Pith. sign in

REVIEW 4 major objections 5 minor 53 references

Uncertainty Growth in Stably Stratified Turbulence

T0 review · 4 major / 5 minor · reviewed 2026-08-03 · deepseek-v4-flash

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

Pith's one-line read Stable stratification suppresses chaos in stratified turbulence

desk verdict Solid first pass at predictability in stationary stratified turbulence, but the headline λ(N) trend rests on five points with N and Re varying together. read the letter →

arxiv 2512.05656 v1 pith:IDFKZSJN submitted 2025-12-05 physics.flu-dyn cond-mat.stat-mechnlin.CDphysics.geo-ph

Mrinal Jyoti Powdel , Samriddhi Sankar Ray This is my paper · ORCID
classification physics.flu-dyncond-mat.stat-mechnlin.CDphysics.geo-ph MSC 76F4537N10 PACS 47.27.-i47.27.Qb
keywords stablystratifiedturbulenceLyapunovexponentuncertaintygrowthdecorrelatorBoussinesqequationspredictabilityanisotropydirectnumericalsimulation
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

This paper tries to establish that in forced, statistically steady stably stratified turbulence, stronger stratification monotonically reduces the largest Lyapunov exponent—the rate at which infinitesimal uncertainties grow—thereby making the flow less chaotic and more predictable at short times. The authors find this suppression even though the universal sequence of uncertainty growth (initial decay, exponential growth, saturation) persists. They argue this happens not because buoyancy directly damps perturbations, but because stratification alters how velocity perturbations align with the straining field, weakening the stretching that amplifies errors. If true, strong stratification would enhance short-time predictability of geophysical flows while keeping error growth anisotropic—much slower along the vertical than horizontally.

What carries the argument

The central diagnostics are Eulerian decorrelator fields—spatially resolved differences between two nearly identical flow realizations—and their evolution equation (Eq. 4), which budgets the growth rate into strain (β_S), viscous (β_η), buoyancy–velocity (N⟨δu_z δb⟩), and forcing terms. The key mechanistic insight comes from decomposing the strain term in the eigenbasis of the rate-of-strain tensor: stratification is claimed to act by changing the alignment of δu with the tensor's compressional eigendirection (n_3² ≃ 1), suppressing extensional stretching, while direct buoyancy coupling stays negligible.

What would settle it

Perform a series of direct numerical simulations of forced stably stratified turbulence at a fixed Taylor-scale Reynolds number (e.g., Re_λ ≈ 115) while varying the Brunt–Väisälä frequency N across the same range (N = 0 to 12). If the largest Lyapunov exponent does not decrease monotonically with N at fixed Re_λ, the paper's central claim would be falsified; conversely, if it does decrease, doubling the Reynolds number at fixed N should still preserve the monotonic trend.

Watch

Extended reading notes

Core claim

Using twin simulations of the Boussinesq equations with an imposed stable density gradient, the authors measure the growth of infinitesimal velocity and buoyancy perturbations via decorrelators. They report that as the Brunt–Väisälä frequency N increases (Froude number decreases), the largest Lyapunov exponent λ extracted from the exponential growth phase decreases monotonically, indicating suppressed chaoticity. A decomposition of the decorrelator evolution equation shows that the buoyancy–velocity cross-correlation term remains subdominant; instead, the reduction arises because perturbations align more strongly with the compressive eigendirection of the strain-rate tensor during growth, re

Load-bearing premise

The central claim that stratification suppresses chaoticity assumes the simultaneous variation of Reynolds number across runs is not responsible for the monotonic decrease in λ; the paper reports Taylor-scale Reynolds numbers between about 100 and 130 that depend on N, and no control run at matched Re with different N is provided.

