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The Asymptotic State of Decaying Turbulence

T0 review · 3 major / 5 minor · reviewed 2026-08-02 · deepseek-v4-flash

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Pith's one-line read The energy-decay exponent of homogeneous turbulence is fixed by the low-wavenumber shape of the initial spectrum — k² gives n≈5/4, k⁴ gives n≈10/7 — not by a universal constant.

desk verdict A carefully executed, very long DNS study whose empirical decay exponents are convincing; the LKB case still has an acknowledged, unresolved logical gap in its interpretation. read the letter →

arxiv 2602.12501 v2 pith:AFD4HFPP submitted 2026-02-13 physics.flu-dyn

Akash Rodhiya , Katepalli R. Sreenivasan This is my paper · ORCID
classification physics.flu-dyn PACS 47.27.Gs
keywords decayingturbulencedirectnumericalsimulationenergydecayexponentlow-wavenumberspectrumenstrophyloop-spacetheoryspectralself-similaritystructurefunction
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 uses exceptionally long direct numerical simulations — up to 200,000 initial eddy-turnover times — to settle what happens to homogeneous turbulence once it is left to decay. With the initial energy spectrum controlled at low wavenumbers, the kinetic energy settles into clean power-law decay in every case, but the exponent depends on how the spectrum starts: about 5/4 when the low-wavenumber spectrum scales as k², and 10/7 when it scales as k⁴. The authors conclude that there is no universal energy-decay exponent; the decay is controlled by the 'permanence of large eddies' and is contaminated by low-wavenumber 'boundary effects'. What is universal is the internal structure: the growth of a spectral-moment length scale L_M and the shape function of the second-order structure function both match a recent parameter-free loop-space theory, for both initial conditions. The paper suggests universality may live in enstrophy decay rather than energy decay.

What carries the argument

The argument is carried by the low-wavenumber tail of the initial energy spectrum and the principle of the 'permanence of large eddies': conserved large-scale invariants — linear momentum for k² spectra, angular momentum for k⁴ spectra — dictate the asymptotic decay exponent. The paper also relies on a robust length scale defined by spectral moments, L_M = ∫ kE dk / ∫ k²E dk, which grows as t^{1/2} and collapses energy and structure-function data across times and initial conditions. A dynamic regridding scheme with a conservative resolution threshold (k_max·η ≥ 3, η the dissipation scale) extends simulations to unprecedented durations.

What would settle it

Run a k⁴-initialized simulation with the low-wavenumber plateau extended to much smaller initial integral scales and monitor both the local spectral slope at the lowest resolved wavenumbers and the decay exponent; if n stabilizes at 10/7 while the low-k slope has already departed from 4, the exponent is not tied to a surviving invariant. Alternatively, a simulation initialized with a steeper, invariant-free low-k tail (say k⁶) that still decays as t^{-10/7} would show the exponent is not selective for the k⁴ state at all.

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Extended reading notes

Core claim

Given enough time and strictly controlled initial spectra, decaying homogeneous turbulence shows unambiguous power-law energy decay whose exponent is set by the low-wavenumber scaling of the initial spectrum, not by a universal constant. The k² initialization yields n ≈ 1.25–1.31, approaching the theoretical 5/4 at the highest Reynolds numbers; the k⁴ initialization yields n ≈ 10/7, persisting for thousands of turnover times. The k² low-wavenumber form survives, whereas the k⁴ form erodes early even though the 10/7 exponent persists — a contradiction the authors note but do not resolve. A recent loop-space theory matches the k² case in decay rate, in the t^{1/2} growth of the spectral-moment

Load-bearing premise

The paper assumes the decay exponent is set by the permanently surviving low-wavenumber part of the initial spectrum ('permanence of large eddies'), yet in the k⁴ simulations that prescribed low-wavenumber slope erodes within tens of eddy-turnover times while the 10/7 exponent persists for thousands — so the attribution of the decay to the angular-momentum invariant is unsupported if the k⁴ form does not survive.

Editorial extensions

If this is right

  • If the initial low-wavenumber spectrum is k², energy decays as t^{-5/4} and matches the loop-space theory; if it is k⁴, decay locks onto t^{-10/7} and stays there for thousands of eddy-turnover times — so no single decay exponent exists.
  • A k^{-5/3} inertial range imposed at t=0 is not sustained without forcing; it disappears within a few turnover times and gives way to a k^{-1} intermediate region, while the low-wavenumber part remains self-similar when plotted against the integral scale.
  • The internal structure — the shape of the second-order structure function and the t^{1/2} growth of L_M — is the same for both initial spectra, so a universal attractor exists for internal structure even though the energy-decay rate is not universal.
  • Enstrophy decays with measurably different exponents for the two spectra, but its relation to L_M is close to the universal slope for the k⁴ case, making enstrophy a candidate for the 'right' universal quantity.
  • Excluding the first few wavenumbers from the energy integral systematically steepens the measured decay exponent, so finite computational or experimental domains can contaminate decay measurements.

