  In this paper, we consider turbulence from a geometric perspective based on
the vorticity equations for incompressible viscous fluid flows. We derive
several quantitative statements about the statistics of turbulent flows. In
particular we establish an entropy decay inequality which involves the
turbulent kinetic energy and $L^{1}$-enstrophy, and we identify the small time
scale of the vorticity broken down and the vorticity creation under the
universality assumption of small time scales of turbulence flows.
