From Triadic Interactions to Kolmogorov Scaling: A Deterministic, Scale-Resolved Formulation of Energy Flux

Erik Bertram

REVIEW 3 major objections 4 minor 39 references

From Triadic Interactions to Kolmogorov Scaling: A Deterministic, Scale-Resolved Formulation of Energy Flux

T0 review · 3 major / 4 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read By decomposing the Navier-Stokes energy flux into triadic interactions between dyadic scales, this paper shows that the Kolmogorov -5/3 spectrum follows formally from a scale-invariant flux and a power-law ansatz, without statistical assump

desk verdict The paper is a clear, honest synthesis of known triadic/Littlewood–Paley machinery, but the central consistency claim with Kolmogorov −5/3 fails because the number of admissible triads in §7.1 is off by a factor of K^3. read the letter →

arxiv 2607.16381 v1 pith:7R7Q7L6H submitted 2026-07-17 physics.flu-dyn astro-ph.GAastro-ph.SR

Erik Bertram This is my paper · ORCID

classification physics.flu-dynastro-ph.GAastro-ph.SR MSC 76F0535Q3076D05

keywords turbulenceNavier-StokestriadicinteractionsenergyfluxscalelocalitydyadicdecompositionKolmogorovscalingcascade

topics Navier-Stokes Regularity

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 develops an exact, deterministic decomposition of the nonlinear energy flux in the three-dimensional incompressible Navier-Stokes equations into triadic interactions between dyadic scale shells. Under a smoothness assumption (velocity in H^s with s>5/2), the expansion is absolutely convergent and the estimates show that local, comparable-scale interactions dominate. Assuming a scale-invariant flux and a power-law form for Fourier coefficients, the triadic structure produces a self-consistency condition that selects the exponent alpha=4/3, which is equivalent to the Kolmogorov -5/3 energy spectrum. The result is formal and conditional, but it shows that the classical turbulent scaling law is consistent with the deterministic structure of the equations without statistical assumptions.

What carries the argument

The central object is the triadic flux representation Π(K) = sum over |k|≤K of sum over p+q=k of T(k,p,q), organized by dyadic Littlewood-Paley shells. The key identity is the per-triad scaling |T| ~ K^(1-3α) for local interactions, combined with the asserted combinatorial count of ~K^3 admissible triads at scale K. The dyadic decomposition also yields estimates that suppress nonlocal interactions under the smoothness assumption s>5/2, making local interactions the dominant contributors. The framework is entirely deterministic and does not rely on statistical averaging.

What would settle it

Enumerate triples (k,p,q) of integer vectors in a 3D periodic box with k+p+q=0 and all three magnitudes between K and 2K, and compare the count to K^3. If the count is not proportional to K^3, the self-consistency argument for α=4/3 fails; a direct numerical measurement of the scale dependence of the local triad contribution to Π(K) would also settle it.

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Extended reading notes

Core claim

The paper's central claim is that the triadic convolution structure of the Navier-Stokes nonlinearity, together with a scale-invariant energy flux and a power-law ansatz for Fourier coefficients, imposes a self-consistency condition on the scaling exponent. For local interactions, each triad contributes roughly K^(1-3α) when the coefficients scale as K^(-α); summing over the ~K^3 triads at scale K gives a flux scaling K^(4-3α). Requiring the flux to be independent of K in the inertial range forces α=4/3, which corresponds to the Kolmogorov energy spectrum E(k) ~ k^(-5/3). The author stresses this is a conditional, formal consistency result, not a proof of turbulence.

Load-bearing premise

The derivation depends on the assertion that the number of admissible local triads at wavenumber K scales as K^3; if the true count grows differently, the formal exponent 4/3 would not follow.

Editorial extensions

If this is right

If the formal derivation holds, the Kolmogorov -5/3 spectrum emerges directly from the triadic convolution constraint plus scale-invariant flux, with no statistical closure needed.
The dominance of local interactions gives a structural justification for the neighbor-only couplings used in shell models.
The quantitative bounds show nonlocal triads are suppressed under smoothness, reinforcing scale locality as a property of the nonlinearity itself.
The same scaling argument connects the Onsager critical threshold s=1/3 to a regime where triadic contributions become scale-invariant, linking the framework to anomalous dissipation.
Because the flux representation is exact and absolutely convergent, it can in principle be evaluated directly on single numerical velocity fields, enabling deterministic checks of the cascade.

