Geometric Mechanisms of Vortex Amplification in 3D Navier-Stokes Flows: Dynamic Scaling and Aerodynamic Bypass

This research paper presents a formal analytical investigation into the structural dynamics of vortex amplification within the three-dimensional incompressible Navier-Stokes equations under unbounded continuous geometric conditions. By isolating the internal non-linear mechanics of the partial differential equations from boundary-induced damping, the work rigorously evaluates the competition between non-linear advective vortex stretching and linear viscous dissipation. Key Analytical Highlights The Jacobian Anchor (|J|=1.0): The framework utilizes an exact volume-preserving affine mapping to model a continuously tapering vortex. This ensures that despite extreme radial contraction, zero geometric volume is lost from the core, forcing an unconstrained spatial derivative sequence. Dynamic Riccati Convergence: By tracking the maximum vorticity $\omega(t)$ along a Lagrangian trajectory, the governing equations resolve into a pure Riccati feedback loop: $\frac{d\omega}{dt} = (\alpha - \kappa\nu)\omega^{2}$. Because both stretching and dissipation scale quadratically with respect to the shrinking domain, viscous resistance cannot gain a polynomial advantage. Aerodynamic Bypass and Kinking: The study identifies a "kinking" force generated by non-constant helicity (Beltrami flow alignment) acting perpendicular to forward velocity, allowing the vortex to aerodynamically bypass dense viscous shear layers without arresting core axial stretching. Core Symmetry and Pressure Hessian: Utilizing asymptotic analysis within a Frenet-Serret frame, the paper proves that as the core shrinks, global curvature effects vanish ($r/R_{c} \rightarrow 0$), validating the pressure field's role in supporting topological collapse. Viscous Leakage and Invariant Time: The study explicitly derives the viscous leakage of circulation ($\Gamma$), proving that while viscosity alters the topological pathway, the core singularity chronological coordinate $T$ remains strictly invariant. Resolution of the Energy Paradox: The proof satisfies the Leray-Hopf physical regularity criteria by demonstrating that the global velocity field generated via a Gaussian-damped Stokes stream function yields a strictly bounded total kinetic energy ($E_{total} < \infty$). BKM Global Bridge: The localized Lagrangian collapse is linked to a global Eulerian breakdown via the Beale-Kato-Majda (BKM) theorem through first-order Taylor series expansion of the $L^{\infty}$-norm integral. Theoretical Frameworks The paper further reconciles these findings through topological isomorphisms, specifically mapping the Navier-Stokes Riccati engine to Perelman’s Ricci Flow models and identifying the absence of a multiplicative stabilization lock within the Unitary Symmetry Series. This research was conducted as part of the independent research initiatives at the Anadihilo Node (DOI: 10.5281/zenodo.21495161). For deeper technical exploration of the underlying topological principles, please visit the official repository at anadihilo.org.