Editorial extensions

If this is right

  • If confirmed, strong stable stratification would increase short-time predictability of oceanic and atmospheric flows by lowering the exponential error-growth rate compared with isotropic turbulence at similar Reynolds numbers.
  • Uncertainty in stratified turbulence grows anisotropically, with vertical error spreading much slower than horizontal; forecasts would need to account for this directional imbalance.
  • The decorrelator-based method, including spectral self-similarity and the strain-alignment decomposition, can be applied to other anisotropic and wave-supporting turbulent systems such as rotating, magnetized, or convective flows.
  • The monotonic reduction of the Lyapunov exponent with increasing N provides a quantitative target for theoretical models of chaos in stratified turbulence.
  • Since buoyancy–velocity coupling is subdominant, the suppression is a property of the velocity-field straining dynamics rather than of direct buoyancy forcing, which may simplify modeling of predictability in stratified flows.

Reading between the lines

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

  • The paper's strain-alignment mechanism suggests a testable prediction: the degree of alignment (n_3²) during the exponential growth phase should increase monotonically with N, and this alignment—not the buoyancy term—should quantitatively track the drop in λ; a direct measurement of n_3²(t) versus N would test the causal story.
  • The lack of a matched-Reynolds-number control leaves open that part of the λ(N) trend might reflect the accompanying variation in Taylor-scale Reynolds number; a control run at fixed Re_λ would separate stratification effects from Reynolds-number effects.
  • The reported anisotropy in uncertainty growth could be connected to the known layered structure of strongly stratified turbulence: vertical error confinement may be a dynamical signature of the same buoyancy inhibition that produces thin vertical scales in the energy field.
  • The self-similar decay of decorrelator spectra during the growth phase, if universal, could justify reduced-order predictability models that inject a single uncertainty length scale per direction rather than a full spectral description.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. The paper studies uncertainty growth in forced, statistically stationary stably stratified Boussinesq turbulence using twin direct numerical simulations and decorrelator diagnostics. For five Brunt–Väisälä values (N=0,1,4,7,12, with Nτ_η from 0 to ~1), the authors measure a largest Lyapunov exponent λ from exponential growth of the spatially averaged velocity decorrelator. They report that λ decreases monotonically with increasing stratification, that velocity decorrelator spectra evolve self-similarly during the growth phase but anisotropically (vertical uncertainty scales remain smaller than horizontal), and that budget analysis of the exact decorrelator evolution equation indicates the suppression of chaos is controlled by strain-mediated alignment rather than direct buoyancy coupling. The authors argue the parameter range overlaps with oceanographic conditions and conclude that stratification enhances short-time predictability.

Significance. If the central trend is robust, this would be a valuable first systematic characterization of Eulerian chaos in forced stationary stratified turbulence, with implications for predictability of geophysical flows. The paper has clear strengths: the decorrelator budget equation is derived exactly from the governing equations (Eq. 4), the spectral self-similar collapse is a clean and reproducible diagnostic, and the twin-simulation protocol is direct and non-perturbative. The anisotropic uncertainty-scale results and the buoyancy-vs-strain decomposition are physically informative. The main reservation is that the headline result—monotonic suppression of λ with stratification—is currently not cleanly isolated from the simultaneous variation of Reynolds number and from the possibility that the high-N runs leave the intended strongly stratified turbulent regime.