Reading between the lines

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

The asserted K^3 count of admissible local triads is a load-bearing step that the paper does not derive; a direct count of triples (k,p,q) with k+p+q=0 and all legs of order K in three dimensions would settle whether the exponent is 4/3 or something else.
If the correct count were K^6, the self-consistency condition would become Π ~ K^(7-3α), giving α=7/3 and a spectrum E(k) ~ k^(-8/3) under the same assumptions.
The framework suggests a numerical test: measure the scale dependence of local triad contributions to Π(K) in direct numerical simulations; they should scale as K^(-2/3) if α=4/3.
The same dyadic-triadic machinery could be applied to other quadratic nonlinearities with triadic structure, such as those in magnetohydrodynamics or rotating flows, to derive analogous consistency conditions for their inertial-range spectra.

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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

The paper is a clearly written, honest synthesis of standard triadic interaction and dyadic decomposition tools. The exact flux decomposition in §3–4 is correct, and the convergence bounds in §5 under s > 5/2 are fine, though not new. The nonlocal locality estimates in §6 are also correctly stated and properly attributed to Aluie & Eyink. The paper explicitly says it is not deriving Kolmogorov from first principles, and it flags its own conditionality. Credit where due: it is a well-organized exposition of known mechanics.

The soft spot is load-bearing. In §7.1, the scaling argument assumes |v_k| ~ K^{-α}, estimates a single triad as K^{1-3α}, and then asserts that the number of admissible triads with |k|,|p|,|q| ~ K scales like K^3. That count is wrong. The shell contains ~K^3 wavevectors k. For each k, the number of p with |p|~K and |k−p|~K is also ~K^3, since the dyadic shells have thickness proportional to their radius. So the total number of ordered triples is ~K^6, not K^3. Correcting this makes Π_local(K) ~ K^{7-3α}, and constant flux gives α = 7/3, which would produce a spectrum E(K) ~ K^{-8/3} (or worse, depending on how you convert mode amplitudes to spectral density). This does not match Kolmogorov. The same inconsistency shows up in Proposition 3: the shell-energy ansatz ∥v^{(j)}∥² ~ 2^{-αj} is incompatible with the mode-level ansatz |v_k| ~ K^{-α} because a shell contains ~2^{3j} modes.

There is also a minor notational issue in §7.2: e(K) is called “energy in a dyadic shell” but defined as K²|v_k|², which is actually the spectral density; the subsequent division by K is ad hoc. These are not just presentation choices—they contribute to the error.

So the advertised “formal consistency with −5/3” rests on an arithmetic mistake. The paper still has value as a review of the triadic flux formalism, but its new contribution does not survive scrutiny. If the authors redo the count correctly and find some other route to −5/3, that would be worth seeing; as written, the central claim fails.

I would have a knowledgeable referee confirm the counting error. The paper is not a waste of refereeing time, but it should not be accepted in its current form.

Referee Report

3 major / 4 minor

Summary. The paper constructs an exact triadic decomposition of the nonlinear energy flux for the three-dimensional incompressible Navier–Stokes equations on the torus, organizes it by dyadic (Littlewood–Paley) shells, and derives convergence and locality bounds under H^s regularity with s > 5/2. It then claims that, under a scale-invariant flux assumption and a power-law ansatz for Fourier coefficients, the triadic structure yields a self-consistency condition formally compatible with the Kolmogorov −5/3 spectrum. The first half (Sections 2–6) is mostly standard: the exact flux representation, absolute convergence for s > 5/2, and the suppression of separated-scale interactions are correct but not new. The central scaling argument in Section 7, however, contains a basic arithmetic error in the triad count, and the spectral normalization is inconsistent. Once the count is corrected, the derived exponent changes and the advertised Kolmogorov consistency fails.

