Vorticity packing effects on long-time turbulent transport in decaying two-dimensional incompressible Navier–Stokes fluids

Recent high-resolution, high-Reynolds number simulations have shown that the initial total circulation, quantified by the vorticity packing fraction (VPF), strongly influences the late-time Eulerian statistical equilibria of decaying incompressible two-dimensional Navier–Stokes turbulence [Biswas and Ganesh, “Long time fate of two-dimensional incompressible high Reynolds number Navier–Stokes turbulence: A quantitative comparison between theory and simulation,” Phys. Fluids 34, 065101(2022)], revealing a transition from point-vortex dominated to patch-vortex (finite-size) equilibria with increasing vortex packing, and emphasizing the role of the classical exclusion principle (i.e., incompressibility) and total circulation in determining the final statistical states. The present study examines how the associated Lagrangian tracer transport evolves with VPF across the early (linear–nonlinear turbulence onset), intermediate (turbulence development), and late (coherent dipole evolution) stages, and how it correlates with the corresponding Eulerian states across different classes of initial vorticity field configurations. Turbulence, triggered by the Kelvin–Helmholtz instability and sustained by inverse energy cascades, forms large-scale coherent vortices that govern long-time transport. Tracer dynamics, analyzed via mean square displacement and position–velocity probability distribution functions including kurtosis, skewness, and their correlations, reveal that increasing VPF accelerates turbulence onset, drives a transition from sub- to superdiffusive transport with increasing isotropy in the intermediate stage, and determines late-time behavior dominated by either orbital coherent vortex trapping (sub-diffusive) or linear translational dipole motion (superdiffusive). These trends are governed primarily by VPF and are largely independent of the initial vorticity field configuration. These distinct long-time transport characteristics, evolving from sub- to superdiffusive behavior with increasing vorticity packing, demonstrate a strong correspondence between the transition from point-vortex– to finite-size-vortex–dominated Eulerian equilibria and the underlying Lagrangian transport in decaying, incompressible two-dimensional Navier–Stokes turbulence.