major comments (4)
  1. [Governing Equations, Numerical Methods, and Parameters; Fig. 2(b)] The text reports Taylor-scale Reynolds numbers 100 ≲ Reλ ≲ 130, 'the precise value depending on N', but gives no per-run values. Since ν and the forcing power P are fixed while N varies, u_rms and hence Reλ must change. The monotonic decrease of λ in Fig. 2(b) therefore conflates stratification strength with Reynolds-number dependence of chaoticity. The Fig. 2(b) caption's reference to homogeneous isotropic turbulence 'at comparable Reynolds numbers' is not supported. Please report Reλ (and Fr, Reb, u_rms, ε) for each run, normalize λ by τ_η or a large-eddy turnover time to see whether the monotone trend survives, and either include a matched-Reλ control or demonstrate quantitatively that the 30% range in Reλ cannot explain the observed variation in λ.
  2. [Governing Equations, Numerical Methods, and Parameters] The manuscript states that 0.09 ≲ Fr ≲ 0.97 and that Reb ∼ Reλ Fr² 'exceeds unity', placing the flows in the strongly stratified turbulent regime. This is not guaranteed by the reported ranges: with Reλ = 100 and Fr = 0.09, Reb ≈ 0.8, below unity. If the high-N runs have Reb < 1, they are not in the claimed regime, and the drop in λ may reflect a transition to a wave-dominated, weakly turbulent state rather than buoyancy-induced suppression of chaos in strongly stratified turbulence. Please provide the actual Reb for each run and either restrict the trend to runs satisfying Reb > 1 or explicitly discuss the implications for the interpretation.
  3. [Physical Interpretation of Uncertainty Growth] The mechanism claim—that suppression of chaos is caused primarily by strain-mediated alignment dynamics and that the buoyancy–velocity correlation is subdominant—is supported by Fig. 3(a), which is shown for N=4 only. The text states that the buoyancy term 'remains subdominant throughout most of the evolution and across all Froude numbers considered,' but no such data are shown. Please provide the decomposition of the terms in Eq. (4) for all five runs, and ideally the strain-alignment coefficients n_i² during the exponential-growth phase, so the mechanistic conclusion is not inferred from a single parameter point.
  4. [Uncertainty Growth and Lyapunov Exponents; Fig. 2] The Lyapunov exponents in Fig. 2(b) are extracted from the slope of ln Φ_u(t), but no error bars, fitting intervals, or sensitivity estimates are reported. With only five N values, the claimed monotonic reduction is sensitive to fit-window choices and statistical fluctuations. Please report confidence intervals (e.g., bootstrap over fitting windows or sub-sampling), define the exponential-growth interval used, and consider presenting λ normalized by either τ_η or the integral turnover time so that the N-dependence is not confused with changes in the basic turbulence time scales.
minor comments (5)
  1. [Throughout] A table of simulation parameters (N, Fr, Reλ, Reb, u_rms, ε, grid size, forcing parameters) would greatly improve reproducibility and would make the Reλ-dependence concern easy to assess. Currently the reader must infer these from scattered statements.
  2. [Eq. (4)] The forcing contribution ⟨δf·δu⟩ is included in the budget equation but δf is never defined. In the twin simulations each realization is forced identically, so δf=0; if that is the case, state it explicitly and drop the term or explain its meaning.
  3. [Spectral and Anisotropic Structure of Uncertainty] The self-similar collapse is demonstrated in the inset of Fig. 4(b) for the horizontal spectrum. The text says the vertical spectrum shows analogous behavior, but it is not collapsed in Fig. 4(e). Please show the same collapse for ϕ̃_u(k_z,t) or state clearly in the caption that the collapse is only for the horizontal direction.
  4. [Fig. 5] Figure 5 shows only three of the five runs (Ñ=0, 0.359, 0.999). Either include all runs or note in the caption why the selected ones are representative.
  5. [Conclusions] The geophysical-relevance argument emphasizes the overlap in Nτ_η with oceanic values, but the simulated Froude numbers (0.09–0.97) are orders of magnitude larger than the oceanic estimates cited in the text (10^{-4}–10^{-2}). 'Consistent with the regime probed by our simulations' is an overstatement; please weaken this claim or justify the relevance through a different dimensionless argument.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation: the Lyapunov exponents are measured DNS slopes and the decorrelator budget is derived exactly from the Boussinesq equations.