Significance. If correct, the paper would offer a deterministic, self-consistency route from the triadic structure of the Navier–Stokes nonlinearity to the Kolmogorov −5/3 law, without statistical assumptions. The exact triadic decomposition and the s > 5/2 convergence/localization estimates are useful but standard, and they are not sufficient to establish the advertised scaling. The paper is transparent about the conditional nature of the argument, and the formal exact decomposition is a strength. However, the scaling conclusion is invalid because it relies on a triad count that is off by a factor K^3, and the energy-spectrum normalization is internally inconsistent. These are load-bearing flaws, not presentation issues.

major comments (3)

[§7.1, Eq. (66)] The claim that 'the number of admissible triads with |k|∼K and |p|,|q|∼K scales like K^3' is incorrect. In the flux definition (30), Π(K) is summed over all |k|≤K; the number of k in the shell |k|∼K is ∼K^3, and for each such k the constraint p+q=k with |p|,|q|∼K leaves ∼K^3 choices of p (intersection of two dyadic annuli of volume O(K^3)). Hence the number of triples is ∼K^6, not K^3. This changes Eq. (66) to Π_local(K)∼K^6·K^{1−3α}=K^{7−3α}, giving α=7/3 instead of 4/3. With α=7/3, the resulting spectrum is E(K)∼K^{-11/3} (or K^{-8/3} depending on normalization), not Kolmogorov's −5/3. The advertised consistency therefore rests on a factor-K^3 arithmetic error.
[§7.2, Eqs. (70)–(73)] The relation between the Fourier amplitude and the energy spectrum is inconsistent. Equation (70) sets e(K)∼K^2|v_k|^2 and calls e(K) the energy in a dyadic shell. The energy in a dyadic shell of width O(K) is instead ∼K^3|v_k|^2; K^2|v_k|^2 is the spectral density E(k) (energy per unit wavenumber). With |v_k|∼K^{-4/3}, one obtains spectral density K^{-2/3} and shell energy K^{1/3}, neither of which corresponds to E(k)∼K^{-5/3}. The step E(K)∼e(K)/K is an ad hoc renormalization that does not repair the mismatch. A consistent calculation with |v_k|∼k^{-4/3} gives E(k)∼k^{-2/3}, not −5/3.
[Proposition 3] The dyadic-shell ansatz ∥v(j)∥²_{L2}∼2^{-αj} is not equivalent to the mode-level ansatz |v_k|∼k^{-α} used in §7.1, because a dyadic shell contains ∼2^{3j} Fourier modes. If ∥v(j)∥²∼2^{-αj}, then |v_k|∼2^{-(α+3)j/2} for k∼2^j. The proposition asserts that triadic scaling selects α=4/3 without a proof, but under a consistent translation this choice would correspond to |v_k|∼k^{-7/3}, not the k^{-4/3} of §7.1. This inconsistency undermines the claim that the framework selects a unique exponent.

minor comments (4)

[§6.3, Eq. (58)] The Onsager-critical observation |T_jmn|∼2^{(1−3s)j} is per single interaction, not per shell. At s=1/3, the per-triad term is scale-invariant, but the number of triads in a shell grows as 2^{3j} (or more), so the cumulative flux would not be scale-invariant. The connection to Onsager's criterion is misleading unless the mode count is folded in.
[Title page] The header 'Accepted on 14 July 2026 for publication in Physica D' and the acknowledgments thanking referees are unusual for a submitted manuscript and should be removed; they also complicate the review record.
[Notation] The symbol α is used for the mode-amplitude exponent in §7.1 (Eq. 63) and for the shell-energy exponent in Proposition 3, with different meanings. Please disambiguate (e.g., α for mode amplitude, β for shell energy) to avoid confusion.
[§2.4, paragraph before Eq. (12)] The statement that different dyadic discretizations 'influence prefactors but not the scaling exponents' is contradicted by the miscount in §7.1: the exponent of K depends directly on the mode count in the shell. The claim should be qualified or corrected.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central scaling result is explicitly labeled a conditional, formal consistency argument rather than a first-principles prediction.

