Towards Field Theory of Turbulence
Abstract
We revisit the problem of stationary distribution of vorticity in threedimensional 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 $\delta(z)$ 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 BiotSavart 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.
 Publication:

arXiv eprints
 Pub Date:
 May 2020
 arXiv:
 arXiv:2005.01231
 Bibcode:
 2020arXiv200501231M
 Keywords:

 High Energy Physics  Theory;
 Mathematical Physics;
 Nonlinear Sciences  Chaotic Dynamics;
 Physics  Fluid Dynamics
 EPrint:
 36 pages, 8 figures, two Appendixes (minimal surfaces and Instanton analysis added). Second revision: added detailed discussion of invariance and timeindependence 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 This paper is superseded by my new review paper "Clebsch Confinement and Instantons in Turbulence" arXiv:2007.12468