Scale-Dependent Anisotropy in Two-Dimensional Spectra of Wind-Turbine Wake Boundary-Layer Flows

Abstract. Accurate characterization of anisotropic turbulence in wind-turbine wakes requires spectral models that retain both directional information and its variation across scales. Although one-dimensional spectra are widely used to compare measured and simulated wake turbulence, they collapse the underlying wavevector structure and therefore cannot fully describe the geometry of anisotropic energy organization. We propose a two-dimensional spectral framework based on a bidirectional, scale-dependent tensorial remapping of the planar wavenumber domain. The model represents the strong anisotropy observed at energy-containing scales and its gradual relaxation toward smaller-scale isotropic behavior. This transition is described by two directional stretch factors that depend jointly on the streamwise and spanwise wavenumbers. The corresponding Jacobian correction ensures that the spectral density is transformed consistently under wavevector mapping. The scale dependence of the directional stretching is represented by using constrained Bernstein polynomial surfaces. A cumulative softplus parameterization guaranties boundedness, smoothness, and monotonic decay throughout the optimization. The model is applied independently to large-eddy simulation datasets representing wind-turbine wakes under high- and low-turbulence inflow conditions, together with a reference atmospheric boundary-layer case without turbine forcing. Parameters are estimated using a staged, coarse-to-fine constrained optimization of the planar spectra of the streamwise and wall-normal velocity components. Component-energy constraints, Jacobian admissibility, and surface-smoothness regularization are included to maintain physical consistency and numerical robustness. The fitted spectra are also used to generate statistically equivalent synthetic turbulent fields to evaluate spatial correlations, integral scales, structure functions, and one-point statistics. The resulting framework provides an interpretable and computationally practical method for quantifying how turbine wakes and inflow turbulence intensity modify the directional, scale-dependent organization of turbulent energy.