Pith. sign in

REVIEW 2 major objections 4 minor 39 references

Autoregressive rollout error in latent-space reduced-order models of bluff-body wakes is accumulated phase drift

T0 review · 2 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Latent rollout error in wake reduced-order models is accumulated phase drift, not unstructured noise.

desk verdict Rollout error in these periodic-wake ROMs is mostly linear phase drift; the central claim holds, but the exact phase share needs a synthetic-control check before it is fully secure. read the letter →

arxiv 2608.07189 v1 pith:AD67NVVQ submitted 2026-08-07 physics.flu-dyn

Suvam Samanta , Sachidananda Behera This is my paper · ORCID
classification physics.flu-dyn
keywords reduced-ordermodelsrollouterrorphasedriftbluff-bodywakecylinderconvolutionalautoencoderLSTMamplitude-phasedecomposition
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Across circular-cylinder wakes at Re=300–800 and a square-cylinder wake at Re=100, the paper finds that 95–98% of the autoregressive rollout error of a convolutional-autoencoder-LSTM reduced-order model is phase error, not amplitude error. The network reproduces the limit-cycle amplitude within 0.15% but traverses the cycle at a slightly wrong rate. This phase drift accumulates linearly to only about 0.02–0.03 rad over the rollout yet accounts for the long-horizon error growth. Because the drift is linear, it can be corrected offline with one parameter per latent coordinate, removing 72–83% of latent error in representative runs and 91–98% of correctable field error. The paper also shows that the correction's diagnostic, the signal-to-noise ratio of the phase fit, predicts success before application across 50 networks.

What carries the argument

The central object is the amplitude–phase decomposition of each latent coordinate via the Hilbert analytic signal, which requires the coordinate to be narrowband (98–99% of spectral energy within ±10% of the shedding frequency). The load-bearing identity is e(t)=−2A sin(φ(t)/2) sin(2πf_s t+φ(t)/2), which shows that a correct-amplitude, phase-drifting prediction inevitably produces an error spectrum peaked at the fundamental. The correction itself is a one-parameter phase realignment, $ẍz^{{(d)}}$(t)= $Â^{{(d)}}$(t) cos($ẜḢ^{{(d)}}$(t)-ω_d t)+ $ẍz^{{(d)}}$, applied offline after the rollout without feedback into the network.

What would settle it

A concrete test: take a trained latent-space ROM on a periodic wake and compute the cumulative phase difference φ(t) over the rollout. If the error spectrum is broadband rather than sharply peaked at the vortex-shedding frequency, or if the amplitude and phase contributions to the error variance do not partition the error to within a fraction of a percent (with a negligible cross term), the central claim fails. Applying the same decomposition to a genuinely three-dimensional wake or to a flow without a dominant frequency should produce a phase share well below 95% and a cross term of order unity.

Watch

Extended reading notes

Core claim

The paper claims that the rollout error of latent-space reduced-order models of periodic bluff-body wakes is a coherent phase drift rather than unstructured compounding noise. The error's power spectrum peaks sharply at the vortex-shedding frequency of each flow, and an amplitude–phase decomposition attributes 97–98% of the error variance to the phase of the predicted oscillation and only 2.0–2.3% to its amplitude. The model learns the geometry of the attractor almost exactly and misjudges only the traversal rate. The accumulated phase error φ(t) grows linearly with time, so each latent coordinate is characterized by a single drift rate. The identity e(t)≈−Aφ(t)sin(2πf_s t) explains why so small a phase slip produces an error peak at the shedding frequency, and it is the algebraic basis for the offline, one-parameter phase-realignment correction that removes most of the correctable error.

Load-bearing premise

Each latent coordinate must be narrowband around a single dominant frequency so that the Hilbert analytic signal gives a well-defined instantaneous phase; at Re=1000 or in genuinely broadband flows this condition fails and the 95–98% phase share becomes an artifact of the chosen phase definition.

