pith. sign in

arxiv: 2606.09205 · v1 · pith:NKHQRRGInew · submitted 2026-06-08 · ⚛️ physics.flu-dyn

On the spatial statistics of free-surface turbulence and the complementarity of 'dimples' and 'scars'

Daniel R. Kjellevold , J{\o}rgen R. Aarnes , Omer M. Babiker , Simen {\AA}. Ellingsen , Ingelin Steinsland This is my paper

Pith reviewed 2026-06-27 15:07 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn
keywords free-surface turbulencedimplesscarsPoisson point processsurface divergencevertical vorticityspatial statisticsDNS
0
0 comments X

The pith

Dimples connect locally to vertical vorticity while scars connect locally to surface divergence in free-surface turbulence.

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

The paper models dimples and scars as inhomogeneous Poisson point processes whose intensities are driven by the local variance of surface divergence and vertical vorticity. Maximum-likelihood fitting to six DNS runs quantifies spatial support radii and time lags, revealing that dimples tie strongly to vorticity at short distances but need a spatially global model to estimate divergence. Scars show the opposite pattern, coupling locally to divergence but requiring global context for vorticity. This complementarity clarifies how each surface feature can be used to infer different aspects of sub-surface flow. A reader would care because the spatial structure had remained unquantified even though temporal links were already known.

Core claim

Dimples and scars are modelled as inhomogeneous Poisson point processes with intensity fields driven by the local variance of surface divergence β and vertical vorticity ω. Parameters including spatial support radius r and time lag τ are estimated by maximum likelihood against DNS data. The results show dimples have strong local connection to ω but weak spatial connection to β, so a spatially global model is required for dimples to estimate divergence; scars couple locally to β but globally to ω.

What carries the argument

Inhomogeneous Poisson point-process model whose intensity fields are driven by local variance of surface divergence β and vertical vorticity ω, with spatial support radius r and time lag τ fitted by maximum likelihood.

Load-bearing premise

The intensity fields of the Poisson processes for dimples and scars are driven by the local variance of surface divergence and vertical vorticity.

What would settle it

New DNS or laboratory data showing identical spatial correlation lengths for dimple-vorticity and dimple-divergence pairs across multiple realisations would falsify the claimed complementarity.

read the original abstract

The air--water interface governs the exchange of heat and gas between natural water bodies and their surrounding environment. Turbulence beneath the free surface imprints characteristic features: near-circular depressions (`dimples') and elongated indentations (`scars'). Recent studies have shown that these are linked to sub-surface flow features in a temporal sense. For instance, rapid increases in mean-square surface divergence due to upwelling events, precede dimple count surges. Yet the spatial structure of these connections remains unquantified. We employ spatial statistics to consider the spatial and temporal correlations between dimples and scars and two key velocity-derived fields, surface divergence $\beta=\partial_x u+\partial_y v$ and vertical vorticity $\omega=\partial_x v-\partial_y u$. The dimples and scars are modelled as inhomogeneous Poisson point-processes, with intensity fields driven by the local variance of $\beta$ and $\omega$. Parameters, including spatial support radius $r$ and time lag $\tau$, are estimated by maximum likelihood against six DNS datasets, quantifying the spatial and temporal connection between dimples, scars, surface divergence and vorticity. Our results demonstrate a clear complementarity: Dimples show strong local connection to the vertical vorticity field but has weak spatial connection with surface divergence and a spatially ``global'' model is required for dimples to work as estimators of surface divergence; scars, in a similar but opposite manner, couple locally to surface divergence but globally to the vertical vorticity. The complementarity sheds new light on the way dimples and scars may be used to infer fluxes across the surface, e.g., in remote sensing contexts.

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 models dimples and scars as inhomogeneous Poisson point processes with intensity fields driven by the local variance of surface divergence β and vertical vorticity ω. Parameters r (spatial support radius) and τ (time lag) are estimated via maximum likelihood on six DNS datasets to quantify spatial and temporal connections, leading to the claim of complementarity: dimples connect strongly and locally to ω but weakly to β (requiring a spatially global model), while scars connect locally to β but globally to ω.

