  The Navier-Stokes equations and their various approximations can be described
in terms of near identity maps, that are diffusive particle path
transformations of physical space. The active velocity is obtained from the
diffusive path transformation and a virtual velocity using the Weber formula.
The active vorticity is obtained from the diffusive path transformation and a
virtual vorticity using a Cauchy formula. The virtual velocity and the virtual
vorticity obey diffusive equations, which reduce to passive advection formally,
if the viscosity is zero. Apart from being proportional to the viscosity, the
coefficients of these diffusion equations involve second derivatives of the
near identity transformation and are related to the Christoffel coefficients.
If and when the near-identity transformation departs excessively from the
identity, one resets the calculation. Lower bounds on the minimum time between
two successive resettings are given in terms of the maximum enstrophy.
