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A Schur decomposition of the velocity gradient tensor (VGT) for homogeneous,isotropic turbulence (HIT) is undertaken and its physical consequencesexamined. This decomposition permits the normal parts of the tensor(represented by the eigenvalues) to be separated explicitly from the non-normaleffects. Given the restricted {E}uler approximation to the VGT dynamics iswritten in terms of the isotropic part of the pressure Hessian and theinvariants of the characteristic equation of the VGT (in turn expressed interms of the eigenvalues), the non-normal terms are related to the non-localaspects of the dynamics and the anisotropic part of the pressure Hessian. Usinga direct numerical simulation of HIT, we show that the norm of the non-normalpart of the tensor is of a similar order to the normal part, highlighting theimportance of non-local effects. In fact, beneath the discriminant function ina Q-R plot, all enstrophy arises from the non-normal term, meaning thatvorticity and intermediate strain eigenvector alignment in this region is animmediate consequence of non-normality. A non-normal term appears in theexpressions for both enstrophy and total strain and cancels when calculatingthe second invariant of the VGT, while the self-amplification of non-normalityand the normal straining of non-normality appear in the strain production andenstrophy production equations and cancel when calculating the third invariant.However, these terms are significant for understanding the full VGT dynamics,explaining how flow structures evolve to a disc-like state despite the straineigenvalues sometimes indicating opposite (rod-like) behaviour, as well asexplaining vorticity and strain alignments in HIT.

The velocity gradient tensor for homogeneous, isotropic turbulence  (HIT), with explicit consideration of local and non-local effects using a  Schur decomposition