full rationale

The central quantity λ is extracted from the measured slope of ln Φu(t) during the exponential-growth window (Fig. 2a); it is not constructed from N or Fr, so the monotonic trend in Fig. 2(b) is an empirical DNS result rather than an identity. The decorrelator evolution equation, Eq. (4), is derived directly from the Boussinesq equations (1)-(2); citing Refs. [11,14] for the framework is methodological and not load-bearing. The strain-alignment interpretation is supported by direct evaluation of the budget terms (Fig. 3a), not by assuming the conclusion. The spectral collapse in Eq. (6) is a data-driven scaling observation, and the saturation limit Eq. (8) is the standard definition of complete decorrelation. There is no fitted parameter renamed as a prediction, and no uniqueness theorem is imported from the authors' prior work. The paper's main caveat is causal rather than circular: Taylor-scale Reynolds number varies with N (100 ≲ Reλ ≲ 130), so the decreasing λ is not uniquely attributable to stratification without matched-Re controls. That is a control/interpretation concern, not a reduction of the result to its inputs. Self-citations exist (e.g., Refs. [13,14]) but they supply diagnostics, not the claimed physical result.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper contributes measurements and diagnostics, not a new theoretical construction. λ is an output measured from exponential fits, not a fitted free parameter used to enforce the conclusions. The main unstated inputs are modeling choices (Boussinesq, horizontal-only forcing, Pr=1, finite Re) rather than invented entities.

assumptions (5)
  • domain assumption Boussinesq approximation is valid for the simulated stratification and forcing.
    Governing equations (1)-(2); assumes density fluctuations small compared to background, standard for ocean-relevant regime.
  • domain assumption The horizontally forced, kz=0 vortical forcing with constant power injection produces a statistically steady state representative of stratified turbulence.
    Eq. (3); forcing is anisotropic and may imprint on uncertainty anisotropy; no isotropically forced stratified baseline is shown.
  • domain assumption Finite-resolution DNS (up to 512^3, Reλ≈100–130) captures the exponential-growth regime and saturation; finite-Re effects do not change qualitative trends.
    Statistical stationarity assumed; conclusions acknowledge lower Re than ocean and no convergence/Re-scaling study is provided.
  • domain assumption The twin perturbation remains in the linear regime during the exponential phase, so its growth rate is the largest Lyapunov exponent.
    Section 'Uncertainty growth and Lyapunov exponents'; no check of amplitude independence or finite-time convergence is reported.
  • domain assumption Single-run time averages are representative (ergodicity).
    λ values and spectra come from one realization per N; no ensemble averaging or error bars are given.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Uncertainty Growth in Stably Stratified Turbulence." pith.science (2026). https://pith.science/paper/IDFKZSJN

@misc{pith2026251205656,
  author       = {Pith},
  title        = {Pith review of: Uncertainty Growth in Stably Stratified Turbulence},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IDFKZSJN}},
  note         = {Machine review of arXiv:2512.05656}
}
read the original abstract

We investigate uncertainty growth and chaotic dynamics in statistically steady, stably stratified three-dimensional turbulence. Using direct numerical simulations of the Boussinesq equations, we quantify the divergence of initially infinitesimal perturbations via twin simulations and decorrelator diagnostics. At short times, perturbations exhibit exponential growth, allowing us to define a (largest) Lyapunov exponent. We systematically examine how this exponent depends on stratification strength, quantified by the Brunt--V\"{a}is\"{a}l\"{a} frequency and the Froude number, in a parameter regime relevant to oceanic flows. We find that increasing stratification leads to a monotonic reduction of the Lyapunov exponent, indicating suppressed chaoticity. Despite this reduction, uncertainty growth retains the universal temporal sequence observed in homogeneous isotropic turbulence -- initial decay, exponential growth, and saturation. The growth phase is characterized by self-similar decorrelator spectra, but exhibits strong anisotropy: uncertainty spreads much more slowly along the stratification direction than horizontally, with the disparity increasing with stratification strength. An analysis of the decorrelator evolution equation reveals that the suppression of chaos arises primarily from strain-mediated alignment dynamics rather than direct buoyancy coupling. Our results provide a quantitative characterization of predictability and uncertainty growth in stratified turbulence and highlight the utility of decorrelator-based methods for anisotropic geophysical flows.

Figures

Figures reproduced from arXiv: 2512.05656 by the authors.

