Vortex Sheet Turbulence as Solvable String Theory

Abstract

We study steady vortex sheet solutions of the Navier-Stokes in the limit of vanishing viscosity at fixed energy flow. We refer to this as the turbulent limit. These steady flows correspond to a minimum of the Euler Hamiltonian as a functional of the tangent discontinuity of the local velocity parametrized as $Δ\vec v_t =\vec \nabla Γ$. This observation means that the steady flow represents the low-temperature limit of the Gibbs distribution for vortex sheet dynamics. The normal displacement $δr_\perp$ of the vortex sheet as a Hamiltonian coordinate and $Γ$ as a conjugate momentum. An infinite number of Euler conservation laws lead to a degenerate vacuum of this system, which explains the complexity of turbulence statistics and provides the relevant degrees of freedom (random surfaces). The simplest example of a steady solution of the Navier-Stokes equation in the turbulent limit is a spherical vortex sheet, which we investigate. This family of steady solutions provides an example of the Euler instanton advocated in our recent work, which is supposed to be responsible for the dissipation of the \NS{} equation in the turbulent limit. We further conclude that one can obtain turbulent statistics from the Gibbs statistics of vortex sheets by adding Lagrange multipliers for the conserved volume inside closed surfaces, the rate of energy pumping, and energy dissipation. The effective temperature in our Gibbs distribution goes to zero as $\mbox{Re}^{-\frac{1}{3}}$ with Reynolds number in the turbulent limit. \textbf{The Gibbs statistics in this limit reduces to the solvable string theory in two dimensions (so-called $c=1$ critical matrix model)}. This opens the way for non-perturbative calculations in the Vortex Sheet Turbulence, some of which we report here.