A Gauge-Theoretic Action Principle for Viscous Incompressible Fluids

Abstract: We propose a novel action principle for two dimensional incompressible fluid dynamics that naturally incorporates both vorticity and viscous dissipation via gauge field couplings. The action features a Chern Simons like term, $\epsilon^{\mu\nu\rho} A_\mu \partial_\nu A_\rho$, capturing the topological structure of vorticity, alongside a quadratic term $(\epsilon^{\mu\nu\rho} \partial_\nu A_\rho)^2$ representing viscous damping. Incompressibility is enforced through a Lagrange multiplier, while coupling to an external potential allows applications in geophysical flows. We derive the equations of motion, recovering the vorticity formulation of the two-dimensional incompressible Navier Stokes equations and explicitly identifying the kinematic viscosity. This gauge theoretic framework leads to a Helmholtz type equation for vorticity linking topological and dissipative phenomena in viscous incompressible fluids. Analysis of Noether symmetries reveals conserved charges arising from gauge invariance and spatial translations, while viscosity explicitly breaks time reversal symmetry within this topological setting. Furthermore, the velocity vorticity gauge correspondence naturally suggests a Lindblad operator structure, providing a pathway toward a quantum description of viscous dissipation and allowing quantization of dissipative hydrodynamics. This framework also highlights how vorticity emerges as a natural Lindblad operator, capturing the transition from coherent rotational motion to thermal disorder as decoherence in the gauge field ensemble.