Significance. If the modeling and fitting hold, the work supplies a quantitative statistical link between observable free-surface features and sub-surface turbulence structures, with implications for inferring interfacial fluxes (e.g., via remote sensing). Credit is due for the use of multiple independent DNS realizations and explicit maximum-likelihood parameter estimation rather than ad-hoc thresholds.

major comments (2)
  1. [modeling approach] The modeling choice that intensity λ is a function of local variance of β/ω (abstract and modeling approach) is load-bearing for the local-vs-global complementarity distinction. No sensitivity tests to alternative intensity drivers (e.g., the signed field itself, absolute value, or different functional forms) are described; if this choice is misspecified, the quantitative asymmetry between dimples and scars cannot be trusted.
  2. [results] The MLE fits for r and τ that underpin the 'strong local connection' vs 'weak spatial connection' statements (results section) lack reported uncertainties, goodness-of-fit diagnostics, or cross-validation across the six datasets; without these, it is unclear whether the estimated parameters robustly support the claimed complementarity.
minor comments (2)
  1. Define the precise computation of 'local variance' (including how the radius r enters the averaging kernel) in the intensity model; the current description leaves the exact functional form ambiguous.
  2. Add error bars or confidence intervals derived from the MLE to all reported parameter values and to any plots of intensity fields or point-process realizations.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their positive assessment of the significance of our work and for the constructive major comments. We provide point-by-point responses below and will incorporate revisions to address the concerns raised.

read point-by-point responses

  1. Referee: [modeling approach] The modeling choice that intensity λ is a function of local variance of β/ω (abstract and modeling approach) is load-bearing for the local-vs-global complementarity distinction. No sensitivity tests to alternative intensity drivers (e.g., the signed field itself, absolute value, or different functional forms) are described; if this choice is misspecified, the quantitative asymmetry between dimples and scars cannot be trusted.

    Authors: The selection of local variance as the intensity driver is grounded in the physical expectation that the magnitude of fluctuations in β and ω, rather than their signed values, correlates with the occurrence of dimples and scars. Signed fields could introduce cancellations that obscure the connection. Nevertheless, we recognize the value of sensitivity analysis. In the revised manuscript, we will conduct and report tests using alternative drivers including the absolute values |β| and |ω|, as well as the signed fields, and compare the resulting MLE parameters and model fits to assess robustness of the complementarity findings. revision: yes

  2. Referee: [results] The MLE fits for r and τ that underpin the 'strong local connection' vs 'weak spatial connection' statements (results section) lack reported uncertainties, goodness-of-fit diagnostics, or cross-validation across the six datasets; without these, it is unclear whether the estimated parameters robustly support the claimed complementarity.

    Authors: We agree that providing uncertainties and validation metrics is essential. The current manuscript reports point estimates from MLE but omits these details. We will revise the results section to include bootstrap confidence intervals for r and τ, goodness-of-fit assessments such as residual analysis or deviance statistics for the Poisson process models, and cross-validation results by training on five datasets and testing on the sixth, repeated across all combinations. This will quantify the robustness of the local-global distinctions. revision: yes

Circularity Check

0 steps flagged

No significant circularity: explicit modeling choice fitted to independent data

full rationale

The paper states an explicit modeling assumption (inhomogeneous Poisson processes with intensity driven by local variance of β and ω) and estimates parameters r and au via MLE on six DNS datasets. The complementarity conclusion is a direct statistical output of those fits, not a reduction of the result to the inputs by construction. No self-citation chain, uniqueness theorem, or ansatz smuggling supports the central claim; the model is falsifiable against the external DNS benchmarks. This is standard empirical spatial statistics with independent data, yielding a self-contained derivation (score 0).

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 0 invented entities

The central claim rests on treating dimples and scars as inhomogeneous Poisson point processes whose intensity is set by local variance of β and ω, with two fitted parameters (spatial radius r and time lag τ) estimated from DNS; no new physical entities are introduced.

free parameters (2)
  • spatial support radius r
    Controls the spatial scale over which local variance of β or ω influences point intensity; estimated by maximum likelihood.
  • time lag τ
    Controls the temporal offset between velocity fields and point-process observations; estimated by maximum likelihood.
axioms (1)
  • domain assumption Dimples and scars can be modeled as inhomogeneous Poisson point processes with intensity driven by local variance of surface divergence and vorticity.
    Stated directly in the abstract as the chosen statistical model.

