Stratification, turbulence organization, and pressure-strain effects on surface-layer turbulence anisotropy

TL;DR

This work addresses how shear and thermal stratification shape Reynolds-stress anisotropy in the daytime atmospheric surface layer (ASL) by combining velocity-variance budgets from multiple near-surface datasets with reduced and extended Rotta-type closures. It demonstrates that simple isotropic-redistribution closures fail near the wall and that both turbulent and pressure transport, plus rapid isotropization of production and buoyancy terms, are needed to reproduce observed energy partitioning, especially the rise of spanwise variance and the suppression of wall-normal variance in the dynamic-convective regime. The extended Model E—with transport and rapid terms and wall-blocking adjustments—consistently improves agreement with measurements across heights and sites, revealing the persistent anisotropy across roughly three stratification regimes and linking variance dynamics to turbulence organization and coherent-structure transitions. The findings highlight the nonlocal nature of pressure transport and the crucial role of rapid distortion in ASL anisotropy, offering guidance for improved closure schemes and LES parameterizations in high-Reynolds-number, stratified wall-bounded flows. Overall, the work advances understanding of anisotropy drivers in the ASL and informs the development of anisotropic, transport-inclusive closures for accurate momentum transport modelling.

Abstract

At large scales, the Reynolds stress tensor exhibits notable anisotropy, a key feature of all wall-bounded turbulent flows. Yet, how the drivers of this anisotropy evolve with shearing and thermal stratification in the atmospheric surface layer (ASL) remains a daunting challenge for theory and models alike. Here, the velocity variance budgets are used to explore the evolution of anisotropy in the daytime ASL close to the surface, region known to be problematic for large eddy simulations. A special focus is placed on the importance of slow and rapid pressure-strain correlations and the role of transport on partitioning the turbulent kinetic energy among the velocity components. Results obtained from near-surface observations of four datasets over flat and horizontally homogeneous terrain show persistent anisotropy over a wide range of flux Richardson numbers $R_{if}$ and wall-normal distances, and highlight the importance of different processes in three distinct flow regimes, roughly related to dynamic ($|R_{if}|\ll1$), dynamic-convective ($|R_{if}|\sim1$) and convective ($|R_{if}|\gg1$) regimes of the ASL. In particular, close to the surface in the dynamic-convective regime, a drop in wall-normal velocity variance and a substantial increase of spanwise velocity variance are shown to result from the increasing role of pressure transport and rapid distortion, related to turbulence organization. This behaviour is not captured by the classic Rotta closure but requires the inclusion of both rapid pressure-strain and transport terms. In all regimes wall blocking is found to influence turbulence close to the surface, thus requiring the adoption of an anisotropic Rotta model to accommodate its effects.

Figures (19)

• Figure 1: Degree of energy anisotropy $y_{B}$ as a function of flux Richardson number $R_{if}$ for the Cabauw tower data. Coloured points represent individual averaging periods for the four measurement heights, where the full coloured lines are the bin averages computed at logarithmically spaced $R_{if}$ and shading is the interquartile range. Solid black line corresponds to the predictions of reduced Model R (Eq. \ref{['eq:Reduced']} with $c = 0.9$), while the dashed black curve corresponds to the prediction of the reduced model with adjusted Rotta constant ($c = 3$) and added wall-blocking $[a_u,a_v,a_w] = [1.25, 1.15, 0.58]$ (Model Ra).
• Figure 2: Relation between the local stability parameter $\zeta=z/\Lambda$ and the flux Richardson number $R_{if}$ for the Cabauw tower as a function of the degree of anisotropy $y_B$ (colour). Dots correspond to observational averaging periods. Curves correspond to the different scaling relations for $\Phi_m$: full black curve Hogstrom96, dashed black curve KaderYaglom1990, dotted black curve the O'KEYPS equation LumleyPanofsky1964, and coloured full curve Mosso2024 that includes the degree of anisotropy into scaling (colour of the curves). Vertical coloured ranges separated by thin dotted lines correspond to the three subranges of KaderYaglom1990: dynamic (blue, $-\zeta < 0.04$), dynamic-convective (yellow, $-\zeta =[0.12 - 1.2]$), and convective (orange, $-\zeta > 2$)
• Figure 3: Terms of the TKE budget (colour) normalized by the dissipation rate as a function of $R_{if}$ for the average of 3 to 10 m levels at Cabauw, METCRAX, AHATS, and M2HATS towers. Lines correspond to logarithmically spaced bin averages, while the shading is the interquartile range. For variable names, see Eq. \ref{['eq:Reynolds']}. Note the significance of the vertical pressure transport $\Pi^t_{ww}$ with decreasing $R_{if}$ at all sites.
• Figure 4: Bin averages of a) streamwise $\overline{u'^2}/e$, b) spanwise $\overline{v'^2}/e$, and c) wall-normal $\overline{w'^2}/e$ velocity variance ratio as a function of $R_{if}$ for the Cabauw tower. Four measurement heights are shown in colours. Full lines show bin averages computed at the logarithmically spaced $R_{if}$, while shading corresponds to the interquartile range. Full black curve corresponds to the predictions of reduced Model R (Eq. \ref{['eq:Reduced']}) with $c = 0.9$, while the dashed curves correspond to the predictions of the reduced Model Ra with an adjusted Rotta constant $c = 3$ and wall-blocking added (Eq. \ref{['eq:Reduced_anis']}). Here the wall-blocking and Rotta constants were obtained from a robust linear fit for the upper levels (60 - 180 m, black) and first level (3 m, brown) separately.
• Figure 5: Bin averages of (a,d,g,j) streamwise $\overline{u'^2}/e$, (b,e,h,k) spanwise $\overline{v'^2}/e$ and (c,f,i,l) wall-normal $\overline{w'^2}/e$ velocity variance ratio as a function of $R_{if}$ for Cabauw (a-c), METCRAX (d-f), AHATS (g-i), and M2HATS (j-l) towers. Full coloured lines are logarithmically spaced bin averages for each measurement height (colours), while shading is the interquartile range. Black points are bin averages of all the data, irrespective of height. The dashed curve corresponds to the predictions of Model Ra (Eq. \ref{['eq:Reduced_anis']}) where the model anisotropy and Rotta constants were obtained from a robust linear fit for the upper levels (60 - 180m, black) and first level (3m, brown) of the Cabauw dataset. Vertical dashed line corresponds to $-R_{if} = 1$