  The vorticity random field of turbulent flow is singled out as the main
dynamical variable for the description of turbulence, and the evolution
equation of the probability density function (PDF) of the vorticity field has
been obtained. This PDF evolution equation is a mixed type partial differential
equation (PDE) of second order which depends only on the conditional mean
(first order) of the underlying turbulent flow, which is in contrast with
Reynolds' mean flow equation which relies on a quadratic statistics. Therefore
the new PDF PDE may provide new closure scheme based on the conditional linear
statistics, and some of them will be described in the present paper too.
