Effects of strong turbulence for water waves

E.A. Kochurin; E.A. Kuznetsov

REVIEW 3 major objections 5 minor 37 references

Effects of strong turbulence for water waves

T0 review · 3 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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

Pith's one-line read Wave breaking sets the water-wave turbulence spectrum to slope -4

desk verdict Plausible numerical evidence for strong-turbulence -4 spectra, but the missing resolution/damping convergence study leaves the scaling possibly set by the filter. read the letter →

arxiv 2507.22557 v2 pith:QXKISIOY submitted 2025-07-30 physics.flu-dyn

E.A. Kochurin , E.A. Kuznetsov This is my paper · ORCID

classification physics.flu-dyn PACS 47.35.-i47.27.-i

keywords waterwaveturbulencebreakingcuspsingularitystrongconformalvariablespower-lawspectrumintermittencysurfacegravitywaves

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

The paper claims that once plane-symmetric deep-water gravity waves are driven beyond a weakly nonlinear regime, the energy spectrum of the turbulent surface is controlled by wave breaking: sharp cusps form on crests and their sudden, random appearances imprint a universal $\omega^{-4}$ law on the frequency spectrum and a $k^{-4}$ law on the wavenumber spectrum. Because both spectra share the same exponent, the effective dispersion relation in this state is linear, $\omega \sim k$, instead of the linear gravity-wave law $\omega = \sqrt{gk}$. These numerical results are put forward as the first direct confirmation of the strong-turbulence spectra predicted in [2], and they come with strongly non-Gaussian, power-law tails in the probability density of surface steepness that signal intermittency. The message matters because it separates the weak-turbulence picture, in which many resonant wave interactions transfer energy, from a strongly nonlinear picture in which actual breaking events set the statistical law.

What carries the argument

The carrying object is the cusp singularity that forms on a wave crest immediately before breaking. At a fixed spatial point the surface height $\eta(t)$ stays smooth until the breaking time $t_0$, when $d^2\eta/dt^2$ acquires a delta-function spike; after averaging over many such random events with jump amplitudes $\Gamma_i$ and occurrence rate $\nu$, the singular part of the Fourier transform yields exactly $E_\omega = g\nu \langle\Gamma^2\rangle/(2\pi\omega^4)$. Because the same cusp produces a jump in the slope $\eta_x$ as a function of $x$, the same power law shows up in $k$-space, and the agreement of the two exponents implies $\omega\sim k$. The numerical scheme uses a conformal mapping of the fluid region onto a half-plane, reducing the free-boundary problem to two coupled equations for the surface coordinate and velocity potential, with random low-frequency forcing, viscous damping $\gamma_0 k^2$, extra strong damping for $k\ge k_d=5000$, and a low-pass filter removing $k\ge N/3$ each step.

What would settle it

Run the same simulation with the dissipation threshold $k_d$ moved well below 5000 and well above it, and with the low-pass cutoff changed, and see whether the measured $k^{-4}$ and $\omega^{-4}$ slopes and the $|\eta_x|^{-7/2}$ tail survive; if the exponents move with the damping scale or filter, the spectrum is shaped by the numerics rather than by wave breaking itself. Alternatively, a wave-tank experiment with breaking-dominated waves that resolves both spatial and temporal spectra would directly test the linear $\omega\sim k$ ridge and the equal -4 exponents.

Watch

Extended reading notes

Core claim

In a one-dimensional, plane-symmetric model of deep-water surface gravity waves, written in conformal variables so the free surface is tracked exactly, the authors drive the system with random low-wavenumber forcing and damp high-wavenumber motion. For a wide range of pumping amplitudes they never see the weak-turbulence regime; instead the surface develops narrow shock-like fronts where the slope $\eta_x$ jumps. At these fronts the second time derivative of the surface elevation behaves as a sum of random delta functions, and Fourier transforming that singular part gives the spectrum $E_\omega \propto \omega^{-4}$; the same algebraic exponent $-4$ appears in the wavenumber spectrum $E_k$, and the spatiotemporal spectrum shows that disturbances move along straight lines $\omega \sim k$, not along the linear gravity wave dispersion curve. The authors claim these results are the first direct numerical confirmation of the strong-turbulence spectra predicted in [2], and that the measured probability density of steepness, with tails $\propto |\eta_x|^{-7/2}$, matches the intermittency expected for such cusp-dominated turbulence.

