  We consider cascade models of turbulence which are obtained by restricting
the Navier-Stokes equation to local interactions. By combining the results of
the method of extended self-similarity and a novel subgrid model, we
investigate the inertial range fluctuations of the cascade. Significant
corrections to the classical scaling exponents are found. The dynamics of our
local Navier-Stokes models is described accurately by a simple set of Langevin
equations proposed earlier as a model of turbulence [Phys. Rev. E {\bf 50}, 285
(1994)]. This allows for a prediction of the intermittency exponents without
adjustable parameters. Excellent agreement with numerical simulations is found.
