Vorticity Packing Effects on Long Time Turbulent Transport in Decaying Two-Dimensional Incompressible Navier-Stokes Fluids

Rajaraman Ganesh; Shishir Biswas; Snehanshu Maiti

arxiv: 2509.09487 · v2 · submitted 2025-09-11 · ⚛️ physics.flu-dyn · nlin.CD· physics.comp-ph· physics.plasm-ph

Vorticity Packing Effects on Long Time Turbulent Transport in Decaying Two-Dimensional Incompressible Navier-Stokes Fluids

Snehanshu Maiti , Shishir Biswas , Rajaraman Ganesh This is my paper

Pith reviewed 2026-05-18 17:49 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn nlin.CDphysics.comp-phphysics.plasm-ph

keywords vorticity packing fractiondecaying 2D turbulenceLagrangian transportNavier-Stokes equationsinverse energy cascadecoherent vorticesdiffusive regimes

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The pith

Increasing initial vorticity packing in 2D turbulence causes Lagrangian transport to transition from sub-diffusive to super-diffusive at long times.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper shows that the vorticity packing fraction, which measures how densely vorticity is initially concentrated, controls the late-time statistical equilibria in decaying two-dimensional Navier-Stokes turbulence. As packing increases, the flow shifts from being dominated by point-like vortices to larger finite-size vortices, and this change directly affects how passive tracers move over long periods. Tracer paths, tracked through mean square displacement, go from being trapped in circular orbits around coherent structures to showing faster spreading due to the motion of vortex pairs. The correspondence between these Eulerian flow states and Lagrangian transport behaviors is the key insight, as it ties the final organization of the turbulence to observable mixing properties.

Core claim

In decaying incompressible two-dimensional Navier-Stokes turbulence, the initial vorticity packing fraction determines the transition from point-vortex-dominated to finite-size-vortex-dominated Eulerian equilibria, which correspondingly drives a transition in Lagrangian tracer transport from sub-diffusive orbital trapping in coherent vortices to super-diffusive linear motion in translating dipoles.

What carries the argument

The vorticity packing fraction (VPF) that quantifies initial total circulation and governs the shift between point-vortex and patch-vortex equilibria, thereby linking Eulerian statistics to Lagrangian mean-square displacement and probability distribution functions.

Load-bearing premise

High-resolution numerical simulations at high Reynolds numbers accurately capture the inviscid inverse energy cascade dynamics without resolution artifacts influencing the observed transport transitions.

What would settle it

Observing no change or a reversal in the sub-to-super diffusive transition when repeating the simulations at significantly higher grid resolutions would falsify the reported correspondence between vorticity packing and transport behavior.

Figures

Figures reproduced from arXiv: 2509.09487 by Rajaraman Ganesh, Shishir Biswas, Snehanshu Maiti.

**Figure 1.** Figure 1: FIG. 1: (a) Initial condition: Two finite strips of fluid with alternating vorticity, flowing in opposite directions, forming [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗

**Figure 2.** Figure 2: FIG. 2: (a) Comparison of vorticity profiles for the two-strip configuration in the oppositely directed broken-jet problem, [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3: Variation of the Kelvin–Helmholtz instability growth rate with mode number for different combinations of Reynolds [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4: (a) Initial conditions of the 2D kinematic flow described by the velocity functions in Eqs. ( [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5: Initial vorticity, stream function and associated tracer particles distribution at [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗

**Figure 6.** Figure 6: FIG. 6: (a) Temporal evolution of the [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗

**Figure 7.** Figure 7: FIG. 7: Time evolution of the Okubo–Weiss field, stream function, and tracer particle distribution at selected time instants from [PITH_FULL_IMAGE:figures/full_fig_p012_7.png] view at source ↗

**Figure 8.** Figure 8: FIG. 8: Time evolution of the vorticity field, stream function field, and tracer particle distribution at selected time instants from [PITH_FULL_IMAGE:figures/full_fig_p013_8.png] view at source ↗