Load-bearing premise

The central claim assumes that the numerical damping applied to short waves and the low-pass filter used to prevent overturning wave shapes do not influence the measured spectral exponents; the paper does not report tests with different damping strengths or resolutions.

Editorial extensions

If this is right

If these spectra are the genuine strong-turbulence state, then in the plane-symmetric setup the classical Kolmogorov-Zakharov weak-turbulence cascade does not appear even at small pumping amplitudes, so weak-turbulence theory is not the right starting point for one-dimensional gravity waves.
An $\omega^{-4}$ frequency law and a $k^{-4}$ wavenumber law with linear dispersion mean that measuring only the spectral slope cannot distinguish weak from strong turbulence; the linear $\omega\sim k$ signature is the distinguishing observable.
The same cusp mechanism should make strong water-wave turbulence highly intermittent, with rare large steepness events following $\sim |\eta_x|^{-7/2}$ tails, so wave-breaking statistics, not Gaussian fluctuations, dominate extreme events.
Because this behavior is shared with Burgers turbulence and with sound turbulence, the $\omega^{-4}$/$k^{-4}$ result connects surface gravity waves to a broader family of nondispersive shock-dominated turbulent systems.

Reading between the lines

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

If the -4 slope is a genuine physical state, then the spectral collapse to $\omega \sim k$ suggests that in the inertial range the wave field behaves as a set of nearly non-dispersive shock fronts, so theories of strong turbulence that start from weak-wave interactions are missing the relevant degrees of freedom.
The numerical regularization could, in principle, set the cusp shape and therefore the high-frequency tail; a natural test is to vary the damping threshold $k_d$ and the low-pass cutoff and check whether the $-4$ exponents and the $-7/2$ PDF tails remain unchanged.
A laboratory experiment with breaking-dominated waves that measures both frequency and wavenumber spectra simultaneously could test the predicted $\omega \sim k$ ridge and the equality of the two -4 exponents, which would corroborate this mechanism beyond numerics.

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.

Desk Editor's Note

read the letter

Read it. Short numerical paper claiming that plane-symmetric deep-water gravity wave turbulence in a conformal-variable Euler solver settles into a strongly nonlinear regime dominated by wave breaking, with E_omega ~ omega^-4 and E_k ~ k^-4, linear omega~k dispersion, and -7/2 steepness PDF tails, and that weak-turbulence spectra are not observed. The genuinely new element is the direct numerical realization: shock fronts moving at nearly constant velocity, and spectra and tails that match the earlier theory in [2] and the experimental papers [16-18]. That is a real and useful observation, and the equations and numerical method are standard.

The soft spots are real but not fatal. The central one is the absence of any convergence study. The inertial range in Figs. 5-6 extends almost to the low-pass filter at k = N/3, and the enhanced damping for k >= kd = 5000 is three orders of magnitude larger than the background. The stated purpose of that regularization is to stop the cusps from overturning. Since the -4 law is generated by cusp-like slope discontinuities, the regularization could be setting the very scaling it is supposed to reveal. No run with a different N, gamma0, kd, or filter location is shown, so the stress-test concern is not answered by the paper.

Second, the paper says a 'wide range of pumping amplitudes' was probed and weak turbulence was never seen, but only one set of parameters is presented. The negative weak-turbulence result therefore rests on an unshown scan. No error bars or ensemble averages are reported anywhere, which is common for a Letter but still limits how much weight the exponents can carry.

Third, the claim 'first direct numerical confirmation' is a little aggressive given the previous numerical literature on the same model, but the self-citation to [2] is legitimate: the theory is parameter-free and the simulation is not fitted to it.

For all that, the qualitative picture is credible and the paper is doing something real. The right next step is to give the authors a chance to run the convergence tests and show a few pumping amplitudes. I would send it to review, with a request for those additions. If the -4 slope survives resolution and damping changes, this becomes a solid confirmation; as it stands, it is a well-posed observation with a load-bearing missing control.

