A Unified Multi-Scale Vorticity Transport Equation for Large Eddy Simulation: Theoretical Framework and Physical Foundations

Large Eddy Simulation (LES) remains the dominant tool for high-Reynolds-number turbulent flows, yet widely used subgrid-scale closures, such as the Smagorinsky model, exhibit three fundamental deficiencies: purely dissipative dynamics (ignoring backscatter), isotropic treatment of stresses, and the Markovian assumption which neglects long-range temporal correlations. In this work, we present a unified multi-scale vorticity transport equation that addresses these deficiencies within a single, consistent theoretical framework. The proposed equation generalizes classical vorticity transport by incorporating: (i) a Caputo fractional time derivative to capture non-Markovian memory effects; (ii) an adaptive subgrid viscosity closure permitting both forward scatter and localized backscatter; (iii) a Coriolis source term for rotating-frame applications; (iv) a baroclinic torque term coupling density and pressure gradients; (v) an anisotropic backscatter tensor replacing conventional scalar eddy viscosity; (vi) an intermittency-sensitive hyperviscosity term; and (vii) a vorticity-strain coupling term for curved flows.