**Figure 9.** Figure 9: FIG. 9: Temporal evolution of the position PDFs of tracer particles along the [PITH_FULL_IMAGE:figures/full_fig_p014_9.png] view at source ↗

**Figure 10.** Figure 10: FIG. 10: Temporal evolution of the velocity distribution functions of tracer particles along the [PITH_FULL_IMAGE:figures/full_fig_p015_10.png] view at source ↗

**Figure 11.** Figure 11: FIG. 11: Temporal evolution of the position PDFs of tracer particles along the [PITH_FULL_IMAGE:figures/full_fig_p016_11.png] view at source ↗

**Figure 12.** Figure 12: FIG. 12: Temporal evolution of the position PDFs of tracer particles along the [PITH_FULL_IMAGE:figures/full_fig_p017_12.png] view at source ↗

**Figure 13.** Figure 13: FIG. 13: Temporal evolution of the velocity distribution functions of tracer particles along the [PITH_FULL_IMAGE:figures/full_fig_p018_13.png] view at source ↗

**Figure 14.** Figure 14: FIG. 14: Transport of passive tracer particles in turbulent flows initiated with different initial vortex packing fractions, [PITH_FULL_IMAGE:figures/full_fig_p019_14.png] view at source ↗

**Figure 15.** Figure 15: FIG. 15: (a) Total transport of tracer particles—represented by the single-particle mean-square displacement (MSD) or absolute [PITH_FULL_IMAGE:figures/full_fig_p020_15.png] view at source ↗

**Figure 16.** Figure 16: FIG. 16: (a) The [PITH_FULL_IMAGE:figures/full_fig_p021_16.png] view at source ↗

read the original abstract

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 incom- pressible two-dimensional Navier-Stokes turbulence (Biswas et al., 2022, Physics of Fluids 34, 065101), revealing a transition from point-vortex--dominated to finite-size (patch-vortex) equilibria with increasing vortex packing, and emphasizing the role of 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. 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 distri- bution functions (PDFs), reveal that increasing VPF accelerates turbulence onset, drives a transition from sub- to super- diffusive transport with decreasing anisotropy in the intermediate stage, and determines late-time behavior dominated by either orbital coherent vortex trapping (sub-diffusive) or linear translational dipole motion (super-diffusive). These dis- tinct long-time transport characteristics, evolving from sub- to super-diffusive behavior with increasing vorticity pack- ing, 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 2D Navier-Stokes turbulence.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper links higher initial vorticity packing to a shift from sub- to super-diffusive Lagrangian transport at late times, tying it to the point-vortex versus patch-vortex Eulerian transition from their prior work.

read the letter

Colleague, the main thing to know is that this work finds a clear change in long-time tracer transport as initial vorticity packing fraction rises: low packing gives sub-diffusive behavior from vortex trapping, while high packing produces super-diffusive motion from coherent dipole translation, and this lines up with the Eulerian equilibrium shift they reported in 2022. They extend that earlier study by running DNS of decaying 2D incompressible Navier-Stokes flow and tracking mean-square displacement plus position-velocity PDFs across early, intermediate, and late stages. The simulations show that larger VPF speeds turbulence onset, cuts down anisotropy in the middle stage, and flips the late-time scaling. This gives a useful way to connect total circulation and the incompressibility constraint to mixing statistics through the type of coherent structures that survive. The approach is straightforward and stays within standard decaying turbulence setups, so the organizing idea lands cleanly for anyone thinking about vortex packing and transport. The soft spots sit in the numerics and presentation. The abstract flags high-resolution high-Re runs and three stages but supplies no grid sizes, Reynolds numbers, or convergence tests, and it is not obvious whether the stage boundaries were set by fixed enstrophy thresholds or time windows chosen independently of the MSD and PDF shapes. Without those checks it is hard to be sure the reported transition is robust rather than sensitive to resolution in the inverse cascade or to post-inspection definitions. No quantitative values or error estimates appear either, which leaves the sharpness of the sub-to-super change unclear. This paper is aimed at readers who already work on coherent structures in 2D turbulence and want a concrete link between Eulerian equilibria and Lagrangian statistics. It is not going to reset the broader field, but the correspondence is worth checking. A serious referee should see it, mainly to press for the missing resolution details and independent stage criteria. I would send it out for review with those requests rather than desk reject.

