  In the present work, we show the existence and uniqueness of the
two-dimensional turbulent vorticity term under conditions given on the weak
solutions to the classical incompressible Navier-Stokes problem in a compact
domain $R$ with periodic boundary conditions and initial conditions in Sobolev
space $H^m(R)$. To achieve this, we derive a pressureless incompressible system
and calculate kinetic energy quantities on the turbulent vorticity from the
properties of the classical curl. Finally, since the system obtained is
algebraic, we experimented with the possibilities of reducing it by applying
the Rosenfeld-Groebner algorithm.
