Eulerian Lagrangian relations in decaying two dimensional incompressible Navier Stokes fluids across initial vorticity packing and Reynolds number

Abstract

Recent studies Vorticity packing effects on long time turbulent transport in decaying two dimensional incompressible Navier Stokes fluids, Phys. Fluids 38, 045159 (2026) demonstrated that, at a fixed high Reynolds number (Re), the initial vorticity packing fraction (VPF) governs the coupled Eulerian flow evolution and Lagrangian tracer particle transport in decaying two dimensional incompressible Navier Stokes fluids, revealing a strong Eulerian Lagrangian relationship during the nonequilibrium inverse cascade regime and a direct Eulerian Lagrangian correspondence in the late time coherent vortex quasi equilibrium regime, wherein increasing VPF drives transitions from point vortex to patch vortex equilibria and from subdiffusive to superdiffusive transport. In the present work, we investigate how these Eulerian Lagrangian connections evolve across a broad (VPF, Re) parameter space. The results show that the Eulerian Lagrangian relationship remains largely preserved during the nonequilibrium inverse-cascade regime, where transport increases systematically with VPF and remains primarily controlled by VPF despite secondary Re dependent oscillatory modulation. In contrast, the late-time coherent-vortex quasi-equilibrium regime exhibits Eulerian statistical equilibria that remain largely insensitive to Re, while the corresponding tracer-particle transport displays a strong Re dependence characterized by strong oscillatory and nonmonotonic variations across the (VPF, Re) parameter space, substantially weakening the VPF-ordered transport hierarchy observed in the inverse-cascade regime. Consequently, for the parameter range, spatial resolutions, and integration times explored in the present study, the direct Eulerian Lagrangian correspondence identified at fixed high Re is not universally maintained across the broader (VPF, Re) parameter space.