  This manuscript is a draft of work in progress, meant for network
distribution only. It will be updated to a formal preprint when the numerical
calculations will be accomplished. In this draft we develop a consistent
closure procedure for the calculation of the scaling exponents $\zeta_n$ of the
$n$th order correlation functions in fully developed hydrodynamic turbulence,
starting from first principles. The closure procedure is constructed to respect
the fundamental rescaling symmetry of the Euler equation. The starting point of
the procedure is an infinite hierarchy of coupled equations that are obeyed
identically with respect to scaling for any set of scaling exponents $\zeta_n$.
This hierarchy was discussed in detail in a recent publication [V.S. L'vov and
I. Procaccia, Phys. Rev. E, submitted, chao-dyn/9707015]. The scaling exponents
in this set of equations cannot be found from power counting. In this draft we
discuss in detail low order non-trivial closures of this infinite set of
equations, and prove that these closures lead to the determination of the
scaling exponents from solvability conditions. The equations under
consideration after this closure are nonlinear integro-differential equations,
reflecting the nonlinearity of the original Navier-Stokes equations.
Nevertheless they have a very special structure such that the determination of
the scaling exponents requires a procedure that is very similar to the solution
of linear homogeneous equations, in which amplitudes are determined by fitting
to the boundary conditions in the space of scales. The re-normalization scale
that i necessary for any anomalous scaling appears at this point. The Holder
inequalities on the scaling exponents select the renormalizaiton scale as the
outer scale of turbulence $L$.