Editorial extensions

If this is right

  • A model can be excellent by every conventional metric—near-zero one-step validation error, correct amplitude, visually perfect reconstructions—and still fail on long horizons because of phase drift.
  • The error spectrum peaking at the shedding frequency is not a learned property of the network; it follows algebraically from the amplitude–phase decomposition once the amplitude is known to be correct.
  • Because the phase drift is linear, the correction improves with prediction horizon, unlike an error-fitted envelope, and the correction degrades far less than a trivial periodic-extension baseline when the flow is not exactly periodic.
  • The calibration-window R² of the phase fit is a prospective diagnostic that separates correction successes from failures (r=0.85) and prevents harmful application on poorly resolved drifts.
  • On a stationary limit cycle the correction only matches the trivial baseline of repeating the last shedding period; its genuine advantage appears on coherent but non-periodic flows, where it removes roughly half of the rollout error while tiling fails by an order of magnitude.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The phase-drift mechanism, once identified on periodic cylinder wakes, may extend to any autoregressive surrogate of an oscillatory system (e.g., modal or spectral weather emulators) where a single dominant frequency exists; the paper's own harmonic-propagator result supports that the mechanism does not depend on the LSTM architecture.
  • The R² diagnostic could be used as an early-stopping or model-selection criterion during training, since it predicts correction efficacy without needing ground truth beyond the calibration window.
  • In genuinely three-dimensional or broadband flows, where the analytic-signal phase is not uniquely defined, the paper's own breakdown at Re=1000 suggests the claim would need a different phase estimator—such as a band-passed or geometric phase—before it can be tested.
  • A time-varying drift rate, estimated from a sliding-window phase derivative, is the natural extension and would likely recover more than half of the error on the modulated-inflow wake that the paper reports as a constant-rate residual.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The paper studies autoregressive rollout error in convolutional-autoencoder-LSTM reduced-order models of two-dimensional bluff-body wakes. Using a Hilbert analytic-signal decomposition, it reports that 95-98% of the rollout-error variance is phase error, that the predicted limit-cycle amplitude is correct to within 0.15%, and that the accumulated phase difference grows linearly to only 0.02-0.03 rad over the rollout. It then proposes a one-parameter-per-coordinate phase-realignment correction, fitted on a calibration window, which removes 72-83% of latent error in representative runs and 91-98% of the correctable field error, with an R^2-based diagnostic that predicts success (r=0.85 over 50 networks). The paper includes two geometries, several Reynolds numbers, seed ensembles, DMD and harmonic-model baselines, a documented breakdown case at Re=1000, and a non-stationary modulated-inflow test where periodic tiling fails but the phase correction retains about half of the error.

Significance. If the central diagnosis is correct, the paper reframes compounding rollout error in latent-space ROMs from unstructured noise to a single interpretable defect: a slow drift of the phase of an otherwise correctly learned limit cycle. This is a substantive and practically useful reframing, and the one-parameter, self-diagnosing correction is a natural consequence. The paper's strengths are its honesty and breadth: two shedding mechanisms, multiple Reynolds numbers and latent dimensions, 50- and 40-network seed ensembles, open code and case files, explicit acknowledgment of the Re=1000 breakdown, and a non-stationary test that separates coherence from exact periodicity. The main weakness is that the quantitative headline rests on a Hilbert phase estimator that has not been validated against a known injected phase drift, and several quantitative tables are single-run entries despite large documented seed spread.