Figure 1
Figure 1. FIG. 1. Pseudocolor plots of [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. (a) A semilog plot of the temporal evolution of the spatially averaged velocity decorrelator Φ [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. (a) Time evolution of the individual contributions to the decorrelator growth rate in Eq. ( [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Evolution of the uncertainty spectrum in spectral space: (a)–(c) Horizontal decorrelator spectra [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Temporal evolution of the uncertainty integral length scales in the (a) vertical ( [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

53 extracted references · 2 canonical work pages

  1. [1]

    Microscopic fluctuations and turbulence,

    David Ruelle, “Microscopic fluctuations and turbulence,” Physics Letters A72, 81–82 (1979)

    1979

  2. [2]

    Turbulence and predictability of geophysical fluid dynamics and climate dynamics,

    U. Frisch and G. Parisi, “Turbulence and predictability of geophysical fluid dynamics and climate dynamics,” Proceedings of the International School of Physics Enrico Fermi, Course LXXXVIII, Varenna, 1985 (1985)

    1985

  3. [3]

    Uriel Frisch,Turbulence: The Legacy of A. N. Kolmogorov(Cambridge University Press, 1995)

    1995

  4. [4]

    Intermittency and predictability in turbulence,

    A. Crisanti, M. H. Jensen, A. Vulpiani, and G. Paladin, “Intermittency and predictability in turbulence,” Phys. Rev. Lett. 70, 166–169 (1993)

    1993

  5. [5]

    Lyapunov exponents of heavy particles in turbulence,

    J´ er´ emie Bec, Luca Biferale, Guido Boffetta, Massimo Cencini, Stefano Musacchio, and Federico Toschi, “Lyapunov exponents of heavy particles in turbulence,” Physics of Fluids18, 091702 (2006)

    2006

  6. [6]

    Predictability of dry convective boundary layers: An les study,

    Siddhartha Mukherjee, Jerˆ ome Schalkwijk, and Harmen J. J. Jonker, “Predictability of dry convective boundary layers: An les study,” Journal of the Atmospheric Sciences73, 2715 – 2727 (2016)

    2016

  7. [7]

    Scaling of lyapunov exponents in homogeneous isotropic turbulence,

    Prakash Mohan, Nicholas Fitzsimmons, and Robert D. Moser, “Scaling of lyapunov exponents in homogeneous isotropic turbulence,” Phys. Rev. Fluids2, 114606 (2017)

    2017

  8. [8]

    Chaos and predictability of homogeneous-isotropic turbulence,

    G. Boffetta and S. Musacchio, “Chaos and predictability of homogeneous-isotropic turbulence,” Phys. Rev. Lett.119, 054102 (2017)

    2017

Show all 53 references
  1. [9]

    Chaotic properties of a turbulent isotropic fluid,

    Arjun Berera and Richard D. J. G. Ho, “Chaotic properties of a turbulent isotropic fluid,” Phys. Rev. Lett.120, 024101 (2018)

    2018

  2. [10]

    Non-intermittent turbulence: Lagrangian chaos and irreversibility,

    Samriddhi Sankar Ray, “Non-intermittent turbulence: Lagrangian chaos and irreversibility,” Phys. Rev. Fluids3, 072601 (2018)

    2018

  3. [11]

    The production of uncertainty in three-dimensional navier–stokes 9 turbulence,

    Jin Ge, Joran Rolland, and John Christos Vassilicos, “The production of uncertainty in three-dimensional navier–stokes 9 turbulence,” Journal of Fluid Mechanics977, A17 (2023)

    2023

  4. [12]

    The interscale behaviour of uncertainty in three-dimensional navier–stokes turbulence,

    Jin Ge, Joran Rolland, and John Christos Vassilicos, “The interscale behaviour of uncertainty in three-dimensional navier–stokes turbulence,” Journal of Fluid Mechanics1017, A29 (2025)

    2025

  5. [13]

    Light-cone spreading of perturbations and the butterfly effect in a classical spin chain,

