  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
$\Delta \vec v_t =\vec \nabla \Gamma$. This observation means that the steady
flow represents the low-temperature limit of the Gibbs distribution for vortex
sheet dynamics. The normal displacement $\delta r_\perp$ of the vortex sheet as
a Hamiltonian coordinate and $\Gamma$ 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.
