Renormalized hydrodynamics in polar chiral active matter: Spectral scaling and disorder-driven vortex clustering in phase-coupled, motile oscillators

Active turbulence in overdamped chiral systems presents a complex challenge, namely, the frequent exhibition of nonuniversal spectral scaling, creating large-scale coherent structuring that seemingly defies standard inertial fluid descriptions. In this study, we investigate the hydrodynamic limit of a two-dimensional polar chiral active fluid modeled as an ensemble of locally coupled, motile Kuramoto-Sakaguchi oscillators. By introducing a renormalized fluid element (RFE) operator, we coarse-grain microscopic phase singularities, and in so doing, we isolate the macroscopic transport dynamics. We demonstrate that while the raw phase field consistently exhibit a steep, dissipative energy spectrum, associated with enstrophy injection at the microscale, the RFE-filtered field reveals a dual behavior characterized by an inverse energy cascade. When the intrinsic frequency dispersion, drawn from a scale-free power-law distribution, is broad enough to seed fluctuations at all resolved scales, this hidden cascade acts akin to a topological heat pump, driving the system toward a state of macroscopic vortex clustering, structurally analogous to supersonic shallow water dynamics. Conversely, a narrow frequency dispersion results in kinetic arrest, forming an active vortex glass. These results suggest that overdamped phase-slaved active matter can sustain effective inertial cascades, providing a mathematical framework for understanding scale-dependent energy transport in driven chiral systems.