  We study the spatial statistics of velocity gradient volatility (i,e., the
energy dissipation rate) in turbulent flow. We extend the Kolmogorov-Obukhov
theory but also narrow its scope. The models are log normal, with verification
from finely resolved large eddy and direct numerical simulations. They are
parameterized by a mean and a covariance operator. Addressing applications to
large eddy simulations, the mean and the covariance depend on the resolved
scale solution. Removing this resolved scale dependence by a locally defined
rescaling of the turbulent statistics yields a universal theory for the subgrid
statistics, specifically for the mean and variance of turbulent fluctuations,
i.e., the log of the energy dissipation rate {\epsilon}. The variance of the
velocity gradient statistics is found to be log normal, in accordance with
Kolmogorov 1962. Rescaling by the resolved scale mean and variance removes the
influence of the resolved scales on the subgrid scales, justifying conceptually
the universality we observe. The universality is the basis of new power scaling
laws. The new power laws, in turn, allow a simple parameterization of the mean
and covariance of the subgrid rescaled dissipation rate, i.e., velocity
gradient volatility. We treat the coarse grid resolved space- time location as
a random variable. A restriction of the theory to a small range of unresolved
but stochastically modelled scales is proposed, with renormalization group
ideas offered to overcome this restriction.
