On the mechanism and conditions for <i>k</i>⁻¹ scaling in turbulent velocity spectra

We present a theoretical approach that derives the wavenumber $k^{-1}$ spectral scaling in turbulent velocity spectra using random field theory without assuming specific eddy correlation forms or Kolmogorov’s inertial-range scaling. We argue for the mechanism by Nikora (1999 Phys. Rev. Lett. 83 (4), 734), modelling turbulence as a superposition of eddy clusters with eddy numbers inversely proportional to their characteristic length scale. Statistical mixing of integral scales within these clusters naturally yields the $k^{-1}$ scaling as an intermediate asymptotic regime. Building on the spectrum modelling introduced in Jetti et al. (2025 b Z. Angew. Math. Physik. 74 (3), 123), we develop and apply an integral formulation of the general velocity spectrum that reproduces the $k^{-1}$ regime observed in field spectra, thereby bridging theoretical derivation and empirical observations. The model is validated using wind data at a coastal site, and tidal data in a riverine environment where the –1 scaling persists beyond the surface layer logarithmic region. The results confirm the robustness of the model at various flow conditions, offering new insights into the spectral energy distribution in geophysical and engineering flows.