full rationale

The paper's scaling chain (§7.1–7.2) is transparent about its inputs: it assumes a scale-invariant flux Πlocal(K)≈ε (Eq. 62) and a formal power-law ansatz |v_k|~k^{-α} (Eq. 63), then imposes the constant-flux condition to select α=4/3 (Eqs. 65–69). It repeatedly states that this is not a derivation from first principles: “This argument is formal and relies on the assumption that the scaling ansatz and constant-flux condition are valid in the regime considered. It does not constitute a derivation of turbulent scaling from the Navier–Stokes equations” (§7.1); “under a scale-invariant flux assumption, the classical scaling law is consistent, at a formal level” (§8.2); and the introduction says “The present results should be understood as structural and conditional rather than as a rigorous derivation of turbulent behavior from first principles.” Because the assumptions are explicitly stated and the advertised result is a self-consistency statement, this is not a hidden fit or a definitional identification of the conclusion with the premise. No self-citations are load-bearing, and no author-specific uniqueness or ansatz is imported. The most serious concern in the manuscript—the asserted triad count “the number of admissible triads with |k|∼K and |p|,|q|∼K scales like K^3” before Eq. (66)—is a potential correctness/arithmetic issue, not a circularity: even if the count is wrong, the derivation would be invalid rather than equivalent to its inputs. That is outside the circularity pass, so the score remains 0.

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

The central claim depends on three external inputs: high regularity (s>5/2), a constant-flux assumption, and a power-law ansatz. The first is a mathematical convenience that excludes the turbulent regime, while the latter two are the very phenomenology the argument purports to explain. The triad count that yields α=4/3 is an additional unvalidated step. Thus the paper's contribution is a conditional synthesis, not an independent derivation.

assumptions (5)

domain assumption Velocity field belongs to C([0,T]; H^s) with s>5/2 (§5.1)
Required for Fourier-coefficient decay and absolute convergence of the triadic sums; incompatible with the lower regularity expected in fully developed turbulence.
ad hoc to paper Scale-invariant flux: Π(K) ≈ ε in an inertial range (§7.1, Eq. 62)
This is the key assumption that turns the flux expression into a constraint on α; it is the turbulent state the argument claims to make plausible.
ad hoc to paper Power-law ansatz: |v_k| ~ k^{-α} (§7.1, Eq. 63)
The scaling form is assumed, not derived; without it the triadic flux has no definite K-dependence.
standard math Littlewood–Paley decomposition into dyadic annuli (§2.4)
Standard harmonic-analysis tool; the paper uses it to organize interactions into scales.
domain assumption Navier–Stokes equations on the periodic torus with divergence-free condition (§2.1)
The entire framework is set in T^3 with periodic boundary conditions; extensions to other domains are not worked out.

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Cite this review

Pith. "Pith review of From Triadic Interactions to Kolmogorov Scaling: A Deterministic, Scale-Resolved Formulation of Energy Flux." pith.science (2026). https://pith.science/paper/7R7Q7L6H

@misc{pith2026260716381,
  author       = {Pith},
  title        = {Pith review of: From Triadic Interactions to Kolmogorov Scaling: A Deterministic, Scale-Resolved Formulation of Energy Flux},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7R7Q7L6H}},
  note         = {Machine review of arXiv:2607.16381}
}

abstract

We develop a deterministic, scale-resolved formulation of energy transfer in the three-dimensional incompressible Navier-Stokes equations based on an explicit triadic decomposition of the nonlinear term in Fourier space. Using a systematic dyadic localization of the velocity field, we derive an exact representation of the nonlinear energy flux across scales and organize it in terms of interactions between well-defined scale components. Under suitable smoothness assumptions, we obtain an absolutely convergent triadic expansion and quantitative bounds that distinguish local and nonlocal contributions in scale space. This framework provides a transparent and fully explicit description of how energy transfer is mediated by triadic interactions and how scale locality emerges as a structural property of the nonlinearity. Building on this formulation, we revisit the classical inertial-range picture of turbulence from a deterministic perspective. We show that, under a scale-invariant flux assumption, the Kolmogorov $-5/3$ scaling is formally consistent with the triadic energy-transfer mechanism at a structural level. The result does not rely on statistical assumptions, but instead follows from the structural properties of the Navier-Stokes equations combined with a scale-resolved representation of the energy flux. The present work thus provides a coherent synthesis of triadic interaction analysis, dyadic scale decomposition, and classical turbulence phenomenology, offering a deterministic framework that clarifies how Kolmogorov-type scaling constraints arise in the scale-resolved structure of the underlying equations.