major comments (2)
  1. [Sec. IIC2, Sec. IIID, Sec. IIIE] The Hilbert analytic-signal decomposition is the sole estimator behind the quantitative headline (95-98% phase share, linear drift of 0.02-0.03 rad) and behind the correction Eq. (11), yet no positive control is reported in which a known phase drift is injected into a true or synthetic latent trajectory and recovered by the pipeline. The measured drift is only about an order of magnitude above the quoted estimator jitter, and the reported end-effect check in Sec. IIC2 only trims 5-20 samples from the start of the calibration window, not from the end of the rollout where the correction extrapolates and where Hilbert edge artifacts are largest. Because Eq. (11) is defined through the same Hilbert phase, the reported 72-83% latent and 91-98% field improvements could in part consist of removing an estimator artifact rather than a network defect. Please add a synthetic phase-injection study (known linear and, if feasible, slowly varying phase drift, with varying drift magnitude and record length) reporting the bias and variance of the recovered omega_d and of the phase share, together with a quantification of end-of-record edge effects on the extrapolated correction.
  2. [Sec. IIIK and Tables II, VII, VIII, IX] The seed study in Sec. IIIK shows that drift magnitude varies widely across random initializations (error-growth ratio mean 5.7, standard deviation 4.0 over 40 seeds), yet several quantitative comparisons that feed the headline claims are single-run entries: Table II (phase share and amplitude fidelity), Table VII (calibration-length dependence), Table VIII (linear-baseline error ratios), and Table IX (periodic-tiling comparison). The manuscript acknowledges this in Sec. IVD(g) and labels the entries representative, but the abstract and conclusions present '95-98%', '0.15%', and the correction percentages as general findings. Please replace or supplement these single-run entries with ensemble medians and dispersion (e.g., 5-95% ranges) over the existing seed ensembles, or explicitly rephrase the affected conclusions to the qualitative claim that is ensemble-supported. Without this, the reader cannot distinguish typical performance from a favorable run.
minor comments (4)
  1. [Table III] The column headers contain the run-together tokens 'nphase' and 'n acc'; separate the network count n from the improvement column so the ensemble columns can be parsed unambiguously.
  2. [Fig. 5(b)] The y-axis of panel (b) is labeled only ' [rad]' with no variable name; label it as phi(t) [rad].
  3. [Sec. IIIK] The statement that 'the decomposition returned a phase share above 95%' across seeds should state whether 95% is the observed minimum, a rounded lower bound, or a threshold, so that it matches the abstract's '95 to 98%' wording.
  4. [Abstract] The abstract's Reynolds-number range 'Re = 100 to 800' conflates the circular-cylinder range (300-800) with the square-cylinder case (Re=100); please state the ranges per geometry for clarity.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the phase/amplitude decomposition is measured from data, the correction is scored only on extrapolation, and Eq. (20) is explicitly framed as a consequence rather than independent evidence.

full rationale

I find no circular step that reduces a claimed prediction or first-principles result to its own inputs by construction. The central diagnostic—95–98% phase error with correct amplitude—is obtained by applying the Hilbert analytic-signal representation to the measured true and predicted latent trajectories (Sec. IIC2); the amplitude and phase variances are measured quantities, not fitted parameters, and the paper explicitly reports and bounds the Hilbert end-effect sensitivity without selecting the trim by extrapolation performance ('we do not select the trim by extrapolation performance, which would be circular given that the extrapolation region is also the scoring region'). The correction of Eq. (11) fits one drift rate per latent coordinate on a calibration window and is evaluated only over n > Ncal, with the paper stating 'scoring on the calibration window would be circular' and defining I and G over the extrapolation region. The R2 diagnostic is computed from the same calibration regression and is related to out-of-calibration efficacy, a prospective use of calibration information; the paper discloses that the 0.3 threshold is an in-sample operating point and that conditioning on R2 selects networks with the most correctable error. Eq. (20) is not a hidden circularity: the paper explicitly says the spectral peak and the amplitude–phase partition are not independent and that the square-cylinder result 'confirms an algebraic consequence, not an independent hypothesis,' so it does not use Eq. (20) as independent evidence for the diagnosis. There are no load-bearing self-citations, no uniqueness theorem imported from the authors, and no ansatz smuggled in by citation. The absence of a synthetic phase-injection control is a legitimate validation gap, but it is a question of estimator verification, not a reduction of the central claim to its inputs by definition.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

The paper introduces no new physical or architectural entities; the 'phase drift' is a measured property of existing latent trajectories, and the correction operates on existing latent coordinates. The load-bearing free parameters are the per-coordinate drift rate, the calibration window, and the R2 gate, all explicitly fitted or chosen and all disclosed.

