  We consider equilibrium statistics for high Reynolds number isotropic
turbulence in an incompressible flow driven by steady forcing at the largest
scale. Motivated by shell model observations, we develop a similarity theory
for the inertial range from clearly stated assumptions. In the right variables,
the theory is scaling invariant, but in traditional variables it shows
anomalous scaling. We obtain the underlying probability density function, the
scaling exponents, and the coefficients for the structure functions. An
inertial range length scale also emerges.
