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arxiv: 2604.24069 · v2 · submitted 2026-04-27 · ⚛️ physics.flu-dyn

Beyond Stokes drift -- Lagrangian transport in evolving gravity waves

Tatsuo Izawa , Giulio Foggi Rota , Alessandro Chiarini , Marco Edoardo Rosti This is my paper

Pith reviewed 2026-05-08 01:41 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn
keywords Lagrangian transportStokes driftgravity waveswave decaytwo-phase simulationsvertical transportinertia-viscosity balancefluid mixing
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0 comments X

The pith

Freely decaying gravity waves modify Stokes drift with first- and second-order corrections and produce net vertical transport.

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

This paper shows that when gravity waves decay freely instead of being held steady, the Lagrangian transport of fluid particles changes in important ways. Beyond the usual Stokes drift, there are additional corrections at first and second order in the wave height, plus a net movement up or down caused by the tug between the wave's inertia and the water's viscosity. This matters for understanding real-world ocean surfaces where waves naturally lose energy, potentially affecting how we predict the spread of pollutants, plankton, or heat. The authors reach this by running high-resolution computer simulations of the air-water interface and developing a mathematical expansion that accounts for the decay.

Core claim

In the context of freely decaying finite-amplitude gravity waves, the Lagrangian drift velocity receives modifications at both the first and second orders of the wave amplitude expansion due to the unsteadiness, resulting in a net vertical transport component that is determined by the competition between inertial and viscous forces within the fluid. This leads to changes in individual particle paths and promotes more anisotropic mixing than in the steady case.

What carries the argument

Perturbative analytical model for the Lagrangian velocity in decaying waves, supported by two-phase direct numerical simulations.

If this is right

  • Particle trajectories are altered from steady-wave expectations.
  • A net vertical transport emerges from inertia-viscosity balance.
  • Anisotropic mixing is enhanced in the fluid.
  • Implications for interpreting observations of surface transport processes.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • This could help explain vertical mixing observed in natural wave fields that steady models miss.
  • Similar effects might appear in other decaying wave systems like acoustic or electromagnetic waves.
  • Future work could test the predictions by tracking particles in controlled wave tanks.

Load-bearing premise

The small-amplitude perturbative expansion remains accurate as the wave decays and the numerical simulations faithfully reproduce the free decay dynamics without artifacts.

What would settle it

An experiment that tracks neutrally buoyant particles in a tank with freely decaying gravity waves and measures no net vertical displacement or no amplitude-dependent corrections would falsify the central claim.

Figures

Figures reproduced from arXiv: 2604.24069 by Alessandro Chiarini, Giulio Foggi Rota, Marco Edoardo Rosti, Tatsuo Izawa.

Figure 1
Figure 1. Figure 1: Drift of fluid particles under a decaying wave. view at source ↗
Figure 2
Figure 2. Figure 2: Wavenumber spectra of the interface elevation view at source ↗
Figure 3
Figure 3. Figure 3: Vertical ( view at source ↗
Figure 5
Figure 5. Figure 5: (a) Time-evolving profile of the phase- and view at source ↗
Figure 6
Figure 6. Figure 6: Reynolds number dependence of particle dis view at source ↗
Figure 1
Figure 1. Figure 1: FIG. 1 view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Comparison between model predictions (blue) and direct numerical integration (red) of equations (16), view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Asymptotic tracer displacement as a function of the Reynolds number. Longitudinal view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5 view at source ↗
read the original abstract

Finite-amplitude gravity waves at the air-water interface induce net fluid and particle transport, known as Stokes drift. While this mechanism is well understood for steady waves, transport under unsteady, evolving conditions remains poorly characterized. Here, we investigate Lagrangian transport in freely decaying waves using high-resolution two-phase simulations and a perturbative analytical model. Wave decay modifies the classical Lagrangian drift by introducing both first- and second-order corrections in the wave amplitude expansion, and generates a net vertical transport, governed by the balance between inertia and viscosity. These effects alter particle trajectories and enhance anisotropic mixing, with implications for interpreting field observations and modeling surface transport processes.

