Vorticity and helicity decompositions and dynamics with real Schur form of the velocity gradient

The real Schur form (RSF) of a generic velocity gradient field $\nabla \mathbf{u}$ is exploited to expose the structures of flows, in particular our field decomposition resulting in two vorticities with only mutual linkage as the topological content of the global helicity (accordingly decomposed into two equal parts). The local transformation to RSF may indicate alternative (co)rotating frame(s) for specifying the objective argument(s) of the constitutive equation. When $\nabla \mathbf{u}$ is uniformly of RSF in a fixed Cartesian coordinate frame, i.e., $u_x = u_x(x,y)$ and $u_y = u_y(x,y)$, but $u_z = u_z(x,y,z)$, the model, with the decomposed vorticities both frozen-in to $\mathbf{u}$, is in between two-dimensional-three-component (2D3C) and 3D3C ones and may help curing the pathology in the helical 2D3C absolute equilibrium, making the latter effectively work in more realistic situations.