free parameters (3)
  • phase drift rate omega_d per latent coordinate = -1.03e-3 to +0.65e-3 rad/time unit across Table III cases
    Slope of the linear regression of the unwrapped phase difference phi(t) over the Ncal=150 calibration window; one parameter per latent coordinate, used by the correction in Eq. (11).
  • R2 acceptance threshold = 0.3
    Empirical operating point used to gate whether the correction is applied; set before the main ensembles were trained but not validated on a dedicated held-out set, as the paper itself notes in Sec. IIIG.
  • calibration window length Ncal = 150 steps (~8 shedding cycles)
    Hyperparameter controlling how much ground truth the correction requires; shorter windows degrade or invert the correction, as shown in Table VII.
assumptions (6)
  • domain assumption Incompressible Navier-Stokes equations solved in 2D laminar mode with pimpleFoam, with boundary conditions as described
    Sec. IIA; the entire periodic-wake dataset depends on these settings, and the higher-Reynolds-number 2D cases are not physically realizable 3D wakes.
  • domain assumption Latent coordinates are narrowband around a single dominant frequency, so the Hilbert analytic signal gives a unique instantaneous phase
    Sec. IIC2; verified empirically (98-99% in-band energy) but the amplitude-phase decomposition presupposes it, and it fails at Re=1000.
  • domain assumption Predicted amplitude equals true amplitude, so a common A can appear in Eqs. (18)-(19)
    Sec. IVA; empirically supported by the 0.15% RMS amplitude deviation, but the analytical derivation of Eq. (20) requires this equality.
  • domain assumption Phase drift phi(t) is linear over the calibration and extrapolation windows
    Sec. IIID and Fig. 5(b); extrapolation of the fitted drift rate is exact only if linearity holds, and the non-stationary case shows residual error when it does not.
  • domain assumption Compression error and temporal error combine in quadrature for the ratio rho in Eq. (15)
    Sec. IIH; the paper treats rho as an interpretable diagnostic rather than an exact decomposition, acknowledging that orthogonality is not guaranteed.
  • domain assumption Two-dimensional suppression of spanwise instabilities is an acceptable testbed for the mechanism
    Sec. IVD(b); the survival of the phase-drift mechanism in genuine three-dimensional wakes is left open.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Autoregressive rollout error in latent-space reduced-order models of bluff-body wakes is accumulated phase drift." pith.science (2026). https://pith.science/paper/AD67NVVQ

@misc{pith2026260807189,
  author       = {Pith},
  title        = {Pith review of: Autoregressive rollout error in latent-space reduced-order models of bluff-body wakes is accumulated phase drift},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AD67NVVQ}},
  note         = {Machine review of arXiv:2608.07189}
}
read the original abstract

Autoregressive reduced order models suffer from compounding long horizon rollout errors, typically treated as unstructured noise. We demonstrate that for bluff body wakes across Re=100 to 800, this rollout error is highly structured and reveals what these models actually learn. For a convolutional autoencoder LSTM model, 95 % to 98% of the error is pure phase error, peaking sharply at the vortex shedding frequency. The network reproduces the attractor geometry almost exactly, matching limit cycle amplitudes within 0.15%, but traverses the cycle at slightly the wrong rate. This timing error, accumulating to just a few thousandths of a cycle over the entire rollout, drives the long horizon error even while one step validation errors appear virtually perfect. Because phase error drifts linearly, it can be corrected offline without retraining using just one parameter per latent coordinate, fitted on a short calibration window. The signal to noise ratio of this phase fit serves as a diagnostic that reliably predicts correction success (r=0.85 across 50 networks). While simple periodic baselines match this performance on stationary limit cycles, they fail by over an order of magnitude when applied to wakes driven by slowly varying inflows. Conversely, our phase correction requires only phase coherence, successfully removing roughly half of the rollout error during non stationary flow.

Figures

Figures reproduced from arXiv: 2608.07189 by the authors.

Figure 1
Figure 1. FIG. 1. Flow configurations, both on the fully developed [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. The reduced-order model and the proposed correction. A convolutional autoencoder maps the flow field to a latent [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Power spectrum of the latent rollout error, circular [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Power spectrum of the latent rollout error, square [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Amplitude–phase decomposition of the rollout error [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Latent-space error over the rollout, circular cylinder, [PITH_FULL_IMAGE:figures/full_fig_p010_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Field-space RMSE against the solver snapshots, cir [PITH_FULL_IMAGE:figures/full_fig_p011_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. Linear-propagator baselines. (a) Latent error against time for the circular cylinder at [PITH_FULL_IMAGE:figures/full_fig_p015_8.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

39 extracted references · 30 canonical work pages

  1. [1]

