Scalewise invariant analysis of the anisotropic Reynolds stress tensor for atmospheric surface layer and canopy sublayer turbulent flows

Anisotropy in the turbulent stress tensor, which forms the basis of invariant analysis, is conducted using velocity time series measurements collected in the canopy sublayer (CSL) and the atmospheric surface layer (ASL). The goal is to assess how thermal stratification and surface roughness conditions simultaneously distort the scalewise relaxation towards isotropic state from large to small scales when referenced to homogeneous turbulence. To achieve this goal, conventional invariant analysis is extended to allow scalewise information about relaxation to isotropy in physical (instead of Fourier) space to be incorporated. The proposed analysis shows that the CSL is more isotropic than its ASL counterpart at large, intermediate, and small (or inertial) scales irrespective of the thermal stratification. Moreover, the small (or inertial) scale anisotropy is more prevalent in the ASL when compared to the CSL, a finding that cannot be fully explained by the intensity of the mean velocity gradient acting on all scales. Implications to the validity of scalewise Rotta and Lumley models for return to isotropy as well as advantages to using barycentric instead of anisotropy invariant maps for such scalewise analysis are discussed.

Figure 1

The measured components of the anisotropy tensor $a_{ij}$ are shown as a function of the absolute value of the stability parameter $\left|\zeta \right|=\left|\right(z-d)/L|$ [(a), (b), (c), and (e)]. Measurements of the ASL (desert) are red and those of the CSL (forest) are blue. Circles show stable conditions, diamonds are used for near neutral stratification conditions, and crosses denote unstable conditions. The $a_{33}$ shown in panel (c) are significantly larger in the CSL compared to the ASL at a confidence level of 95%. The black dashed line shows the expected value for isotropic turbulence and the solid blue line in panel (e) shows a linear regression of $a_{33}$ for the CSL. The lower right panel (f) shows turbulent kinetic energy $k$ normalized with $u_{*}$ and the lower left panel (g) shows ${\sigma }_{u3}{({\sigma }_{u1}+{\sigma }_{u2})}^{-1}$ together with the expectation for near neutral conditions as dashed lines [5]. Note the larger ${\sigma }_{u3}{({\sigma }_{u1}+{\sigma }_{u2})}^{-1}$ for the CSL when compared to the ASL.

Figure 2

Normalized length scale $L_{u3}{z}^{-1}$ (a) and the length scale ratios $L_{u3}{L}_{u1}^{-1}$ (b) and $L_{u3}{L}_{u2}^{-1}$ (c) are shown as a function of the absolute value of the stability parameter $\left|\zeta \right|=\left|\right(z-d)/L|$. Measurements of the ASL (desert) are red and those of the CSL (forest) are blue. Unstable stratification is shown as crosses, near neutral as diamonds, and stable as circles. The dashed line in panels (b) and (c) shows $L_{u3}{L}_{u1}^{-1}=0.1$ reported from other experiments [57, 58].

Figure 3

Ensemble averaged of normalized structure function $\frac{1}{2}{D}_{11}{\overline{u_1{u}_1}}^{-1}$ (left column), $\frac{1}{2}{D}_{22}{\overline{u_2{u}_2}}^{-1}$ (middle column), and $\frac{1}{2}{D}_{33}{\overline{u_3{u}_3}}^{-1}$ (right column) are shown for the ASL (top, red) and CSL (bottom, blue). The black dotted line is $y=1$ and the black dashed line shows the slope $r^{2/3}$ for Kolmogorov scaling [Eq. (18)]. The error bars show the standard deviation of the ensemble.

Figure 4

Local isotropy attained by the ratios $D_{11}{D}_{33}^{-1}$ (left column) and $D_{22}{D}_{33}^{-1}$ (right column) for the ASL (top row, red) and CSL (bottom row, blue). The three lines show one example interval for stable (solid), neutral (dashed), and unstable (dotted) conditions. The black dashed line shows the expected ratio for locally isotropic turbulence based on K41.

Figure 5

The top row shows the trajectories of all 30 minute runs for the ASL (a) and the CSL (b) together with starting points color coded according to their stability class (black is unstable, dark grey is near neutral, and light grey is stable). The middle row shows return-to-isotropy trajectories in the BAM for three sample cases with unstable, neutral, and stable stratification of the ASL (c) and CSL (d) together with model trajectories [Eq. (24)]. The insets show the same three trajectories in the AIM representation. The bottom row shows the mean distance between modeled and measured trajectories in BAM, with the standard deviation as error bars, for the ASL (e) and CSL (f) together with the part of each deviation which can be explained by the measurement errors (black).

Figure 6

Anisotropy measures $F$ [left column, Eq. (14)] and $C_{\mathrm{ani}}$ [right column, Eq. (15)] are shown for ASL (top row, red) and CSL (bottom rom, blue) as an ensemble average with standard deviation across all $\zeta$ to highlight the role of surface roughness. The black dashed lines show three regimes defined by reaching 90% of maximum isotropy or 10% of anisotropy.

Figure 7

The starting scales of the return to isotropy $r_{\mathrm{ani}}$ (a) and $r_{\mathrm{ani}}{L}_{u3}^{-1}$ (c) and scales $r_{\mathrm{iso}}$ (b) and $r_{\mathrm{iso}}{L}_{u3}^{-1}$ (d) at which isotropy is reached are plotted against the temperature length scale $L_{uT}$. Circles indicate stable, diamonds near-neutral, and crosses unstable stratification. Blue symbols show the CSL over the forest canopy and red symbols the ASL over the desert surface.