Quaternionic geometric framework for viscous vorticity and coherent structure formation in nonlinear flows

Formulation of a rigorous framework for the dynamics of dissipative and vortical fluids continues to pose a challenge in mathematical physics, particularly towards an explicit limiting pathway from the reversible microscopic Liouville evolution to the irreversible Navier–Stokes regime. In this work, we present a geometric framework in which nonlinear vorticity and macroscopic dissipation emerge as an inherent consequence of a generalized Quaternionic Liouville Superoperator. By promoting the phase space algebra to a non-commutative Quaternionic (H) Hilbert module, we identify at level of representation the imaginary unit not as global scalar constant, but as a local spatial versor. We demonstrate that macroscopic fluid vorticity ω=∇×u and viscous stress tensor are obtained via explicit representation of the non-commutative brackets of this algebra, thereby identifying them with components of an associated non-commutative connection curvature rather than phenomenological additions. Furthermore, we derive a Quaternionic Hydrodynamic Schrödinger Equation (QHSE) and establish that its non-Hermitian evolution acts as rigorous contraction semigroup. This dissipative generator provides a norm-contractive closure mechanism for topological relaxation. Specifically, we show that SU(2) degrees of freedom enable a transition from rigid frustration to topological plasticity, allowing the fluid to bypass traditional U(1) Abrikosov lattice constraints. Validated through Split-Step Fourier Method (SSFM) numerical integration, these results characterize geometric pattern formation of the non-commutative structure, serving as unified analytical and computational framework for relating algebraic quantum foundations to macroscopic hydrodynamic attractors, without presupposing definitive correspondence.