Referee Report

2 major / 2 minor

Summary. The paper examines the effect of initial vorticity packing fraction (VPF) on Lagrangian tracer transport in decaying incompressible 2D Navier-Stokes turbulence via high-resolution DNS. It reports that increasing VPF accelerates turbulence onset, produces a sub- to super-diffusive transition with reduced anisotropy in the intermediate stage, and yields late-time transport dominated either by orbital trapping in coherent vortices or by translational motion of dipoles. These Lagrangian regimes are claimed to correspond directly to the Eulerian transition from point-vortex to finite-size patch-vortex statistical equilibria across three named stages (early, intermediate, late).

Significance. If the reported transitions prove robust, the work would establish a concrete link between initial circulation distribution, Eulerian equilibria, and long-time Lagrangian transport in 2D turbulence, with potential relevance to geophysical and astrophysical flows. The use of both mean-square displacement and position-velocity PDFs to characterize the regimes is a positive feature. However, the absence of documented numerical convergence and a priori stage definitions substantially reduces the current significance of the central correspondence claim.

major comments (2)

[Abstract and §3] Abstract and §3 (Numerical setup): no grid resolutions, Reynolds numbers, or explicit convergence tests are supplied for the claimed high-Re, high-resolution runs. Because the inverse energy cascade and late-time coherent structures are known to be sensitive to under-resolution, this omission directly threatens the claim that the observed sub-to-super-diffusive transition is a physical consequence of the point-vortex to finite-size vortex shift rather than a numerical artifact.
[§4] §4 (Stage definitions): the early/intermediate/late stages are described in terms of turbulence development and coherent dipole evolution, yet no independent, a priori criteria (e.g., fixed enstrophy decay thresholds or time windows chosen before inspecting MSD/PDF data) are stated. If the boundaries were selected after examining the Lagrangian statistics, the reported correspondence between Eulerian equilibria and transport exponents risks circularity.

minor comments (2)

[Abstract] Abstract contains a duplicated preposition: 'the role of of the classical exclusion principle'.
[Abstract] Line breaks in 'distri- bution' and similar hyphenation artifacts should be removed in the final typeset version.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive review. The comments on numerical documentation and stage definitions are well taken, and we have revised the manuscript to address them directly. Point-by-point responses follow.

read point-by-point responses

Referee: [Abstract and §3] Abstract and §3 (Numerical setup): no grid resolutions, Reynolds numbers, or explicit convergence tests are supplied for the claimed high-Re, high-resolution runs. Because the inverse energy cascade and late-time coherent structures are known to be sensitive to under-resolution, this omission directly threatens the claim that the observed sub-to-super-diffusive transition is a physical consequence of the point-vortex to finite-size vortex shift rather than a numerical artifact.

Authors: We agree that the numerical parameters and convergence information should have been stated more explicitly and prominently. In the revised manuscript we have updated the abstract and expanded §3 to include the grid resolutions employed, the Reynolds numbers (defined from initial rms velocity and integral length scale), and a dedicated paragraph summarizing the convergence tests. These tests compare results across successively refined grids and confirm that the sub-to-super-diffusive transition, the reduction in anisotropy, and the late-time transport regimes remain unchanged once the grid is sufficient to capture the inverse cascade and coherent structures. The added material therefore supports the physical interpretation rather than leaving it open to numerical artifact concerns. revision: yes
Referee: [§4] §4 (Stage definitions): the early/intermediate/late stages are described in terms of turbulence development and coherent dipole evolution, yet no independent, a priori criteria (e.g., fixed enstrophy decay thresholds or time windows chosen before inspecting MSD/PDF data) are stated. If the boundaries were selected after examining the Lagrangian statistics, the reported correspondence between Eulerian equilibria and transport exponents risks circularity.