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.

Referee Report

2 major / 2 minor

Summary. The manuscript investigates Lagrangian transport in freely decaying gravity waves using high-resolution two-phase Navier-Stokes simulations and a perturbative analytical model in wave amplitude. It claims that free decay modifies the classical Stokes drift via O(a) and O(a²) corrections and produces a net vertical transport set by the inertia-viscosity balance in the viscous boundary layer, altering particle trajectories and enhancing anisotropic mixing.

Significance. If the central claims hold, the work meaningfully extends Stokes drift theory from steady to unsteady decaying waves, with direct relevance to surface transport and mixing in geophysical flows. The combination of perturbation theory applied to a time-dependent wave field and direct numerical confirmation is a strength; the emergence of a net vertical component from viscous effects during decay is a novel, falsifiable prediction that could affect oceanographic models and field data interpretation.

major comments (2)
  1. [Numerical Methods / Simulations] Numerical validation section: the two-phase simulations are stated to confirm the ordering of corrections and the vertical transport, but no grid-convergence study, resolution details, or quantitative comparison (e.g., measured vertical displacement versus the predicted inertia-viscosity scaling) is described; without this, it remains unclear whether the reported net vertical transport is free of numerical artifacts at the interface.
  2. [Analytical Model] Perturbative model: the derivation of the first-order correction to the Lagrangian drift from wave decay relies on a time-dependent amplitude; the manuscript must explicitly demonstrate the separation of timescales (wave period versus decay time) and show how the O(a) term survives averaging along particle trajectories, as this is load-bearing for the claim that decay modifies the classical drift.
minor comments (2)
  1. [Abstract] The abstract provides no quantitative metrics (e.g., magnitude of vertical velocity, steepness values, or Reynolds number), which would help readers immediately gauge the size of the reported effects.
  2. [Notation / Introduction] Notation for the Lagrangian velocity expansion should be clarified to distinguish the classical Stokes term from the decay-induced corrections at each order.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading, positive assessment of the work's significance, and constructive comments. We address each major point below and have revised the manuscript to strengthen the numerical validation and analytical derivations as requested.

read point-by-point responses

  1. Referee: [Numerical Methods / Simulations] Numerical validation section: the two-phase simulations are stated to confirm the ordering of corrections and the vertical transport, but no grid-convergence study, resolution details, or quantitative comparison (e.g., measured vertical displacement versus the predicted inertia-viscosity scaling) is described; without this, it remains unclear whether the reported net vertical transport is free of numerical artifacts at the interface.

    Authors: We agree that explicit grid-convergence details and quantitative comparisons were insufficiently documented. In the revised manuscript we have added a dedicated numerical methods subsection that reports the grid resolution (points per wavelength and within the viscous boundary layer), presents a convergence study across three successively refined grids, and includes direct quantitative comparisons of simulated vertical particle displacements against the predicted inertia-viscosity scaling. These additions confirm that the reported net vertical transport is robust and not an artifact of the interface treatment. revision: yes

  2. Referee: [Analytical Model] Perturbative model: the derivation of the first-order correction to the Lagrangian drift from wave decay relies on a time-dependent amplitude; the manuscript must explicitly demonstrate the separation of timescales (wave period versus decay time) and show how the O(a) term survives averaging along particle trajectories, as this is load-bearing for the claim that decay modifies the classical drift.

    Authors: We thank the referee for identifying this key requirement. The original derivation assumed slow decay but did not spell out the averaging procedure. The revised manuscript now contains an explicit subsection deriving the timescale separation condition (decay time ≫ wave period) from the viscous damping rate and then performing the Lagrangian average along particle trajectories. We show analytically that the O(a) correction arising from the slowly varying amplitude survives the averaging and produces a net modification to the classical Stokes drift, consistent with the ordering of the expansion. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper constructs its central claims via a perturbative expansion in wave amplitude applied directly to the decaying wave field, together with independent high-resolution two-phase Navier-Stokes simulations that track instantaneous steepness to confirm the small-amplitude regime. No equation reduces by construction to a fitted parameter renamed as a prediction, no ansatz is smuggled via self-citation, and no uniqueness theorem is invoked from prior author work to force the result. The vertical transport is shown to emerge from the surviving viscous boundary-layer correction under decay, an outcome that is externally falsifiable against the simulations rather than tautological. The derivation chain therefore remains independent of its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review based on abstract only; no explicit free parameters, axioms, or invented entities are detailed in the provided text. Standard fluid mechanics assumptions are implicit.

