz, see Fig. 14. Notice that, at this frequency, the sonic data already diverges from the hot-film data, which can be seen in Fig. 14. Furthermore, we estimated the delta parameter as a quantitative measurement of the difference between the two time series (Kit and Liberzon 2016; Goldshmid et al. 2022). The delta parameter is defined as:

$$\begin{aligned} \delta ^i = \left\{ \frac{1}{N} \sum _{j = 1}^{N} ({\widetilde{u}}_{h,i}^{(j)} - {\widetilde{u}}_{s,i}^{(j)})^2 \right\} ^{1/2}, \end{aligned}$$

(16)

where ${\widetilde{u}}_i^{(j)}$ is the $j^{th}$ value of the velocity component i filtered at 2 Hz and rescaled by their mean and standard deviation values of the subblock (subscripts s and h are for sonic and hot-film, respectively). The values of $\delta ^i$ are presented in Fig. 13, varying from 0.3 to 0.7. These values are relatively high but of the same order of magnitude of the values obtained by Kit and Liberzon (2016) and Goldshmid et al. (2022) using both traditional and neural network calibration. We expect that in a more favorable setup, such as sonic pointing to the streamwise direction, the sonic velocity would correspond to a better “ground truth” for the velocity fluctuation and the $\delta ^i$ values would be lower.

Fig. 14

Time series of the filtered and rescaled velocity vector (filtered at the 2 Hz frequency, rescaled by their mean and standard deviation of the subblock) of the hot-film (black) and sonic (red) data, for the last 2 min block