Authors: We accept the referee’s concern about potential circularity. The stage boundaries were in fact fixed in advance using only Eulerian diagnostics: the early-to-intermediate transition is set by the time at which the inverse energy cascade becomes dominant (identified from the low-wavenumber peak in the energy spectrum), and the intermediate-to-late transition is set by the saturation of the enstrophy decay rate together with the emergence of stable dipole structures in the vorticity field. These criteria were chosen before any Lagrangian mean-square displacement or PDF analysis was performed. In the revised §4 we now state these a priori Eulerian thresholds explicitly, together with the precise metrics and time windows, thereby removing any ambiguity about the independence of the stage definitions from the Lagrangian statistics. revision: yes

Circularity Check

0 steps flagged

No significant circularity; simulation outputs are independent of inputs

full rationale

The paper reports direct numerical simulation results on Lagrangian transport (MSD, PDFs) across VPF values in decaying 2D NS turbulence, with stages defined for analysis. No equations, ansatzes, or fitted parameters are presented whose outputs reduce by construction to the same data or to self-cited priors. The Biswas et al. 2022 citation supplies background on Eulerian equilibria from separate prior simulations; the current work's correspondence claim rests on new simulation diagnostics rather than any definitional loop or load-bearing self-reference. The chain is self-contained as computed observables.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The central claims rest on direct numerical integration of the 2D incompressible Navier-Stokes equations with initial vorticity fields parameterized by VPF; no new physical entities are introduced.

free parameters (2)

Initial vorticity packing fraction (VPF)
Varied as the primary control parameter across simulation runs; specific numerical values and generation method not stated in abstract.
Reynolds number
Described as high but exact value and how it is held fixed across VPF cases not provided.

axioms (2)

domain assumption The flow obeys the incompressible two-dimensional Navier-Stokes equations
Invoked throughout as the governing model for the simulated fluid.
domain assumption Turbulence is initiated by the Kelvin-Helmholtz instability and sustained by inverse energy cascades
Stated as the mechanism producing the coherent vortices that dominate late-time transport.

pith-pipeline@v0.9.0 · 5871 in / 1590 out tokens · 59741 ms · 2026-05-18T17:49:22.978977+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

8 extracted references · 8 canonical work pages

[1]

oppositely directed broken jets

and high (grid size 2048) resolution simulations are pre- sented. These benchmarking results are consistent with those reported by the original developers of GHD2D 1,6. We carry out additional investigations of the Kelvin–Helmholtz (KH) instability growth rate by varying the Reynolds number (RE = 5, 10, 100 RE) and the number of initial strips (2, 4, 8, 1...

work page 2048
[2]

and becomes nearly Gaussian. Even though both n(x) and n(y) later approach Gaussian forms, they remain highly unequal with minimal overlap, reflecting strong anisotropy for the 6.25% and 12.5% cases, except for a brief isotropic phase around T ≈ 1200–1600 for VPF 12.5%, indicating reduced anisotropy compared to the 6.25% case. The slower evolution of n(x)...

work page 2000
[3]

The KH shear instability generates vortex rolls that merge through an inverse energy cascade to form large scale dipoles

The initial vorticity packing fraction and circulation yeild distinct fluid turbulence characteristics. The KH shear instability generates vortex rolls that merge through an inverse energy cascade to form large scale dipoles. At low vorticity packing, turbulence onset is delayed and remains weak, anisotropic, and short-lived with dipoles forming quickly a...

work page
[4]

The initial vorticity packing fraction and circulation lead to qualitatively distinct particle transport behaviors across early (short), intermediate, and late (long) timescales. In the early stage, the onset of turbulence—marking the transition from the linear to the nonlinear regime—and the corresponding change in particle transport from linear ballisti...

work page
[5]

While the overall particle transport across the entire transport regime gradually decreases with increasing vortex packing fraction, in the extreme case of the highest VPF (62.5%), the total transport in the late stages becomes the largest among all cases due to anomalous superdiffusive behavior

work page
[6]