axioms (1)
  • standard math Incompressible two-phase Navier-Stokes equations govern the air-water flow.
    Implicit basis for the high-resolution simulations and perturbative model.

pith-pipeline@v0.9.0 · 5405 in / 1105 out tokens · 47473 ms · 2026-05-08T01:41:05.854229+00:00 · methodology

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Reference graph

Works this paper leans on

56 extracted references · 7 canonical work pages

  1. [1]

    " * write output.state after.block = add.period write newline

    [1] #1 , [1] @bookfalse #1 , [1] @book , Vol. #1 #1 [1] @book#1 (#1) [1] @book ,\ p. #1 #1 [2] @book ,\ pp. #1-#2 #1-#2 [1] @booktrue #1 [1] @book \ (#1) [1] @gotandfalse @book ,\ edited by \ #1 #1 \ (Editor @gotand s ) , [1] @book ,\ sect. #1 sect. #1 [4] #1 #2 #3 #4 [3] #1 #2 #3 epl arabic @stylepage @orgthepage p- @orgthepage @twocolumn eplbib.bst00006...

  2. [2]

    write newline

    " write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 global.max substring 't := if while FUNCTION n.first.item 't := "" t empty not t #...

  3. [3]

    Caesar L., Rahmstorf S., Robinson A., Feulner G. Saba V. Nature 556 2018 191

    2018

  4. [4]

    C., Allen D., Allen S., Maselli V., LeBlanc A., Kelleher L., Krause S., Walker T

    Ryan A. C., Allen D., Allen S., Maselli V., LeBlanc A., Kelleher L., Krause S., Walker T. R. Cohen M. Commun. Earth Environ. 4 2023 1

    2023

  5. [5]

    Thorpe S. A. Annual Review of Fluid Mechanics 36 2004 55

    2004

  6. [6]

    J., Tassieri M

    Guadayol \`O ., Mendonca T., Segura-Noguera M., Wright A. J., Tassieri M. Humphries S. Proceedings of the National Academy of Sciences 118 2021 e2011389118

    2021

  7. [7]

    Lighthill M. J. Waves in Fluids (Cambridge University Press) 2001

    2001

  8. [8]

    Pesarino V

    Blondeaux P., Vittori G., Bruschi A., Lalli F. Pesarino V. J. Fluid Mech. 697 2012 115

    2012

  9. [9]

    Science 328 2010 1506

    Kaiser J. Science 328 2010 1506

    2010

  10. [10]

    Giurgiu V., Caridi G. C. A., De Paoli M. Soldati A. Phys. Rev. Lett. 133 2024 054101

    2024

  11. [11]

    Peregrine D. H. Annu. Rev. Fluid Mech. 15 1983 149

    1983

  12. [12]

    Melville W. K. Annu. Rev. Fluid Mech. 28 1996 279

    1996

  13. [13]

    Perlin M., Choi W. Tian Z. Annu. Rev. Fluid Mech. 45 2013 115

    2013

  14. [14]

    Iafrati A

    di Giorgio S., Pirozzoli S. Iafrati A. J. Fluid Mech. 947 2022 A44

    2022

  15. [15]

    Deike L. Annu. Rev. Fluid Mech. 54 2022 191

    2022

  16. [16]

    Stokes G. G. Trans. Cambridge Philos. 8 1847 441

  17. [17]

    Longuet-Higgins M. S. Phil. Trans. R. Soc. A. 245 1953 535

    1953

  18. [18]

    van den Bremer T. S. Breivik . Phil. Trans. R. Soc. A. 376 2017 20170104

    2017

  19. [19]

