Turbulence as Gibbs Statistics of Vortex Sheets

We study the vortex sheet solutions of the Euler equation, which correspond to the tangent discontinuity of the velocity field.

We observe that stationary flows correspond to the Hamiltonian's minimization by the tangent discontinuity density Γ. This observation means that the stationary flow represents the low-temperature limit of Gibbs distribution of the vortex dynamics. An infinite number of Euler conservation laws leads to a degenerate vacuum of this system, which degeneracy explains the complexity of the turbulent statistics and provides the relevant degrees of freedom (random surfaces).

We find an exact analytic solution with a spherical surface. This solution provides an example of the instanton advocated in our recent work, which is supposed to be responsible for the dissipation in the Navier-Stokes equation in the turbulent limit of vanishing viscosity at fixed energy flow. We further conclude that one can obtain the turbulent statistics from the Gibbs statistics of vortex sheets by adding Lagrange multipliers for the conserved volume inside closed surfaces, energy pumping, and energy dissipation via viscosity anomaly in the enstrophy. The effective temperature in our Gibbs distribution goes to zero as ν25ν25 in the turbulent limit, which opens the way for the quantitative theory of turbulence as low-temperature expansion in this Gibbs ensemble around minimal surfaces.

The instantons correspond to domain walls bounded by strings, studied in cosmology, and observed in liquid Helium. They are known to be topologically stable.