Universal Return to Isotropy of Inhomogeneous Atmospheric Boundary Layer Turbulence

A recalcitrant problem in the physics of turbulence is the representation of the tendency of large-scale anisotropic eddies to redistribute their energy content with decreasing scales, a phenomenon referred to as return to isotropy. An unprecedented dataset of atmospheric turbulence measurements covering flat to mountainous terrain, stratification spanning convective to very stable conditions, surface roughness ranging over several orders of magnitude, and Reynolds numbers that far exceed the limits of direct numerical simulations and laboratory experiments was assembled for the first time and used to explore the scalewise return to isotropy. The multiple routes to energy equipartitioning among velocity components are shown to be universal once the initial anisotropy at large scales, linked to turbulence generation, is accounted for.

Figure 1

Example average scalewise return-to-isotropy trajectory in barycentric coordinates for METCRAX II NEAR tower data [24] with two-component axisymmetric bulk anisotropy. Circles represent the value of nondimensional timescale $\tau /T_{ϵ}$. The limiting states of anisotropy (blue text) and their respective ranges used for clustering (blue kites and a square emanating from the pure states) are indicated. Solid black line is the PSL.

Figure 2

Two-dimensional density plots of the scalewise anisotropy for (a) all data, and data clustered according to bulk anisotropy into (b) one-component, (c) two-component, (d) two-component axisymmetric, and (e) isotropic, for unstable (top) and stable (bottom) stratification. Colored circles show ensemble trajectories for each cluster.

Figure 3

Ensemble return-to-isotropy trajectories in (a) BAM and (b) AIM, for four clusters and two stratifications. Shading shows interquartile range. Black straight line is PSL, gray straight line is the line through the center of BAM, and curved gray lines are nonlinear trajectories of Ref. [30].

Figure 4

Ensemble scalewise eigenvalues corresponding to trajectories in Fig. 3 for unstable stratification: (a) $G({\tau }_n)={\lambda }_1({\tau }_n)-{\lambda }_2({\tau }_n)$, and (c) ${\lambda }_3({\tau }_n)$, and their scaled values normalized by (b) $G^{*}=0.54{\lambda }_1({\tau }_n=1)-0.04$ and (d) ${\lambda }_3^{*}=0.35{\lambda }_3({\tau }_n=1)+0.01$. The nondimensional timescale is defined as ${\tau }_n={\log }_{10}(\tau /T_{\epsilon })$.