Geometric Dynamics of Turbulence

Abstract: Turbulent flows exhibit robust universal features -- including logarithmic mean velocity profiles, scale-invariant energy spectra, anisotropy constraints and strongly non-local transport -- yet a unifying dynamical principle underlying these phenomena remains elusive. We show here that turbulence can be organized around an emergent oscillatory degree of freedom governing the Reynolds stress. Starting from the exact non-local representation of the stress in terms of a propagator, we demonstrate that the spectral structure of the response contains a dominant complex-conjugate pair of poles, implying an effective oscillator coupled to the mean flow. In wall-bounded turbulence, the near-wall Airy structure selects and stabilizes this mode through non-local feedback, yielding the logarithmic velocity profile and fixing the asymptotic von Kármán constant, $κ\simeq 0.39$. In homogeneous turbulence, the same dynamical picture closes the inertial-range energy balance and yields the Kolmogorov constant as $C_k=2/[3(1-2^{-2/3})]\simeq 1.80$ at leading order. The resulting formulation leads to a closed tensorial set of mean-field equations in three spatial dimensions, significantly cheaper than direct numerical simulation yet rich enough to support geometry-dependent reduced dynamics interpretable as distributed networks of interacting oscillators. The associated phase field admits a geometric description connected with Berry phase, anisotropy evolution on the Lumley triangle, and an effective gauge-covariant structure of phase transport. These results suggest that turbulence is governed not by an algebraic closure, but by a dynamical and geometric organization of the mean stress.