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Works this paper leans on

39 extracted references · 1 canonical work pages

[1]

Frisch , title =

U. Frisch , title =. 1995 , address =

1995
[2]

Aluie and G

H. Aluie and G. L. Eyink , title =. Physical Review Letters , volume =. 2010 , doi =

2010
[3]

Aluie and G

H. Aluie and G. L. Eyink , title =. Physics of Fluids , volume =. 2009 , doi =

2009
[4]

Physica D , volume=

Energy transfer in two-dimensional magnetohydrodynamic turbulence: formalism and numerical results , author=. Physica D , volume=. 2001 , publisher=

2001
[5]

The European Physical Journal E , volume=

Disentangling the triadic interactions in Navier-Stokes equations , author=. The European Physical Journal E , volume=. 2015 , publisher=

2015
[6]

Chen and R

J. Chen and R. Garcia-Mayoral , title =. 2511.05428 , archivePrefix =

arXiv
[7]

2503.06915 , archivePrefix =

Spatio-temporal characterization of nonlinear forcing and response in turbulent channel flow , author=. 2503.06915 , archivePrefix =

arXiv
[8]

Akademiia Nauk SSSR Doklady , year = 1941, month = jan, volume =

The Local Structure of Turbulence in Incompressible Viscous Fluid for Very Large Reynolds' Numbers. Akademiia Nauk SSSR Doklady , year = 1941, month = jan, volume =

1941

Show all 39 references

[9]

doi:10.1088/0034-4885/36/11/001 , year =

D C Leslie , title =. doi:10.1088/0034-4885/36/11/001 , year =

doi
[10]

2021 , note =

Turbulence theories and statistical closure approaches , journal =. 2021 , note =. doi:10.1016/j.physrep.2021.07.001 , author =

2021 doi
[11]

Journal of Fluid Mechanics , volume=

Analytical theories of turbulence , author=. Journal of Fluid Mechanics , volume=. 1970 , publisher=

1970
[12]

Reports on Progress in Physics , volume=

Theory of turbulence , author=. Reports on Progress in Physics , volume=
[13]

Journal of Fluid Mechanics , volume=

Some developments in the theory of turbulence , author=. Journal of Fluid Mechanics , volume=. 1981 , publisher=

1981
[14]

Journal of the London Mathematical Society , volume=

Theorems on Fourier series and power series , author=. Journal of the London Mathematical Society , volume=. 1931 , publisher=

1931
[15]

2003 , publisher=

Sobolev spaces , author=. 2003 , publisher=

2003
[16]

2011 , publisher=

Functional analysis, Sobolev spaces and partial differential equations , author=. 2011 , publisher=

2011
[17]

On the distribution of energy in the spectrum of turbulent flow , author=. Dokl. Akad. Nauk SSSR , volume=
[18]

Il Nuovo Cimento (1943-1954) , volume=

Statistical hydrodynamics , author=. Il Nuovo Cimento (1943-1954) , volume=. 1949 , publisher=

1943
[19]

Physics of Fluids A , volume=

The nature of triad interactions in homogeneous turbulence , author=. Physics of Fluids A , volume=. 1992 , publisher=

1992
[20]

Physics of Fluids A , volume=

Local energy transfer and nonlocal interactions in homogeneous, isotropic turbulence , author=. Physics of Fluids A , volume=. 1990 , publisher=

1990
[21]

Physica D , volume=

Locality of turbulent cascades , author=. Physica D , volume=. 2005 , publisher=

2005
[22]

2011 , publisher=

Fourier analysis and nonlinear PDEs , author=. 2011 , publisher=

2011
[23]

1998 , publisher=

Perfect incompressible fluids , author=. 1998 , publisher=

1998
[24]

Nonlinearity , volume=

Inertial energy dissipation for weak solutions of incompressible Euler and Navier-Stokes equations , author=. Nonlinearity , volume=
[25]