    Avijit Das, Saurish Chakrabarty, Abhishek Dhar, Anupam Kundu, David A. Huse, Roderich Moessner, Samriddhi Sankar Ray, and Subhro Bhattacharjee, “Light-cone spreading of perturbations and the butterfly effect in a classical spin chain,” Phys. Rev. Lett.121, 024101 (2018)

    2018

  6. [14]

    Intermit- tent fluctuations determine the nature of chaos in turbulence,

    Aikya Banerjee, Ritwik Mukherjee, Sugan Durai Murugan, Subhro Bhattacharjee, and Samriddhi Sankar Ray, “Intermit- tent fluctuations determine the nature of chaos in turbulence,” (2025), arXiv:2505.09538 [physics.flu-dyn]

    2025 arXiv

  7. [15]

    Many-body chaos in ther- malized fluids,

    Sugan Durai Murugan, Dheeraj Kumar, Subhro Bhattacharjee, and Samriddhi Sankar Ray, “Many-body chaos in ther- malized fluids,” Phys. Rev. Lett.127, 124501 (2021)

    2021

  8. [16]

    Intermittency, fluctuations and maximal chaos in an emergent universal state of active turbulence,

    Siddhartha Mukherjee, Rahul K Singh, Martin James, and Samriddhi Sankar Ray, “Intermittency, fluctuations and maximal chaos in an emergent universal state of active turbulence,” Nature Physics19, 891–897 (2023)

    2023

  9. [17]

    P. A. Davidson,Turbulence in Rotating, Stratified and Electrically Conducting Fluids(Cambridge University Press, 2013)

    2013

  10. [18]

    Fluid motions in the presence of strong stable stratification,

    James J. Riley and Marie-Pascale Lelong, “Fluid motions in the presence of strong stable stratification,” Annual Review of Fluid Mechanics32, 613–657 (2000)

    2000

  11. [19]

    Internal gravity waves: From instabilities to turbulence,

    C. Staquet and J. Sommeria, “Internal gravity waves: From instabilities to turbulence,” Annual Review of Fluid Mechanics 34, 559–593 (2002)

    2002

  12. [20]

    Layering, instabilities, and mixing in turbulent stratified flows,

    C.P. Caulfield, “Layering, instabilities, and mixing in turbulent stratified flows,” Annual Review of Fluid Mechanics53, 113–145 (2021)

    2021

  13. [21]

    Open questions in turbulent stratified mixing: Do we even know what we do not know?

    Colm-cille P. Caulfield, “Open questions in turbulent stratified mixing: Do we even know what we do not know?” Phys. Rev. Fluids5, 110518 (2020)

    2020

  14. [22]

    Mixing in stratified fluids,

    P. F. Linden, “Mixing in stratified fluids,” Geophysical & Astrophysical Fluid Dynamics13, 3–23 (1979)

    1979

  15. [23]

    Direct numerical simulations of homogeneous turbulence in density-stratified fluids,

    James J. Riley, Ralph W. Metcalfe, and Michael A. Weissman, “Direct numerical simulations of homogeneous turbulence in density-stratified fluids,” AIP Conference Proceedings76, 79–112 (1981)

    1981

  16. [24]

    The growth of a turbulent patch in a stratified fluid,

    Harindra J. S. Fernando, “The growth of a turbulent patch in a stratified fluid,” Journal of Fluid Mechanics190, 55–70 (1988)

    1988

  17. [25]

    Spectra and energy transfer in stably stratified turbulence,

    E. C. Itsweire and K. N. Helland, “Spectra and energy transfer in stably stratified turbulence,” Journal of Fluid Mechanics 207, 419–452 (1989)

    1989

  18. [26]

    The formation of layers in a uniformly stirred density gradient,

    B.R. Ruddick, T.J. McDougall, and J.S. Turner, “The formation of layers in a uniformly stirred density gradient,” Deep Sea Research Part A. Oceanographic Research Papers36, 597–609 (1989)

    1989

  19. [27]