Referee Report

3 major / 5 minor

Summary. This manuscript reports direct numerical simulations of plane-symmetric deep-water gravity waves in conformal variables, with random-phase low-wavenumber pumping and high-wavenumber viscous damping. The authors find that, rather than a weak-turbulence Kolmogorov-Zakharov spectrum, the system enters a strongly nonlinear quasi-steady state dominated by wave breaking and cusp formation. They report spatial and frequency spectra consistent with k^-4 and omega^-4 over about three decades, a linear omega ~ k branch in the spatiotemporal spectrum, and power-law tails |eta_x|^-7/2 in the steepness PDF. They interpret these as the first direct numerical confirmation of the strong-turbulence spectra predicted in [2].

Significance. If the reported spectra are genuine, this is a notable result: it gives a numerical test of a parameter-free strong-turbulence theory, links the -4 spectrum to wave-breaking cusps, and shows intermittency via a -7/2 steepness tail. The paper's central strength is that the observed exponents are not fitted and arise from direct integration of the standard potential-flow equations. However, the evidence is currently based on one run with strong artificial regularization, with no convergence study or amplitude scan, so the confirmation claim is not yet established. The result would be more convincing with a systematic study, but the core idea is defensible and the missing tests are within the scope of a revision.

major comments (3)

[Basic equations and numerical scheme (Eqs. (7)-(8); Figs. 5-6)] The central claim that the measured k^-4 and omega^-4 spectra are the strong-turbulence spectra of [2] requires ruling out that they are imposed by the numerical regularization. The run uses N=32768, gamma0=1e-5, kd=5000, and a low-pass filter removing k>=N/3 ~ 1.1e4 at every time step, with gamma0 enhanced by three orders of magnitude for k>=kd; the apparent spectral range in Figs. 5-6 extends close to this cutoff. Because the curvature singularities responsible for the predicted -4 tail are exactly what the enhanced damping and filter prevent from forming, the observed slope may simply be the Fourier signature of events smoothed at the regularization scale. I request convergence runs with at least two higher resolutions, different kd and gamma0 values, and different filter cutoffs, plus a demonstration that the spectral exponent and its plateau width are unchanged outside the dissipation/filter range.
[Abstract and Simulation Results] The abstract and the text state that for a wide range of pumping amplitudes the weak turbulence regime was not detected, but the paper reports a single simulation configuration (F0=0.6e5, kf=3, L0=0.64). No amplitude scan, ensemble averaging, or error bars are shown, so the negative claim about weak turbulence and the claim of a wide amplitude range are not supported by the presented data. Either the scan should be shown (even in summary form), or the claims should be restricted to the single studied amplitude.
[Turbulence spectra (Figs. 5-6) and PDF (Fig. 2)] The paper states that the inertial interval spans almost three decades in both k and omega, but the figures do not include error bars, confidence bands, or a statement of how the fitting interval was chosen. Given that the spectra are time averages from one run and the PDF tail in Fig. 2 is also from a single trajectory, quantitative uncertainties are needed before the exponent -4 and the exponent -7/2 can be asserted as robust. At a minimum, the authors should report the time span used, the number of independent data points in the fitting ranges, and the sensitivity of the fitted exponents to the chosen intervals.

minor comments (5)

[Basic equations and numerical scheme] The phrase 'leas to form mutually ambiguous regions of the boundary shape' contains a typo; it should read 'leads to form multivalued regions of the boundary shape' or 'overturning regions.'
[Introduction and Concluding remarks] The text uses 'power-low exponent' in the introduction; this should be 'power-law exponent.'
[Fig. 2] The abscissa label appears truncated or unclear; please specify that the plotted variable is the steepness eta_x normalized by its standard deviation sigma_{eta_x}.
[Fig. 6] The vertical axis label should be defined explicitly, for example as |eta_omega|^2 or the frequency spectrum E_omega, so that the reader can connect it to Eq. (2).
[Spatiotemporal analysis] The term 'shock fronts' is used for slope discontinuities; a sentence defining the terminology and distinguishing it from compressive shocks would improve clarity.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the predicted strong-turbulence spectrum is re-derived in the text and is not fitted into the simulation; the only self-citation is to a parameter-free theory and is not load-bearing.

