2020/05/04 by Alexander Migdal, Migdal, Alexander
Earth and Planetary Sciences · Engineering · Physics and Astronomy · #Chaotic Dynamics (nlin.CD) #FOS: Physical sciences #Fluid Dynamics (physics.flu-dyn) #Fluid Dynamics and Turbulent Flows #High Energy Physics - Theory (hep-th) #Mathematical Physics (math-ph) #Meteorological Phenomena and Simulations #Solar and Space Plasma Dynamics
paper · pdf · doi:10.48550/arxiv.2005.01231
openalex publication_date 2020/05/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.