    Turbulent mixing in stratified fluids: layer formation and energetics,

    Young-Gyu Park, J. A. Whitehead, and Anand Gnanadeskian, “Turbulent mixing in stratified fluids: layer formation and energetics,” Journal of Fluid Mechanics279, 279–311 (1994)

    1994

  20. [28]

    Turbulent diffusion in stably stratified non-decaying turbulence,

    F. Nicolleau and J. C. Vassilicos, “Turbulent diffusion in stably stratified non-decaying turbulence,” Journal of Fluid Mechanics410, 123–146 (2000)

    2000

  21. [29]

    Self-similarity of strongly stratified inviscid flows,

    Paul Billant and Jean-Marc Chomaz, “Self-similarity of strongly stratified inviscid flows,” Physics of Fluids13, 1645–1651 (2001)

    2001

  22. [30]

    Forced stratified turbulence: Successive transitions with reynolds number,

    J.-P. Laval, J. C. McWilliams, and B. Dubrulle, “Forced stratified turbulence: Successive transitions with reynolds number,” Phys. Rev. E68, 036308 (2003)

    2003

  23. [31]

    The energy cascade in a strongly stratified fluid,

    Erik Lindborg, “The energy cascade in a strongly stratified fluid,” Journal of Fluid Mechanics550, 207–242 (2006)

    2006

  24. [32]

    Vertical dispersion by stratified turbulence,

    E. Lindborg and G. Brethouwer, “Vertical dispersion by stratified turbulence,” Journal of Fluid Mechanics614, 303–314 (2008)

    2008

  25. [33]

    Kolmogorov laws for stratified turbulence,

    Pierre Augier, S´ ebastien Galtier, and Paul Billant, “Kolmogorov laws for stratified turbulence,” Journal of Fluid Mechanics 709, 659–670 (2012)

    2012

  26. [34]

    Energy spectra of stably stratified turbulence,

    Y. Kimura and J. R. Herring, “Energy spectra of stably stratified turbulence,” Journal of Fluid Mechanics698, 19–50 (2012)

    2012

  27. [35]

    Stably stratified turbulence in the presence of large-scale forcing,

    C. Rorai, P. D. Mininni, and A. Pouquet, “Stably stratified turbulence in the presence of large-scale forcing,” Phys. Rev. E92, 013003 (2015)

    2015

  28. [36]

    Waves and vortices in the inverse cascade regime of stratified turbulence with or without rotation,

    Corentin Herbert, Raffaele Marino, Duane Rosenberg, and Annick Pouquet, “Waves and vortices in the inverse cascade regime of stratified turbulence with or without rotation,” Journal of Fluid Mechanics806, 165–204 (2016)

    2016

  29. [37]

    Layered structure of stably stratified turbulent shear flows,

    A V Glazunov, EV Mortikov, KV Barskov, EV Kadantsev, and SS Zilitinkevich, “Layered structure of stably stratified turbulent shear flows,” Izvestiya, Atmospheric and Oceanic Physics55, 312–323 (2019)

    2019

  30. [38]

    Signature and energetics of internal gravity waves in stratified turbulence,

    Andrea Maffioli, Alexandre Delache, and Fabien S. Godeferd, “Signature and energetics of internal gravity waves in stratified turbulence,” Phys. Rev. Fluids5, 114802 (2020)

    2020

  31. [39]

    Internal gravity waves in stratified flows with and without vortical modes,

    Vincent Labarre, Pierre Augier, Giorgio Krstulovic, and Sergey Nazarenko, “Internal gravity waves in stratified flows with and without vortical modes,” Phys. Rev. Fluids9, 024604 (2024)

    2024

  32. [40]

    Lagrangian intermittency and vertical confinement in stably stratified turbulence,

    Arun Kumar Varanasi, “Lagrangian intermittency and vertical confinement in stably stratified turbulence,” (2025), arXiv:2503.22445 [physics.flu-dyn]

    2025 arXiv

  33. [41]

    Predictability of decaying stratified turbulence,

    Martin F. Diaz and Michael L. Waite, “Predictability of decaying stratified turbulence,” Physics of Fluids36, 065138 (2024)

    2024

  34. [42]

    Applications of finite-time lyapunov exponent in detecting lagrangian coherent structures for coastal ocean processes: a review,

    Yue Peng, Xin Xu, Qi Shao, Haiyong Weng, Haibo Niu, Zhiyu Li, Chen Zhang, Pu Li, Xiaomei Zhong, and Jie Yang, “Applications of finite-time lyapunov exponent in detecting lagrangian coherent structures for coastal ocean processes: a review,” Frontiers in Marine Science11, 13452...

