  Expressing the evolution equations for the filtered velocity gradient tensor
(FVGT) in the strain-rate eigenframe provides an insightful way to disentangle
and understand various processes such as strain self-amplification, vortex
stretching and tilting, and to consider their properties at different scales in
the flow. Using data from Direct Numerical Simulation (DNS) of the forced
Navier-Stokes equation, we consider the relative importance of local and
non-local terms in the FVGT eigenframe equations across the scales using
statistical analysis. The analysis of the eigenframe rotation-rate, that drives
vorticity tilting, shows that the anisotropic pressure Hessian plays a key
role, with the sub-grid stress making an important contribution outside the
dissipation range, and the local spinning due to vorticity making a much
smaller contribution. The results also show the striking behavior that the
vorticity tilting term remains highly intermittent even at relatively large
scales. We derive a generalization of the Lumley triangle that allows us to
show that the pressure Hessian has a preference for two-component axisymmetric
configurations at small scales, with a transition to a more isotropic state at
larger scales. Correlations between the sub-grid stress and other terms in the
eigenframe equations are considered, highlighting the coupling between the
sub-grid and nonlinear amplification terms, with the sub-grid term playing an
important role in regularizing the system. These results provide useful
guidelines for improving Lagrangian models of the FVGT, since current models
fail to capture a number of subtle features observed in our results.