pith-pipeline@v0.9.1-grok · 5861 in / 1399 out tokens · 26664 ms · 2026-06-27T15:07:59.560066+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

26 extracted references · 12 canonical work pages

  1. [1]

    Banerjee, S.: Upwellings, downdrafts, and whirlpools: Dominant structures in free surface turbulence. Appl. Mech. Rev.47(6S), 166–172 (1994) https://doi.org/10. 1115/1.3124398

    1994

  2. [2]

    Longuet-Higgins, M.S.: Surface manifestations of turbulent flow. J. Fluid Mech. 308, 15–29 (1996)

    1996

  3. [3]

    Brocchini, M., Peregrine, D.H.: The dynamics of strong turbulence at free surfaces. Part 1. Description. J. Fluid Mech.449, 225–254 (2001)

    2001

  4. [4]

    Babiker, O.M., Bjerkebæk, I., Xuan, A., Shen, L., Ellingsen, S.˚A.: Vortex imprints on a free surface as proxy for surface divergence. J. Fluid Mech.964, 2 (2023)

    2023

  5. [5]

    Aarnes, J.R., Babiker, O.M., Xuan, A., Shen, L., Ellingsen, S.A.: Vortex struc- tures under dimples and scars in turbulent free-surface flows. J. Fluid Mech.1007, 38 (2025) https://doi.org/10.1017/jfm.2025.72

    work page doi:10.1017/jfm.2025.72 2025

  6. [6]

    Physical Review Fluids11, 054802 (2026) https://doi.org/10

    Babiker, O.M., Aarnes, J.R., Semati, A., Ferran, A., Tee, Y.H., Hearst, R.J., Ellingsen, S.˚A.: Experimental investigation relating free-surface features to sub- surface turbulence. Physical Review Fluids11, 054802 (2026) https://doi.org/10. 1103/bmx7-2z3h

    2026

  7. [7]

    Geophysical research letters36(10), 10605 (2009) https://doi.org/10

    Kermani, A., Shen, L.: Surface age of surface renewal in turbulent interfacial transport. Geophysical research letters36(10), 10605 (2009) https://doi.org/10. 1029/2008GL037050

    2009

  8. [8]

    McKenna, S.P., McGillis, W.R.: The role of free-surface turbulence and sur- factants in air–water gas transfer. Int. J. Heat Mass Tran.47(3), 539–553 (2004) 18

    2004

  9. [9]

    Banerjee, S., Lakehal, D., Fulgosi, M.: Surface divergence models for scalar exchange between turbulent streams. Int. J. Multiph. Flow30(7), 963–977 (2004) https://doi.org/10.1016/j.ijmultiphaseflow.2004.05.004

    work page doi:10.1016/j.ijmultiphaseflow.2004.05.004 2004

  10. [10]

    Turney, D.E., Smith, W.C., Banerjee, S.: A measure of near-surface fluid motions that predicts air-water gas transfer in a wide range of conditions. Geophys. Res. Lett.32(4) (2005)

    2005

  11. [11]

    AIChE journal 56(8), 2005–2017 (2010)

    Janzen, J.G., Herlina, H., Jirka, G.H., Schulz, H.E., Gulliver, J.S.: Estimation of mass transfer velocity based on measured turbulence parameters. AIChE journal 56(8), 2005–2017 (2010)

    2005

  12. [12]

    Turney, D.E., Banerjee, S.: Air–water gas transfer and near-surface motions. J. Fluid Mech.733, 588–624 (2013) https://doi.org/10.1017/jfm.2013.435

    work page doi:10.1017/jfm.2013.435 2013

  13. [13]

    Journal of Fluid Mechanics933, 49 (2022)

    Pinelli, M., Herlina, H., Wissink, J.G., Uhlmann, M.: Direct numerical simulation of turbulent mass transfer at the surface of an open channel flow. Journal of Fluid Mechanics933, 49 (2022)

    2022

  14. [14]

    Guo, X., Shen, L.: Interaction of a deformable free surface with statistically steady homogeneous turbulence. J. Fluid Mech.658, 33–62 (2010) https://doi.org/10. 1017/S0022112010001539

    2010

  15. [15]

    Xuan, A., Shen, L.: A conservative scheme for simulation of free-surface turbulent and wave flows. J. Comput. Phys.378, 18–43 (2019) https://doi.org/10.1016/j. jcp.2018.10.046

    work page doi:10.1016/j 2019

  16. [16]

