From bifurcations to state-variable statistics in isotropic turbulence: internal structure, intermittency, and Kolmogorov scaling via non-observable quasi-PDFs

Abstract This article investigates the intrinsic link between skewness and statistical intermittency in velocity and temperature increments within homogeneous isotropic turbulence. The theoretical framework builds upon the author’s previously established closure schemes for the von Kármán–Howarth and Corrsin equations. A transition Taylor–scale Reynolds number is first estimated via a formal bifurcation analysis of the closed von Kármán–Howarth equation. A central thesis of this work is that while the nonlinearity of the Navier–Stokes equations is fundamentally responsible for the emergence of intermittency in velocity and temperature increments, it is insufficient on its own to recover the Kolmogorov scaling law. We demonstrate that the non–observability of bifurcation modes constitutes the missing conceptual link: the concomitant effect of nonlinearity and non–observability not only determines the Kolmogorov scaling and drives an intermittency that grows monotonically with the Taylor–scale Reynolds number, but also enables the analytical determination of the internal structure functions of velocity and temperature differences, along with their corresponding PDFs and statistics. By invoking the principle of the minimum number of parameters for statistical description, as introduced by Fisher (1922), we show that the entire statistics of velocity and temperature increments can be analytically derived. This approach relies on a decomposition into bifurcation modes which, being non-observable quantities, are governed by quasi-probability distribution functions (quasi-PDFs). By admitting negative values, the latter provide the formal mathematical basis required to also represent local energy backscatter. Notably, the analysis recovers the Kolmogorov law –specifically the scaling of the ratio between the velocity standard deviation and the Kolmogorov velocity as R λ 1 / 2 ${R}_{\lambda }^{1/2}$ – as a direct consequence of the non–observability of these modes. Furthermore, our analysis reveals that the bifurcation modes exhibit amplitudes whose third statistical moment scales as R λ − 3 ${R}_{\lambda }^{-3}$ . The results show excellent agreement with benchmark numerical and experimental data existing in the literature, confirming that the proposed bifurcation–based decomposition rigorously captures the multi–scale, intermittent nature of fully developed turbulence.