A Geometric Foundation for the Universal Laws of Turbulence

Abstract

We propose a theoretical framework where the dissipative structures of turbulence emerge from microscopic path uncertainty. By modeling fluid parcels as stochastic tracers governed by the Schr\"odinger Bridge (SB) variational principle, we demonstrate that the Navier--Stokes viscous term is a natural linear, second-order macroscopic operator consistent with isotropic microscopic diffusion. We derive two foundational pillars of turbulence from this single principle. First, we show that the Kolmogorov scale $\eta \sim (\nu^3/\epsilon)^{1/4}$ is not merely a dimensional necessity but a geometric diffusion horizon: it is the scale at which the kinetic energy of a fractal trajectory, scaling as $k \sim \nu/\tau$, balances the macroscopic dissipation rate. Second, we show that the universal law of the wall is the stationary solution to this stochastic process under no-slip constraints. The logarithmic mean profile arises from the scale invariance of the turbulent diffusivity, while finite-Reynolds-number corrections emerge as controlled asymptotic expansions of the stochastic variance. This framework offers a physically grounded derivation of turbulent scaling laws that complements and extends purely phenomenological dimensional analysis.