  In this paper, we prove a new identity for divergence free vector fields,
showing that $\left<-\Delta S,\omega\otimes\omega\right>=0$. This identity will
allow us to understand the interaction of different aspects of the nonlinearity
in the Navier--Stokes equation from the strain and vorticity perspective,
particularly as they relate to the depletion of the nonlinearity by advection.
We will prove global regularity for the strain-vorticity interaction model
equation, a model equation for studying the impact of the vorticity on the
evolution of strain which has the same identity for enstrophy growth as the
full Navier--Stokes equation. We will also use this identity to obtain several
new regularity criteria for the Navier--Stokes equation, one of which will help
to clarify the circumstances in which advection can work to deplete the
nonlinearity, preventing finite-time blowup.
