  We consider a time discretization of incompressible Navier-Stokes equations
with spatial periodic boundary conditions in the vorticity-velocity
formulation. The approximation is based on freezing the velocity on time
subintervals resulting in linear parabolic equations for vorticity.
Probabilistic representations for solutions of these linear equations are
given. At each time step, the velocity is expressed via vorticity using a
formula corresponding to the Biot--Savart-type law. We show that the
approximation is divergent free and of first order. The results are extended to
two-dimensional stochastic Navier-Stokes equations with additive noise, where,
in particular, we prove the first mean-square convergence order of the
vorticity approximation.