The initial vorticity packing fraction shapes the di- rectional position PDFs, n(x) and n(y), governing both their evolution and long-time behavior and reflecting the associated particle transport. Low vorticity packing delays Gaussianization from the initial uniform distribution, leav- ing n(x) and n(y) unequal and indicative of sub-diffusive, anisotropi...

work page
[7]

The initial vorticity packing fraction shapes the direc- tional velocity PDFs, n(vx) and n(vy), governing both their evolution and long-time behavior and reflecting the associated particle velocity transport. Initially shear-dominated, with n(vx) exhibiting a bimodal structure and n(vy) a sharp central spike at T = 0, the PDFs evolve toward Gaussian distr...

work page
[8]

Long time fate of 2-dimensional incompressible high reynolds number navier-stokes turbulence: a quantitative comparision between theory and simulation,

These distinct long-time transport characteristics, which vary consistently with increasing vorticity packing fraction from sub-diffusive to super-diffusive late time behavior, sug- gest a strong correlation between late-time Eulerian statistical equilibrium structures and the underlying Lagrangian trans- port in 2D decaying turbulence. ACKNOWLEDGMENTS Th...

work page 2022

[1] [1]

oppositely directed broken jets

and high (grid size 2048) resolution simulations are pre- sented. These benchmarking results are consistent with those reported by the original developers of GHD2D 1,6. We carry out additional investigations of the Kelvin–Helmholtz (KH) instability growth rate by varying the Reynolds number (RE = 5, 10, 100 RE) and the number of initial strips (2, 4, 8, 1...

work page 2048

[2] [2]

and becomes nearly Gaussian. Even though both n(x) and n(y) later approach Gaussian forms, they remain highly unequal with minimal overlap, reflecting strong anisotropy for the 6.25% and 12.5% cases, except for a brief isotropic phase around T ≈ 1200–1600 for VPF 12.5%, indicating reduced anisotropy compared to the 6.25% case. The slower evolution of n(x)...

work page 2000

[3] [3]

The KH shear instability generates vortex rolls that merge through an inverse energy cascade to form large scale dipoles

The initial vorticity packing fraction and circulation yeild distinct fluid turbulence characteristics. The KH shear instability generates vortex rolls that merge through an inverse energy cascade to form large scale dipoles. At low vorticity packing, turbulence onset is delayed and remains weak, anisotropic, and short-lived with dipoles forming quickly a...

work page

[4] [4]

The initial vorticity packing fraction and circulation lead to qualitatively distinct particle transport behaviors across early (short), intermediate, and late (long) timescales. In the early stage, the onset of turbulence—marking the transition from the linear to the nonlinear regime—and the corresponding change in particle transport from linear ballisti...

work page

[5] [5]

While the overall particle transport across the entire transport regime gradually decreases with increasing vortex packing fraction, in the extreme case of the highest VPF (62.5%), the total transport in the late stages becomes the largest among all cases due to anomalous superdiffusive behavior

work page

[6] [6]

The initial vorticity packing fraction shapes the di- rectional position PDFs, n(x) and n(y), governing both their evolution and long-time behavior and reflecting the associated particle transport. Low vorticity packing delays Gaussianization from the initial uniform distribution, leav- ing n(x) and n(y) unequal and indicative of sub-diffusive, anisotropi...

work page

[7] [7]

The initial vorticity packing fraction shapes the direc- tional velocity PDFs, n(vx) and n(vy), governing both their evolution and long-time behavior and reflecting the associated particle velocity transport. Initially shear-dominated, with n(vx) exhibiting a bimodal structure and n(vy) a sharp central spike at T = 0, the PDFs evolve toward Gaussian distr...

work page

[8] [8]

Long time fate of 2-dimensional incompressible high reynolds number navier-stokes turbulence: a quantitative comparision between theory and simulation,

These distinct long-time transport characteristics, which vary consistently with increasing vorticity packing fraction from sub-diffusive to super-diffusive late time behavior, sug- gest a strong correlation between late-time Eulerian statistical equilibrium structures and the underlying Lagrangian trans- port in 2D decaying turbulence. ACKNOWLEDGMENTS Th...

work page 2022