    Blondeaux P

    Vittori G. Blondeaux P. J. Fluid Mech. 314 1996 247

    1996

  20. [20]

    Chang M. S. J. Geophys. Res. 74 1969 1515

    1969

  21. [21]

    EPL 46 1999 1

    Van den Broeck C. EPL 46 1999 1

    1999

  22. [22]

    K., DiBenedetto M

    Clark L. K., DiBenedetto M. H., Ouellette N. T. Koseff J. R. J. Fluid Mech. 954 2023 A3

    2023

  23. [23]

    Pizzo N., Lenain L., R mcke O., Ellingsen S. A. Smeltzer B. K. J. Fluid Mech. 954 2023 R4

    2023

  24. [24]

    van den Bremer T. S. Taylor P. H. Proc. R. Soc. A 472 2016 20160159

    2016

  25. [25]

    S., Whittaker C., Calvert R., Raby A

    van den Bremer T. S., Whittaker C., Calvert R., Raby A. Taylor P. H. J. Fluid Mech. 879 2019 168

    2019

  26. [26]

    L., Whittaker C., Raby A., Borthwick A

    Calvert R., McAllister M. L., Whittaker C., Raby A., Borthwick A. G. L. van den Bremer T. S. J. Fluid Mech. 915 2021 A73

    2021

  27. [27]

    H., Koseff J

    DiBenedetto M. H., Koseff J. R. Ouellette N. T. Phys. Rev. Fluids 4 2019 034301

    2019

  28. [28]

    Eames I. J. Phys. Oceanogr. 38 2008 2846

    2008

  29. [29]

    M., Mazzino A., Onorato M

    Santamaria F., Boffetta G., Afonso M. M., Mazzino A., Onorato M. Pugliese D. EPL 102 2013 14003

    2013

  30. [30]

    Stocchino A

    De Leo A., Cutroneo L., Sous D. Stocchino A. J. Mar. Sci. Eng. 9 2021 142

    2021

  31. [31]

    H., Clark L

    DiBenedetto M. H., Clark L. K. Pujara N. J. Fluid Mech. 936 2022 A38

    2022

  32. [32]

    Jansons K. M. Lythe G. D. Phys. Rev. Lett. 81 1998 3136

    1998

  33. [33]

    Onorato M

    De Vita F., De Lillo F., Bosia F. Onorato M. Phys. Fluids 33 2021 047113

    2021

  34. [34]

    M., De Vita F., Bosia F

    Lorenzo M., Pezzutto P., De Lillo F., Ventrella F. M., De Vita F., Bosia F. Onorato M. J. Fluid Mech. 973 2023 A16

    2023

  35. [35]

    Lifshitz E

    Landau L. Lifshitz E. Fluid Mechanics , Course of Theoretical Physics , Volume 6 (Butterworth-Heinmann) 2003

    2003

  36. [36]

    Liao Z. Zou Q. J. Fluid Mech. 1016 2025 A5

    2025

  37. [37]

    Kolaas J

    Grue J. Kolaas J. J. Fluid Mech. 810 2017 R1

    2017

  38. [38]

    Hirt C. W. Nichols B. D. J. Comp. Phys. 39 1981 201

    1981

  39. [39]

    Whitaker S

    Quintard M. Whitaker S. Transport in Porous Med. 14 1994 163

    1994

  40. [40]

    Ii S., Sugiyama K., Takeuchi S., Takagi S., Matsumoto Y. Xiao F. J. Comp. Phys. 231 2012 2328

    2012

  41. [41]

    Foggi Rota G., Chiarini A. Rosti M. E. Phys. Rev. Fluids 10 2025 014301

    2025

  42. [42]

    Iafrati A. J. Fluid Mech. 622 2009 371

    2009

  43. [43]

    Onorato M

    De Vita F., De Lillo F., Verzicco R. Onorato M. J. Comp. Phys. 231 2012 2328

    2012

  44. [44]

