Towards Field Theory of Turbulence

Abstract

We revisit the problem of stationary distribution of vorticity in three-dimensional turbulence. Using Clebsch variables we construct an explicit invariant measure on stationary solutions of Euler equations with the extra condition of fixed energy flow/dissipation. The asymptotic solution for large circulation around large loops is studied as a WKB limit (instanton). The Clebsch fields are discontinuous across minimal surface bounded by the loop, with normal vorticity staying continuous. There is also a singular tangential vorticity component proportional to $\hat{I}´(z)$ where $z$ is the normal direction. Resulting flow has nontrivial topology. This singular tangent vorticity component drops from the flux but dominates the energy dissipation as well as the Biot-Savart integral for velocity field. This leads us to a modified equation for vorticity distribution along the minimal surface compared to that assumed in a loop equations, where the singular terms were not noticed. In addition to describing vorticity distribution over the minimal surface, this approach provides formula for the circulation PDF, which was elusive in the Loop Equations.

36 pages, 8 figures, two Appendixes (minimal surfaces and Instanton analysis added). Second revision: added detailed discussion of invariance and time-independence of my measure in GBF space of Clebsch fields to answer questions from sophisticated readers. This paper is superseded by my new review paper "Clebsch Confinement and Instantons in Turbulence" arXiv:2007.12468