Examining Turbulent Flow Anisotropy: Insights from Simplified Variance Budget Analyses

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Anisotropy fundamentally defines the nature of turbulent flows in natural environments, engineering, and technology.  At large scales, turbulence rarely attains an energy state that is equipartitioned between the three velocity components or a state where turbulent stresses disappear. This deviation from isotropy underscores turbulence's essential role in promoting momentum transfer. In canonical boundary layers, turbulence anisotropy emerges primarily from two distinct mechanisms: streamwise energy injections through shear forces and the vertical modulation of turbulent kinetic energy by buoyancy, acting either as a source (unstable stratification) or sink (stable stratification) of turbulence kinetic energy. Close to a solid surface turbulence additionally experiences wall blocking, limiting the energy in the wall-normal direction. Furthermore, in complex terrain, a multitude of factors intricately modify the turbulence anisotropy, thus altering the applicability of traditional similarity scaling. Here, we use simplified Reynolds stress budgets to examine how stratification influences the normal stress components across a comprehensive range of measurement datasets from canonical to highly complex terrain. This reduced set of budget equations assume a balance between shear and buoyancy production and dissipation, and model the return to isotropy using a linear Rotta scheme adjusted by the isotropization of the production. This model provides expressions for normalized velocity variances as function of Richardson number, highlighting the change of anisotropy as stratification becomes progressively more dominant. In canonical terrain, the model is shown to capture the dependence of energy anisotropy on Richardson number away from the surface (heights above 60m), however, it fails in predicting energy anisotropy close to the surface. Furthermore, the increase of terrain complexity leads to a decoupling of the dependence of anisotropy and Richardson number not predicted by the model, and shows a consistent decrease of the contribution of streamwise  velocity variance and increase of spanwise velocity variance to the total TKE budget. Finally, a progressively more important wind turning with height with terrain complexity in neutral stratification causes near-surface turbulence to be more anisotropic over complex than over canonical terrain.  Our findings outline the nuanced role of terrain in shaping turbulence anisotropy, providing avenues for enhanced turbulence modeling and highlighting limitations of conventional approaches in complex environments.