We develop a new theory of circulation statistics in strong turbulence (ν → 0 in the Navier-Stokes equation), treated as a degenerate fixed point of a Hopf equation. We use spherical Clebsch variables to parametrize vorticity in the stationary singular Euler flow. This flow has a tangent velocity gap due to the phase gap in the angular Clebsch variable across a discontinuity surface bounded by a stationary loop C in space. We find a circular vortex with a singular core on this loop, regularized as a limit of the Burgers vortex. We compute anomalous contributions to the Euler Hamiltonian, helicity, and the energy flow, staying finite in the vanishing viscosity limit. The normalization constant in the spherical Clebsch variables is determined from the energy balance between incoming flow and anomalous dissipation. The randomness (spontaneous stochastization) comes from the Gaussian fluctuations of a background velocity due to random locations of remote vortex structures. Assuming weak fluctuations of the background velocity field, we compute the probability distribution of velocity circulation Γ, which decays exponentially with pre-exponential factor 1/√Γ in perfect match with numerical simulations of conventional forced Navier-Stokes equations on periodic lattice 8K3. We also compute effective multifractal indexes for the tails of velocity circulation probability density as a function of conditional probability below that tail. The anomalous dimensions are independent of this probability and decrease as inverse powers of the logarithm of the size of the loop.
This video was produced by the SITE Research Center at New York University, as part of their talk series.
