Nature of Streaky Structures Observed with a Doppler Lidar

Abstract

Observations using a three-dimensional scanning coherent Doppler lidar in an urban area revealed the characteristics of streaky structures above a rough, inhomogeneous surface for a high-Reynolds-number flow. The study focused on two points: (1) the frequency of occurrence and conditions required for the presence of streaky structures, and (2) the universal scaling of the spacing of streaky structures ($\lambda )$. The horizontal snapshots of the radial velocity were visually classified into six groups: Streak, Mixed, Fishnet, No streak, Front, and Others. The Streak category accounted for more than 50% of all possible flows and occurred when the horizontal wind speed was large and the atmospheric stratification was near-neutral. The spacing ($\lambda )$ was estimated from the power spectral density of the streamwise velocity fluctuations along the spanwise direction. The spacing $\lambda$ decreased with an increase in the local velocity gradient. Furthermore, it was revealed that the local velocity gradient normalized by the friction velocity and the boundary-layer height ($z_i )$ comprehensively predicts $\lambda /z_i$ under various experimental and environmental conditions, in terms of the scale of motion (i.e., indoor and outdoor scales), thermal stratification (i.e., from weakly unstable to stable stratification), and surface roughness (i.e., from flat to very rough surfaces).

Appendices

Appendix 1: Investigation of Objectivity of the Visual Classification

To evaluate the objectivity of the visual classification, we calculated the statistics of the radial velocity distribution, which simply expresses the characteristics of the visual criteria in terms of (1) the convergence line, (2) the shapes of the boundary between positive and negative radial velocities, and (3) the homogeneity of the spatial pattern of radial velocity distribution. Characteristic (1) was expressed using the bulk convergence normalized by the horizontal wind speed $({\textit{convergence}}/U)$; $\textit{convergence}$ is the sum of the radial velocities in a circle with a radius of 2025 m divided by the length of the circle. Characteristic (2) was expressed as the variance of the wind direction estimated by the velocity azimuth display method for each radius from 325 to 2025 m. The variance was calculated as the magnitude of a vector that was composed of unit vectors, with the angle of wind direction for each radius. Characteristic (3) was expressed by the standard deviation of the radial velocity fluctuation in a circle with a radius of 2025 m. This fluctuation was calculated as the difference between the radial velocity and the radial component of the mean wind speed estimated by the velocity azimuth display method. Figure 10 shows scatter plots for a combination of the radial velocity statistics coloured according to a visual classification. As expected, the convergence / U values of the category Front were much larger than for all other classes (Fig. 10a). Furthermore, the plots in Fig. 10b are clustered for each category of the visual classification. The plots of $\sigma _{\theta }$ and $\sigma _{v_{r^{\prime }}}$ for each category agree with the criteria in the visual classification. The categories Streak, Mixed, and Fishnet had a larger $\sigma _{v_{r^{\prime }}}$ than those of No streak and Others. In addition, the categories Streak and No streak had a smaller $\sigma _{\theta }$ than those of Mixed, Fishnet, and Others. This visual classification was supported by the objective statistics of the flow pattern. Figure 10c is the same as Fig. 10b, but only the category Streak is plotted, and the cases that were used for the analysis of the spacing $\lambda$ are highlighted using different symbols. The plots used for the analysis of the spacing $\lambda$ are distributed around the upper left of the figure, in which there is almost no contamination of the plots for the other categories. This demonstrated that the cases used for the analysis of the spacing $\lambda$ include only typical streaky flow patterns, which were classified as the category Streak.

Fig. 11

Relationships between the non-dimensional spacing of streaky structures and the length scales. $\lambda$ is the spacing of streaky structures, $l_\mathrm{minpeak}$ is the separation distance from the minimum peak of the two-point correlation, $z_i$ is the boundary layer height, and DL Doppler lidar. The symbols are as follows: filled circle DL, open diamond LES_city (Huda et al. 2016)

Appendix 2: Relationship Between the Spacing of Streaky Structures Calculated by the Power Spectral Density and Length Scales Calculated Using a Two-Point Correlation

The spacing $\lambda$ was calculated from the power spectral density as mentioned in Sect. 3.5. However, the past studies cited above (Table 3) calculated the length scale using a two-point correlation. Therefore, we introduced a function to convert the length scale, which is estimated from the two-point correlation, to the spacing $\lambda$, based on the database of the Doppler lidar and LES_city simulation. The spacing $\lambda$ and the length scale of the LES_city simulation were calculated from the streamwise velocity fluctuations in an area of the same size as that observed by the Doppler lidar. Figure 11 shows the relationships of the non-dimensional spacing $\lambda$ and the length scales from the two-point correlation for the cases of Doppler lidar and the LES_city simulation. The case of the Doppler lidar represents the ABL, which is affected by buoyancy, while that of the LES_city simulation represents the turbulent boundary layer, which is driven by only shear. The length scale $l_{minpeak}$ is the separation distance from the minimum peak of the two-point correlation. It shows a linear relationship and follows a single line regardless of choice of the Doppler lidar or LES. Because the choice of the intercept of the regression line, i.e., zero or the best-fit value, made little difference to the estimation of the spacing $\lambda$, the regression line without the intercept was used in this study, assuming that both the spacing $\lambda$ and $l_{minpeak}$ become zero simultaneously.

Appendix 3: Investigation of the Spurious Correlation in Fig. 8

Some readers may observe that the correlation between $\lambda /z_{i}$ and ${({\Delta U}/{\Delta z})}/{(u_{*}/z_{i})}$ in Fig. 8 is a spurious correlation (Pearson 1896) due to the same denominator ($z_{i})$ being present in both parameters. Hence, the validity of the scaling in Sect. 4.3 is discussed here. Figure 12 shows the same plots as shown in Fig. 8, but the colour of the plots represents the value of $z_{i}$. In the spurious correlation, the distance from the origin of the coordinate to each plot is inversely proportional to the value of $z_{i}$, whereas the equivalent distance of the plots in Fig. 12 is not proportional to the value of $z_{i}$ especially in LES_flat, LES_city simulations and WT_cube results. This supports the validity of the scaling in Sect. 4.3.

Fig. 12

Relationship between the non-dimensional spacing of streaky structures and non-dimensional wind shear. The colour of plots represents value of $z_i$. DL Doppler lidar. The symbols are as follows: open circle DL, open square LES_flat (Lin et al. 1997), open diamond LES_city (Huda et al. 2016), open inverted triangle WT_cube (Takimoto et al. 2013). DL plots include only cases whose mean horizontal wind speed was greater than 3.5 m s$^{-1}.$ The mean horizontal wind speed was taken as the 30-min average of the streamwise velocity component observed by a sonic anemometer at 25 m above ground level