An integral approach to the k^(-1) scaling in turbulent velocity spectra

A k^{-1} power-law scaling is frequently observed in environmental turbulence spectra, particularly within atmospheric boundary layers and open-channel flows, yet its physical origin remains debated. Here, we present a theoretical formulation that reproduces that spectral regime by extending the generalized velocity spectrum model introduced by Jetti et al. (2024), which incorporates Nikora's (1999) eddy clustering hypothesis. Turbulence is modeled as a statistical superposition of eddy clusters spanning a continuum of scales, with the number density of clusters inversely proportional to their characteristic scale. This scale-invariant distribution naturally leads to a logarithmic form in the second-order structure function, consistent with the observed k^{-1} scaling.
The proposed formulation adopts an integral representation of the spectrum, where scale-dependent weights govern the transition between the energy-containing and inertial subranges. This approach provides a unified way of linking theoretical eddy hierarchy models to observed spectral features. Validation against coastal ABL and riverine data demonstrates the model's ability to reproduce the emergence of the k^{-1} regime under various conditions.