    Long Short-Term Memory , journal =

    Hochreiter, Sepp and Schmidhuber, J. Long Short-Term Memory , journal =. 1997 , doi =

    1997

  2. [2]

    and Balachandar, S

    Robichaux, J. and Balachandar, S. and Vanka, S. P. , title =. Physics of Fluids , volume =. 1999 , doi =

    work page 1999

  3. [3]

    International Journal for Numerical Methods in Fluids , volume =

    Sen, Subhankar and Mittal, Sanjay and Biswas, Gautam , title =. International Journal for Numerical Methods in Fluids , volume =

    work page

  4. [4]

    and Norberg, C

    Sohankar, A. and Norberg, C. and Davidson, L. , title =. International Journal for Numerical Methods in Fluids , volume =

    work page

  5. [5]

    Physics of Fluids , volume =

    Maulik, Romit and Lusch, Bethany and Balaprakash, Prasanna , title =. Physics of Fluids , volume =. 2021 , doi =

    work page 2021

  6. [6]

    and Holmes, P

    Berkooz, G. and Holmes, P. and Lumley, J. L. , title =. Annual Review of Fluid Mechanics , volume =. 1993 , doi =

    work page 1993

  7. [7]

    and Afanasiev, Konstantin and Morzy

    Noack, Bernd R. and Afanasiev, Konstantin and Morzy. A hierarchy of low-dimensional models for the transient and post-transient cylinder wake , journal =

    work page

  8. [8]

    and Dawson, Scott T

    Rowley, Clarence W. and Dawson, Scott T. M. , title =. Annual Review of Fluid Mechanics , volume =. 2017 , doi =

    work page 2017

Show all 39 references
  1. [9]

    and Noack, Bernd R

    Brunton, Steven L. and Noack, Bernd R. and Koumoutsakos, Petros , title =. Annual Review of Fluid Mechanics , volume =

  2. [10]

    , title =

    Lee, Kookjin and Carlberg, Kevin T. , title =. Journal of Computational Physics , volume =. 2020 , doi =

    2020

  3. [11]

    Journal of Fluid Mechanics , volume =

    Murata, Takaaki and Fukami, Kai and Fukagata, Koji , title =. Journal of Fluid Mechanics , volume =. 2020 , doi =

    2020

  4. [12]

    Theoretical and Computational Fluid Dynamics , volume =

    Hasegawa, Kazuto and Fukami, Kai and Murata, Takaaki and Fukagata, Koji , title =. Theoretical and Computational Fluid Dynamics , volume =. 2020 , doi =

    2020

  5. [13]

    Fluid Dynamics Research , volume =

    Hasegawa, Kazuto and Fukami, Kai and Murata, Takaaki and Fukagata, Koji , title =. Fluid Dynamics Research , volume =. 2020 , doi =

    2020

  6. [14]

    Physics of Fluids , volume =

    Nakamura, Taichi and Fukami, Kai and Hasegawa, Kazuto and Nabae, Yusuke and Fukagata, Koji , title =. Physics of Fluids , volume =

  7. [15]

    and Byeon, Wonmin and Wan, Zhong Y

    Vlachas, Pantelis R. and Byeon, Wonmin and Wan, Zhong Y. and Sapsis, Themistoklis P. and Koumoutsakos, Petros , title =. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences , volume =. 2018 , doi =

    2018

  8. [16]

    and Gaitonde, Datta V

    Mohan, Arvind T. and Gaitonde, Datta V. , title =. arXiv preprint arXiv:1804.09269 , year =

    arXiv

  9. [17]

    and Welling, Max , title =

    Brandstetter, Johannes and Worrall, Daniel E. and Welling, Max , title =. International Conference on Learning Representations (ICLR) , year =

  10. [18]

    , title =

    Sanchez-Gonzalez, Alvaro and Godwin, Jonathan and Pfaff, Tobias and Ying, Rex and Leskovec, Jure and Battaglia, Peter W. , title =. International Conference on Machine Learning (ICML) , pages =

  11. [19]

    Advances in Neural Information Processing Systems (NeurIPS) , volume =

    Bengio, Samy and Vinyals, Oriol and Jaitly, Navdeep and Shazeer, Noam , title =. Advances in Neural Information Processing Systems (NeurIPS) , volume =