full rationale

The paper's derivation chain is self-contained. The key spectral prediction E_omega proportional to omega^-4 is re-derived in the introduction from the assumed singular structure of d^2 eta/dt^2 = sum Gamma_i delta(t - t_i) followed by Fourier transform and averaging. The wavenumber spectrum E_k proportional to k^-4 is imported from [2], but the numerical experiment independently measures both spectra from the reconstructed surface eta(x,t), and the observed shock fronts and slope discontinuities provide direct evidence of the cusp singularities assumed by the theory. The simulation parameters (F0, kf, L0, gamma0, kd, N/3 filter) are stated as fixed numerical choices; nothing in the text suggests they are tuned to reproduce the -4 exponent, and no constant from the theoretical spectrum enters the equations of motion. The self-citation [2] is to a parameter-free theoretical result with stated assumptions, and the paper re-derives the central frequency spectrum itself, so the citation is not load-bearing circularity. The lack of a resolution or dissipation-strength convergence study is a genuine numerical-validation concern: the enhanced damping for k >= kd and the low-pass filter at N/3 could in principle shape the measured tail. However, that is a correctness or robustness issue, not a reduction of the prediction to the input by construction; the observed -4 plateau and the -7/2 steepness tail are not forced by any fitted parameter in the equations. Therefore no circular step meets the evidentiary standard required here.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

The central numerical result rests on standard water-wave equations and conformal variables, plus a theoretical model of singularities [2], and the authors' choices of forcing and dissipation. No new physical entity is introduced. The most important burden is the unverified assumption that the artificial high-k damping does not affect the measured spectral slope.

free parameters (5)

Pumping amplitude F0 = 0.6e5
Sets the energy input rate; one value shown, despite the claim of a wide range of pumping amplitudes.
Pumping wavenumber kf = 3
Location of low-frequency pumping in Fourier space.
Pumping width L0 = 0.64
Width of the Gaussian pumping envelope in k-space.
Viscous dissipation coefficient gamma0 = 1e-5
Linear damping rate gamma0 k^2 for k < kd; controls dissipation and the quasi-steady energy level.
Dissipation cutoff kd = 5000
Above kd the damping amplitude is increased by three orders of magnitude to stabilize small-scale instability; the choice may affect the inertial range.

assumptions (6)

domain assumption Potential flow with a free surface is governed by the Euler equations with dynamic and kinematic boundary conditions (4)-(5) and a Hamiltonian structure.
Standard model of ideal deep-water gravity waves; not re-derived in this paper.
domain assumption The conformal mapping to the lower half-plane yields the exact equations of motion (7)-(8) for potential deep-water waves.
Relies on [5,6]; the transformation is invoked as established machinery.
domain assumption The singular part of the surface curvature at breakings is modeled as delta functions in d^2 eta/dt^2, Eq. (1), taken from [2].
This is the theoretical basis of the omega^-4 spectrum; the simulation does not directly measure the singularity amplitudes or rates.
domain assumption For one-dimensional gravity waves, four-wave resonances vanish and five-wave interactions determine the weak-turbulence spectrum, as proposed in [7].
Used to interpret the absence of weak turbulence and to identify the expected Kolmogorov-Zakharov spectrum.
ad hoc to paper Random-phase pumping with uniformly distributed phases models incoherent external forcing.
Standard numerical forcing, but the parameter range is not scanned in the paper.
ad hoc to paper The low-pass filter and enhanced damping for k >= kd do not distort the inertial-range spectral slope.
Central to interpreting the observed spectra as physical; no convergence or sensitivity study is provided.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Effects of strong turbulence for water waves." pith.science (2026). https://pith.science/paper/QXKISIOY

@misc{pith2026250722557,
  author       = {Pith},
  title        = {Pith review of: Effects of strong turbulence for water waves},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QXKISIOY}},
  note         = {Machine review of arXiv:2507.22557}
}

abstract

The results of direct numerical simulation of plane-symmetric turbulence of water waves for potential flows within the framework of conformal variables taking into account low-frequency pumping and high-frequency viscous dissipation are presented. In this model, for a wide range of pumping amplitudes, the weak turbulence regime was not detected. It is shown that for typical turbulence parameters, the main effects are the processes of wave breaking, the formation of cusps on wave crests, which make the main contribution to the turbulence spectra with a dependence on frequency and wavenumber with the same exponent equal to $-4$. In this strongly nonlinear regime, the probability density of wave steepness at large deviations has power-law tails responsible for the intermittency of turbulence.

Figures

Figures reproduced from arXiv: 2507.22557 by the authors.