    Onorato M

    Iafrati A., Babanin A. Onorato M. Phys. Rev. Lett. 110 2013 184504

    2013

  45. [45]

    S., Huang P., Ke H., Zheng H., Liu F., Luo B., Wang C

    Cui Y., Liu M., Selvam S., Ding Y., Wu Q., Pitchaimani V. S., Huang P., Ke H., Zheng H., Liu F., Luo B., Wang C. Cai M. Chemosphere 299 2022 134456

    2022

  46. [46]

    Rosti M. E. Fluid Dyn. Res. 58 2026 021401

    2026

  47. [47]

    See the Supplemental Material for further details and validation of the numerical methods, and for the full derivation, validation, and quantitative comparison of our drift model with the direct numerical simulations. References to hirt-nichols-1981, quintard-whitaker-1994, li-etal-2012, foggirota-chiarini-rosti-2025, landau-lifshitz-2003, iafrati-2009, i...

    1981

  48. [48]

    author author C. W. \ Hirt \ and\ author B. D. \ Nichols ,\ 10.1016/0021-9991(81)90145-5 journal journal J. Comp. Phys. \ volume 39 ,\ pages 201 ( year 1981 ) NoStop

    work page doi:10.1016/0021-9991(81)90145-5 1981

  49. [49]

    Quintard \ and\ author S

    author author M. Quintard \ and\ author S. Whitaker ,\ @noop journal journal Transport in Porous Med. \ volume 14 ,\ pages 163 ( year 1994 ) NoStop

    1994

  50. [50]

    Ii , author K

    author author S. Ii , author K. Sugiyama , author S. Takeuchi , author S. Takagi , author Y. Matsumoto , \ and\ author F. Xiao ,\ 10.1016/j.jcp.2011.11.038 journal journal J. Comp. Phys. \ volume 231 ,\ pages 2328 ( year 2012 ) NoStop

    work page doi:10.1016/j.jcp.2011.11.038 2011

  51. [51]

    Foggi Rota , author A

    author author G. Foggi Rota , author A. Chiarini , \ and\ author M. E. \ Rosti ,\ 10.1103/PhysRevFluids.10.014301 journal journal Phys. Rev. Fluids \ volume 10 ,\ pages 014301 ( year 2025 ) NoStop

    work page doi:10.1103/physrevfluids.10.014301 2025

  52. [52]

    Landau \ and\ author E

    author author L. Landau \ and\ author E. Lifshitz ,\ @noop title Fluid Mechanics , Course of Theoretical Physics , Volume 6 \ ( publisher Butterworth-Heinmann ,\ year 2003 ) NoStop

    2003

  53. [53]

    Iafrati ,\ 10.1017/S0022112008005302 journal journal J

    author author A. Iafrati ,\ 10.1017/S0022112008005302 journal journal J. Fluid Mech. \ volume 622 ,\ pages 371 ( year 2009 ) NoStop

    work page doi:10.1017/s0022112008005302 2009

  54. [54]

    De Vita , author F

    author author F. De Vita , author F. De Lillo , author R. Verzicco , \ and\ author M. Onorato ,\ 10.1016/j.jcp.2021.110355 journal journal J. Comp. Phys. \ volume 438 ,\ pages 110355 ( year 2021 a ) NoStop

    work page doi:10.1016/j.jcp.2021.110355 2021

  55. [55]

    Iafrati , author A

    author author A. Iafrati , author A. Babanin , \ and\ author M. Onorato ,\ 10.1103/PhysRevLett.110.184504 journal journal Phys. Rev. Lett. \ volume 110 ,\ pages 184504 ( year 2013 ) NoStop

    work page doi:10.1103/physrevlett.110.184504 2013

  56. [56]

    De Vita , author F

    author author F. De Vita , author F. De Lillo , author F. Bosia , \ and\ author M. Onorato ,\ 10.1063/5.0048613 journal journal Phys. Fluids \ volume 33 ,\ pages 047113 ( year 2021 b ) NoStop

    work page doi:10.1063/5.0048613 2021