Towards Microscopic Theory of Decaying Turbulence

Alexander Migdal

Published 2023-04-26Version 1

We develop a quantitative microscopic theory of decaying Turbulence by studying the dimensional reduction of the Navier-Stokes loop equation for the velocity circulation. We have found a degenerate family of solutions of the \NS{} loop equation\cite{M93, M23PR} for the Wilson loop in decaying Turbulence in arbitrary dimension $d >2$. This family of solutions corresponds to a nonlinear random walk in complex space $\bC^d$, described by an algebraic equation between consecutive positions. The probability measure is explicitly constructed in terms of products of conventional measures for orthogonal group $SO(d)$ and a sphere $\bS^{d-3}$. We compute a prediction for the three-dimensional Wilson loop for a circle of radius $R$ as a universal function of $\frac{R}{\sqrt{2 \nu t}}$.