TOWARDS F IELD T HEORY OF T URBULENCE APREPRINT Alexander Migdal Department of Physics, New York University 726 Broadway, New York NY 10003 November 26, 2021 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 δ(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. 1 Introduction Turbulence is well studied at a phenomenological level using numerical simulations of forced Navier-Stokes equations and fitting the data for distribution of various observables (such as moments of velocity and vorticity fields, as well as velocity circulation). The data suggest multi-fractal scaling laws implying some significant modifications of traditional Kolmogorov scaling by finite size vorticity structures with nontrivial distributions by shape, size and vorticity filling. The microscopic theory, such as an effective Hamiltonian for the Gibbs distribution in ordinary critical phenomena, is missing. It is as though we already know the Newtonian dynamics but do not yet know the Boltzmann distribution. We can simulate the Navier-Stokes equations and average over time, but we lack basic definitions of stationary statistics for vorticity or velocity fields. This statistics would be a fixed point of the evolution of the Hopf functional. If we knew such an analog of the Boltzmann law, we would be able to solve the theory analytically (at least in some extreme regime such as a large circulation limit for large loops). We would also have powerful Monte-Carlo methods with the Metropolis algorithm for fast simulation of this equilibrium statistics. In this paper we are trying to fill this gap. We construct the distribution of vorticity and velocity in three dimensions which is manifestly conserved in Euler dynamics, while describing a steady energy cascade. It involves a two-component Clebsch field, as well as two auxiliary fields: one Bose field and one Majorana Grassmann field, both transforming as vectors in physical space R 3 . In the WKB limit the tails of the PDF for velocity circulation Γ over large fixed loops C are controlled by a classical field φ cl a (r) (instanton) concentrated around the minimal surface bounded by C . The field is discontinuous across the minimal surface which leads to the delta function term for the tangent components of vorticity as a function of normal coordinate. The flux is still determined by the normal component of vorticity, which is smooth. arXiv:2005.01231v3 [hep-th] 13 May 2020