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Statistical equilibrium of circulating fluids

Alexander Migdal
Dec 2022
摘要
We develop a new theory of circulation statistics in strong turbulence,treated as a fixed point of a Hopf equation. Strong turbulence is the limit ofvanishing viscosity in the Navier-Stokes equation. We use spherical Clebschvariables to parametrize vorticity in the stationary singular Euler flow with aphase gap in the angular Clebsch variable across a discontinuity surfacebounded by a stationary loop $C$ in space. We find a circular vortex with asingular core on this loop, regularized as a limit of the Burgers vortex. Wefind anomalous contributions to the Euler Hamiltonian and the energy flow,staying finite in the vanishing viscosity limit. The normalization constant inthe spherical Clebsch variables is determined from the energy balance betweenincoming flow and anomalous dissipation. As a result, we compute the PDF of velocity circulation $\Gamma$, whichdecays exponentially with pre-exponential factor $1/\sqrt{\Gamma}$ in perfectmatch with numerical simulations of conventional forced Navier-Stokes equationson periodic lattice $16K^3$.
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