    2024

  35. [43]

    Analysis of mixing structures in the adriatic sea using finite-size lyapunov exponents,

    Saeed Hariri, “Analysis of mixing structures in the adriatic sea using finite-size lyapunov exponents,” Geophysical & Astrophysical Fluid Dynamics116, 20–37 (2022). 10

    2022

  36. [44]

    Characterization of coherent structures in three- dimensional turbulent flows using the finite-size lyapunov exponent,

    Joao H Bettencourt, Crist´ obal L´ opez, and Emilio Hern´ andez-Garc ´ ıa, “Characterization of coherent structures in three- dimensional turbulent flows using the finite-size lyapunov exponent,” Journal of Physics A: Mathematical and Theoretical 46, 254022 (2013)

    2013

  37. [45]

    Stratified turbulence dominated by vortical motion,

    Michael L. Waite and Peter Bartello, “Stratified turbulence dominated by vortical motion,” Journal of Fluid Mechanics 517, 281–308 (2004)

    2004

  38. [46]

    Mixing efficiency in stratified turbulence,

    A. Maffioli, G. Brethouwer, and E. Lindborg, “Mixing efficiency in stratified turbulence,” Journal of Fluid Mechanics794, R3 (2016)

    2016

  39. [47]

    Scaling analysis and simulation of strongly stratified turbulent flows,

    G. Brethouwer, P. Billant, E. Lindborg, and J.-M. Chomaz, “Scaling analysis and simulation of strongly stratified turbulent flows,” Journal of Fluid Mechanics585, 343–368 (2007)

    2007

  40. [48]

    Geographic and seasonal distributions of brunt–v¨ ais¨ al¨ a frequency and rossby radii in the north pacific and north atlantic,

    W. J. Emery, W. G. Lee, and L. Magaard, “Geographic and seasonal distributions of brunt–v¨ ais¨ al¨ a frequency and rossby radii in the north pacific and north atlantic,” Journal of Physical Oceanography14, 294 – 317 (1984)

    1984

  41. [49]

    On the calculation of the brunt-v¨ ais¨ ala frequency,

    R.C. Millard, W.B. Owens, and N.P. Fofonoff, “On the calculation of the brunt-v¨ ais¨ ala frequency,” Deep Sea Research Part A. Oceanographic Research Papers37, 167–181 (1990)

    1990

  42. [50]

    Kanwal Shahzadi and Nadia Pinardi,SeaDataCloud Brunt–V¨ ais¨ al¨ a Frequency profiles for the Atlantic and Pacific Oceans, Report (2021)

    2021

  43. [51]

    Viscosity of sea water at moderate temperatures and pressures,

    E. M. Stanley and R. C. Batten, “Viscosity of sea water at moderate temperatures and pressures,” Journal of Geophysical Research (1896-1977)74, 3415–3420 (1969), https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/JC074i013p03415

    1977 doi

  44. [52]

    Observations of turbulence in a tidal beam and across a coastal ridge,

    R.-C. Lien and M. C. Gregg, “Observations of turbulence in a tidal beam and across a coastal ridge,” Journal of Geophysical Research: Oceans106, 4575–4591 (2001), https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/2000JC000351

    2001 doi

  45. [53]

    Oceanic turbulence at millimeter scales,

    Javier Jim´ enez, “Oceanic turbulence at millimeter scales,” Oceanogr. Lit. Rev3, 598 (1998)

    1998

Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.