    Khakpour, H.R., Igusa, T., Shen, L.: Coherent vortical structures responsible for strong flux of scalar at free surface. Int. J. Heat Mass Tran.55(19), 5157–5170 (2012) https://doi.org/10.1016/j.ijheatmasstransfer.2012.05.017

    work page doi:10.1016/j.ijheatmasstransfer.2012.05.017 2012

  17. [17]

    ˚A., Kutz, J.N.: Mapping surface height dynamics to subsurface flow physics in free-surface turbulent flow using a shallow recurrent decoder

    Moen, K.S., Aarnes, J.R., Ellingsen, S. ˚A., Kutz, J.N.: Mapping surface height dynamics to subsurface flow physics in free-surface turbulent flow using a shallow recurrent decoder. Preprint arXiv:2510.06202 (2025)

    arXiv 2025

  18. [18]

    Rosales, C., Meneveau, C.: Linear forcing in numerical simulations of isotropic turbulence: Physical space implementations and convergence properties. Phys. Fluids17(9), 095106 (2005) https://doi.org/10.1063/1.2047568

    work page doi:10.1063/1.2047568 2005

  19. [19]

    Jeong, J., Hussain, F.: On the identification of a vortex. J. Fluid Mech.285, 69–94 (1995)

    1995

  20. [20]

    John Wiley & Sons, Ltd, ??? (2007)

    Illian, J.B., Penttinen, A., Stoyan, H., Stoyan, D.: Statistical Analysis and Modelling of Spatial Point Patterns. John Wiley & Sons, Ltd, ??? (2007). https://doi.org/10.1002/9780470725160 . https://onlinelibrary.wiley.com/doi/abs/10.1002/9780470725160

    work page doi:10.1002/9780470725160 2007

  21. [21]

    Springer, ??? (2013)

    Fahrmeir, L., Kneib, T., Lang, S., Marx, B.D.: Regression: Models, Methods and Applications. Springer, ??? (2013). https://doi.org/10.1007/978-3-662-63882-8 . https://link.springer.com/book/10.1007/978-3-662-63882-8

    work page doi:10.1007/978-3-662-63882-8 2013

  22. [22]

    Chapmann & Hall / CRC, ??? (1989)

    McCullagh, P., Nelder, J.A.: Generalized Linear Models. Chapmann & Hall / CRC, ??? (1989). https://doi.org/10.1201/9780203753736 . https://www.taylorfrancis.com/books/mono/10.1201/9780203753736/generalized- linear-models-mccullagh

    work page doi:10.1201/9780203753736 1989

  23. [23]

    Chapman and Hall/CRC, ??? (2015)

    Baddeley, A., Rubak, E., Turner, R.: Spatial Point Patterns: Methodology and Applications with R (1st Ed.). Chapman and Hall/CRC, ??? (2015). https://doi. org/10.1201/b19708 19

    work page doi:10.1201/b19708 2015

  24. [24]

    Shen, L., Zhang, X., Yue, D.K.P., Triantafyllou, G.S.: The surface layer for free- surface turbulent flows. J. Fluid Mech.386, 167–212 (1999) https://doi.org/10. 1017/S0022112099004590

    1999

  25. [25]

    Xuan, Shen, L., Ellingsen, S.A.: Supporting data for: Vortex imprints on a free surface as proxy for surface divergence

    Babiker, O.M., Bjerkebæk, I., A. Xuan, Shen, L., Ellingsen, S.A.: Supporting data for: Vortex imprints on a free surface as proxy for surface divergence. Data- verseNO (2023). https://doi.org/10.18710/ZOJHGJ . https://doi.org/10.18710/ ZOJHGJ

    work page doi:10.18710/zojhgj 2023

  26. [26]

    Vortex structures under dimples and scars in turbulent free-surface flows

    Aarnes, J.R., Babiker, O.M., Xuan, A., Shen, L., Ellingsen, S.˚A.: Replication data for: “Vortex structures under dimples and scars in turbulent free-surface flows” Dataverse, Part 1. DataverseNO (2025). https://doi.org/10.18710/XQ81WH . https://doi.org/10.18710/XQ81WH 20

    work page doi:10.18710/xq81wh 2025