  12. [20]

    Nathan and Brunton, Steven L

    Lusch, Bethany and Kutz, J. Nathan and Brunton, Steven L. , title =. Nature Communications , volume =. 2018 , doi =

    2018

  13. [21]

    and Rowley, Clarence W

    Otto, Samuel E. and Rowley, Clarence W. , title =. SIAM Journal on Applied Dynamical Systems , volume =

  14. [22]

    International Conference on Learning Representations (ICLR) , year =

    Li, Zongyi and Kovachki, Nikola and Azizzadenesheli, Kamyar and Liu, Burigede and Bhattacharya, Kaushik and Stuart, Andrew and Anandkumar, Anima , title =. International Conference on Learning Representations (ICLR) , year =

  15. [23]

    Nature Machine Intelligence , volume =

    Lu, Lu and Jin, Pengzhan and Pang, Guofei and Zhang, Zhongqiang and Karniadakis, George Em , title =. Nature Machine Intelligence , volume =. 2021 , doi =

    2021

  16. [24]

    Recurrent neural network closure of parametric

    Wang, Qian and Ripamonti, Nicol. Recurrent neural network closure of parametric. Journal of Computational Physics , volume =

  17. [25]

    , title =

    Takacs, Lawrence L. , title =. Monthly Weather Review , volume =. 1985 , doi =

    1985

  18. [26]

    and Marsden, Jerrold E

    Rowley, Clarence W. and Marsden, Jerrold E. , title =. Physica D: Nonlinear Phenomena , volume =

  19. [27]

    , title =

    Schmid, Peter J. , title =. Journal of Fluid Mechanics , volume =. 2010 , doi =

    2010

  20. [28]

    and Rowley, Clarence W

    Tu, Jonathan H. and Rowley, Clarence W. and Luchtenburg, Dirk M. and Brunton, Steven L. and Kutz, J. Nathan , title =. Journal of Computational Dynamics , volume =. 2014 , doi =

    2014

  21. [29]

    Journal of Fluid Mechanics , volume =

    Bagheri, Shervin , title =. Journal of Fluid Mechanics , volume =

  22. [30]

    , title =

    Schmid, Peter J. , title =. Annual Review of Fluid Mechanics , volume =. 2022 , doi =

    2022

  23. [31]

    Nathan and Brunton, Steven L

    Kutz, J. Nathan and Brunton, Steven L. and Brunton, Bingni W. and Proctor, Joshua L. , title =. 2016 , doi =

    2016

  24. [32]

    Proceedings of the 36th International Conference on Machine Learning (ICML) , series =

    Rahaman, Nasim and Baratin, Aristide and Arpit, Devansh and Draxler, Felix and Lin, Min and Hamprecht, Fred and Bengio, Yoshua and Courville, Aaron , title =. Proceedings of the 36th International Conference on Machine Learning (ICML) , series =

  25. [33]

    Williamson, C. H. K. , title =. Physics of Fluids , volume =

  26. [34]

    , title =

    Norberg, C. , title =. Journal of Fluids and Structures , volume =

  27. [35]

    and Durran, Dale R

    Weyn, Jonathan A. and Durran, Dale R. and Caruana, Rich , title =. Journal of Advances in Modeling Earth Systems , volume =

  28. [36]

    arXiv preprint arXiv:2202.07575 , year =

    Keisler, Ryan , title =. arXiv preprint arXiv:2202.07575 , year =

    arXiv

  29. [37]

    and Siminos, Evangelos , title =

    Budanur, Nazmi Burak and Cvitanovi\'c, Predrag and Davidchack, Ruslan L. and Siminos, Evangelos , title =. Physical Review Letters , volume =

  30. [38]

    arXiv preprint arXiv:2607.24569 , year =

    Solera-Rico, Alberto and Garc\'ia-Caspue\ nas, Patricia and Sanmiguel Vila, Carlos and Discetti, Stefano , title =. arXiv preprint arXiv:2607.24569 , year =

    arXiv

  31. [39]

    arXiv preprint arXiv:2510.00233 , year =

    Viknesh, Siva and Arzani, Amirhossein , title =. arXiv preprint arXiv:2510.00233 , year =

    arXiv

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.