Bulletin of the American Mathematical Society , volume=

On the Euler equations of incompressible fluids , author=. Bulletin of the American Mathematical Society , volume=
[26]

Annals of Mathematics , volume=

A proof of Onsager's conjecture , author=. Annals of Mathematics , volume=. 2018 , publisher=

2018
[27]

Annals of Mathematics , volume=

Nonuniqueness of weak solutions to the Navier-Stokes equation , author=. Annals of Mathematics , volume=. 2019 , publisher=

2019
[28]

2001 , publisher=

Navier-Stokes equations and turbulence , author=. 2001 , publisher=

2001
[29]

O. V. Besov , title =. Doklady Akademii Nauk SSSR , volume =. 1959 , pages =

1959
[30]

Nonlinearity , volume=

Local 4/5-law and energy dissipation anomaly in turbulence , author=. Nonlinearity , volume=
[31]

Localness of energy cascade in hydrodynamic turbulence. I. Smooth coarse graining , author=. Physics of Fluids , volume=. 2009 , publisher=

2009
[32]

Physics of Fluids , volume=

The joint cascade of energy and helicity in three-dimensional turbulence , author=. Physics of Fluids , volume=. 2003 , publisher=

2003
[33]

Journal of Fluid Mechanics , volume=

Inertial-range transfer in two-and three-dimensional turbulence , author=. Journal of Fluid Mechanics , volume=. 1971 , publisher=

1971
[34]

Physical Review Letters , volume=

Energy and enstrophy transfer in decaying two-dimensional turbulence , author=. Physical Review Letters , volume=. 2003 , publisher=

2003
[35]

Physical Review E—Statistical, Nonlinear, and Soft Matter Physics , volume=

Large-scale flow effects, energy transfer, and self-similarity on turbulence , author=. Physical Review E—Statistical, Nonlinear, and Soft Matter Physics , volume=. 2006 , publisher=

2006
[36]

Physical Review E—Statistical, Nonlinear, and Soft Matter Physics , volume=

Nonlocal interactions in hydrodynamic turbulence at high Reynolds numbers: The slow emergence of scaling laws , author=. Physical Review E—Statistical, Nonlinear, and Soft Matter Physics , volume=. 2008 , publisher=

2008
[37]

Physical Review E , volume=

Improved shell model of turbulence , author=. Physical Review E , volume=. 1998 , publisher=

1998
[38]

Annual Review of Fluid Mechanics , volume=

Shell models of energy cascade in turbulence , author=. Annual Review of Fluid Mechanics , volume=. 2003 , publisher=

2003
[39]

2024 , publisher=

Hydrodynamic Instabilities and Turbulence: Rayleigh--Taylor, Richtmyer--Meshkov, and Kelvin--Helmholtz Mixing , author=. 2024 , publisher=

2024

Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.

[1] [1]

Frisch , title =

U. Frisch , title =. 1995 , address =

1995

[2] [2]

Aluie and G

H. Aluie and G. L. Eyink , title =. Physical Review Letters , volume =. 2010 , doi =

2010

[3] [3]

Aluie and G

H. Aluie and G. L. Eyink , title =. Physics of Fluids , volume =. 2009 , doi =

2009

[4] [4]

Physica D , volume=

Energy transfer in two-dimensional magnetohydrodynamic turbulence: formalism and numerical results , author=. Physica D , volume=. 2001 , publisher=

2001

[5] [5]

The European Physical Journal E , volume=

Disentangling the triadic interactions in Navier-Stokes equations , author=. The European Physical Journal E , volume=. 2015 , publisher=

2015

[6] [6]

Chen and R

J. Chen and R. Garcia-Mayoral , title =. 2511.05428 , archivePrefix =

arXiv

[7] [7]

2503.06915 , archivePrefix =

Spatio-temporal characterization of nonlinear forcing and response in turbulent channel flow , author=. 2503.06915 , archivePrefix =

arXiv

[8] [8]

Akademiia Nauk SSSR Doklady , year = 1941, month = jan, volume =

The Local Structure of Turbulence in Incompressible Viscous Fluid for Very Large Reynolds' Numbers. Akademiia Nauk SSSR Doklady , year = 1941, month = jan, volume =

1941