**Figure 3.** FIG. 3. (Color online) (a) Shape of the liquid boundary, (b) steepness and (c) curvature of the boundary at some moment of [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

**Figure 4.** FIG. 4. (Color online) Evolution of the shape (a), steepness (b) and curvature (c) of the fluid boundary in the time interval [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗

Figures from the paper (2 more)

**Figure 5.** Figure 5: FIG. 5. (Color online) Spatial spectrum of surface perturba [PITH_FULL_IMAGE:figures/full_fig_p005_5.png]

**Figure 6.** Figure 6: FIG. 6. (Color online) Frequency spectrum of surface pertur [PITH_FULL_IMAGE:figures/full_fig_p005_6.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

37 extracted references · 37 canonical work pages

[2]

E. A. Kuznetsov, JETP Lett., (80)(2), 83-89 (2004)

work page 2004
[1]

Zakharov, S.I

V.E. Zakharov, S.I. Badulin, V.V. Geogjaev, A.N. Pushkarev, Weak-turbulent theory of wind-driven sea. Earth and Space Science, 6, 540-556 (2019)

work page 2019
[3]

Zakharov, N.N

V.E. Zakharov, N.N. Filonenko, Sov. Phys. Docl.11, 881- 884 (1967)

work page 1967
[4]

Falcon, N

E. Falcon, N. Mordant, Annu. Rev. Fluid Mech., 54, 1-25 (2022)

work page 2022
[5]

Dyachenko, E.A

A.I. Dyachenko, E.A. Kuznetsov, M.D. Spector, V.E. Za- kharov, Phys. Lett. A, 221, 73-79 (1996)

work page 1996
[6]

Dyachenko, V.E

A.I. Dyachenko, V.E. Zakharov, E.A. Kuznetsov, Fizika Plazmy, 22, 916-928 (1996) [Plasma Phys. Rep., 22(10), 829-840, (1996)]

work page 1996
[7]

Dyachenko, Y.V

A.I. Dyachenko, Y.V. Lvov, V.E. Zakharov, Physica D 87(1-4), 233-261 (1995)

work page 1995
[8]

Ricard, E

G. Ricard, E. Falcon, Europhys. Lett., 135(6), 64001 (2021)

work page 2021

Show all 37 references

[9]

Kochurin, JETP Lett., 118(12), 893-898 (2023)

E.A. Kochurin, JETP Lett., 118(12), 893-898 (2023)

2023
[10]

Kochurin, P.A

E.A. Kochurin, P.A. Russkikh, Physica D, 481, 134763 (2025). 6

2025
[11]

Zakharov, J

V.E. Zakharov, J. Appl. Mech. Tech. Phys.,9(2), 190-194 (1968)

1968
[12]

Zakharov, Some problems of nonlinear theory of sur- face waves

V.E. Zakharov, Some problems of nonlinear theory of sur- face waves. Candidate’s dissertation, Institute of Nuclear Physics of the Siberian Branch of the USSR Academy of Sciences, Novosibirsk (1966)

1966
[13]

Dyachenko, P.M

S.A. Dyachenko, P.M. Lushnikov, A.O. Korotkevich, JETP letters, 98(11), 675-679 (2014)

2014
[14]

Dyachenko, P.M

S.A. Dyachenko, P.M. Lushnikov, A.O. Korotkevich, Stud. Appl. Math., 137(4), 419-472 (2016)

2016
[15]

Korotkevich, P

A.O. Korotkevich, P. M. Lushnikov, A. Semenova, S.A. Dyachenko, Stud. Appl. Math., 150(1), 119-134 (2023)

2023
[16]

Denissenko, S

P. Denissenko, S. Lukaschuk, S. Nazarenko, Phys. Rev. Lett., 99(1), 014501 (2007)

2007
[17]

Lukaschuk, S

S. Lukaschuk, S. Nazarenko, S. McLelland, P. Denis- senko, Phys. Rev. Lett., 103(4), 044501 (2009)

2009
[18]

Nazarenko, S

S. Nazarenko, S. Lukaschuk, Annu. Rev. Condens. Mat- ter Phys., 7(1), 61-88
[19]

Phillips, J

O.M. Phillips, J. Fluid Mech., 2(5), 417-445 (1957)

1957
[20]

Rosenhaus, D

V. Rosenhaus, D. Schubring, arXiv preprint arXiv:2406.18475 (2024)

2024 arXiv
[21]

