The turbulence velocity gradient tensor formed additively by normal and non-normal tensors

We decompose the velocity gradient tensor for turbulence into normal and non-normal parts, and condition our analysis on the strain eigenvector alignments between these tensors. We identify states that always enhance, and always counteract the axisymmetric expansion state, and give a rationale for decomposing the production balance term into its constituents: complex behavior arises when the dominant strain alignments involve the non-normal tensor. Finally, we develop a topological analysis framework where mathematical bounds on two of the three variables leads to an analysis in two planes.