Newell, V.E

A.C. Newell, V.E. Zakharov, Phys. Lett. A, 372(23), 4230-4233 (2008)

2008
[22]

Rosenhaus, G

V. Rosenhaus, G. Falkovich, Phys. Rev. Lett., 133(24), 244002 (2024)

2024
[23]

Dyachenko, V.E

A.I. Dyachenko, V.E. Zakharov, JETP Lett., 88, 307-311 (2008)

2008
[24]

Gurbatov , A.N

S.N. Gurbatov , A.N. Malakhov, A.I. Saichev. Nonlinear random waves and turbulence in nondispersive media: waves, rays, particles. / Manchester University Press
[25]

Yakhot, A

V. Yakhot, A. Chekhlov. Phys. Rev. Lett. 77, 3118 (1996)

1996
[26]

Weinan, E

E. Weinan, E. V. Eijnden. Phys. Rev. Lett. 83 2572 (1999)

1999
[27]

Frisch, J

U. Frisch, J. Bec, Burgulence. In New trends in turbu- lence Turbulence: nouveaux aspects / Berlin, Heidelberg: Springer Berlin Heidelberg. 2002. P. 341

2002
[28]

J. Bec , K. Khanin, Burgers turbulence. Phys. Rep. 447, 1-66 (2007)

2007
[29]

Kochurin, E.A

E.A. Kochurin, E.A. Kuznetsov, Phys. Rev. Lett. 133, 207201 (2024)

2024
[30]

Majda, D.W

A.J. Majda, D.W. McLaughlin, E.Tabak, J. Nonlinear Sci., 7(1), 9-44 (1997)

1997
[31]

Simonis, Y

A. Simonis, Y. Pan, Phys. Rev. E, 110(2), 024202 (2024)

2024
[32]

Chibbaro, F

S. Chibbaro, F. De Lillo, M. Onorato, Phys. Rev. Fluids, 2(5), 052603 (2017)

2017
[33]

Rumpf, T.Y

B. Rumpf, T.Y. Sheffield, Phys. Rev. E, 92(2), 022927 (2015)

2015
[34]

Sheffield, B

T.Y. Sheffield, B. Rumpf, Phys. Rev. E, 95(6), 062225 (2017)

2017
[35]

Rumpf, A.C

B. Rumpf, A.C. Newell, The Competi- tion between Wave Turbulence and Coher- ent Structures. Available at SSRN 5243541. https://papers.ssrn.com/sol3/papers.cfm?abstract id=5243541
[36]

Ricard, E

G. Ricard, E. Falcon, Phys. Rev. Fluids, 8(1), 014804 (2023)

2023
[37]

Kochurin, Water, 17(2), 140 (2025)

E.A. Kochurin, Water, 17(2), 140 (2025)

2025

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [2]

E. A. Kuznetsov, JETP Lett., (80)(2), 83-89 (2004)

work page 2004

[2] [1]

Zakharov, S.I

V.E. Zakharov, S.I. Badulin, V.V. Geogjaev, A.N. Pushkarev, Weak-turbulent theory of wind-driven sea. Earth and Space Science, 6, 540-556 (2019)

work page 2019

[3] [3]

Zakharov, N.N

V.E. Zakharov, N.N. Filonenko, Sov. Phys. Docl.11, 881- 884 (1967)

work page 1967

[4] [4]

Falcon, N

E. Falcon, N. Mordant, Annu. Rev. Fluid Mech., 54, 1-25 (2022)

work page 2022

[5] [5]

Dyachenko, E.A

A.I. Dyachenko, E.A. Kuznetsov, M.D. Spector, V.E. Za- kharov, Phys. Lett. A, 221, 73-79 (1996)

work page 1996

[6] [6]

Dyachenko, V.E

A.I. Dyachenko, V.E. Zakharov, E.A. Kuznetsov, Fizika Plazmy, 22, 916-928 (1996) [Plasma Phys. Rep., 22(10), 829-840, (1996)]

work page 1996

[7] [7]

Dyachenko, Y.V

A.I. Dyachenko, Y.V. Lvov, V.E. Zakharov, Physica D 87(1-4), 233-261 (1995)

work page 1995

[8] [8]

Ricard, E

G. Ricard, E. Falcon, Europhys. Lett., 135(6), 64001 (2021)

work page 2021