Reading between the lines

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

  • A testable extension beyond the paper: initialize a simulation with a much wider k⁴ plateau and check whether n=10/7 persists after the low-k slope erodes; if it does, the exponent is a memory of initial conditions rather than a live invariant, strengthening the case that the energy-decay question is ill-posed.
  • The fast disappearance of k^{-5/3} and the emergence of k^{-1} suggest a non-trivial spectral transfer in unforced turbulence; computing the spectral energy flux as a function of k would show whether the intermediate range is a forward-flux region or a near-equilibrium state.
  • Because the k² case agrees with the loop-space predictions on all tested quantities, one could conjecture that the ensemble describes a maximum-entropy background state, with the k⁴ exponent arising as a boundary correction from the lowest modes — a conjecture the authors gesture at but leave undeveloped.
  • If boundary effects are the main source of exponent variability, grid-turbulence experiments could be re-analyzed by filtering out modes below a cutoff k_0 and testing the universal internal relations; the paper's k_0=0…7 comparison tables provide a ready map for such checks.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. This paper reports a DNS study of freely decaying homogeneous isotropic turbulence with strictly controlled initial spectra: one set with a Birkhoff-Saffman (BS) k^2 low-wavenumber form and one with a Loitsianskii-Kolmogorov-Batchelor (LKB) k^4 form. The simulations span initial Taylor Reynolds numbers 30–145, use multiple realizations per case, and exploit a dynamic grid-coarsening scheme to reach durations up to ~2×10^5 initial eddy-turnover times. The authors report clean power-law decays after transients: n≈1.25 for BS and n≈10/7 for LKB, with associated power laws for integral, Taylor, and Kolmogorov scales. They also report the disappearance of the initial k^-5/3 inertial range and the emergence of a k^-1 intermediate range, compare the data with Migdal's Euler-ensemble theory, and perform a sensitivity analysis in which low wavenumbers are removed from spectral integrals. The central conclusions are that energy decay is strongly influenced by the initial low-wavenumber form and finite-domain 'boundary effects,' that energy decay exponents are not universal, and that universality may reside instead in enstrophy decay and internal spectral shape.

Significance. If substantiated, the paper would be a significant contribution: it provides an unprecedentedly long, ensemble-averaged DNS dataset for decaying turbulence with controlled large-scale initial conditions. The dynamic grid-modification scheme is validated at Re_λ=93, and the simulation parameters are documented in detail. The comparison with Migdal's theory is useful and gives credit to a recent theoretical proposal while identifying its limitation for LKB initial conditions. The main possible weakness is not the data acquisition but the interpretation of the LKB decay exponent: the classical n=10/7 result is tied to persistence of the k^4 low-wavenumber spectrum, yet the authors report that the k^4 slope erodes quickly. This inconsistency is load-bearing for the paper's central non-universality claim. The paper is not a derivation; its value is empirical, and the empirical claims need tighter quantification and internal consistency before the conclusions can be accepted.

major comments (3)
  1. [§3(b), Fig. 9(b)] The LKB leg of the non-universality claim is internally inconsistent. Figure 9(b) shows the local spectral slope at low k departing from the initial value 4 within tens of eddy-turnover times, while Figure 5(b) shows n≈10/7 persisting for thousands of turnover times. The standard derivation of n=10/7 assumes the permanence of the k^4 large-eddy regime via the Loitsianskii invariant. The authors acknowledge the difficulty ('it appears strange that the energy decay should depend so heavily on the very low number regions of the initial energy spectrum', §3(b)) but do not resolve it. Without a mechanism explaining why a non-persistent k^4 initial condition yields exactly the Loitsianskii value, the attribution is unsupported. The paper should quantify the lifetime of the k^4 regime, test sensitivity to the initial low-k slope, and distinguish between a finite-domain artifact, a transient, an
  2. [§3(a), Fig. 3(b), Fig. 5(b), Tables 5–6] The exponents n≈1.25 and 10/7 are the central quantitative results, but they are quoted without uncertainties, fitting ranges, or window-selection criteria. The plateaus are not always long (especially for BS, see inset of Fig. 3) and the values vary with Reynolds number (1.31, 1.28, 1.26, 1.26, 1.25, 1.25). Moreover, Tables 5 and 6 use a different power-law fit window (10<t/T<1000) from the main text (20<t/T<2000). The claim of 'unambiguous power-law decay' requires bootstrap or least-squares uncertainties, sensitivity to the fit window, and a quantitative criterion for choosing the plateau. Without this, the central empirical dichotomy is not as cleanly established as the text implies.
  3. [§5, Eq. (5.1)–(5.3)] The sensitivity analysis removes low wavenumbers from the spectral integrals defining E_b, Ω_b, and L_b. This is a post-processing filter, not a dynamical manipulation of the flow. In a periodic box the physical boundary effect is the absence of wavenumbers below k_min=1, whereas k0 can be raised to 7, excluding dynamically active modes that are not 'boundary' modes in any direct sense. Thus the conclusion that finite-domain boundary effects control the decay exponent is not established by Figs. 16 and Tables 5–6. To make the claim, the paper should compare simulations in boxes of different size at fixed Re_λ and initial integral scale, or otherwise demonstrate dynamical equivalence. At minimum, the conclusion should be rephrased as 'sensitivity to removal of low-wavenumber spectral content' rather than 'boundary effects.'
minor comments (5)
  1. [Fig. 11 caption] The caption for Figure 11 appears to have two '(b)' panels; panel (a) is not clearly labeled. Please correct the label.
  2. [Eq. (5.3) and Tables 5–6] Equation (5.3) defines L_b, but the table headers refer to dlog L_M. Please unify the notation.
  3. [Fig. 16 normalization] In Fig. 16, each curve is normalized by its own initial bulk value. For k0>0, the initial value differs, so the normalized curves may obscure the absolute fraction of energy removed. Please state this explicitly or show a compensated comparison.
  4. [§3(b), Fig. 8(b)] The k^-1 intermediate scaling is described as 'perceptible' for LKB, but the local-slope plot in Fig. 9(b) shows a value somewhat steeper than -1. A quantitative estimate of the fitted slope over a defined range would strengthen the claim.
  5. [§2(b), Fig. 2] The grid-modification validation is performed only for Re_λ=93. Since the method is used for all long runs, a statement about expected validity at other Reynolds numbers or after repeated regriddings would be useful.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: decay exponents are measured and compared with independent theories; the acknowledged LKB discrepancy is a physical gap, not a circular reduction.

full rationale

This is an empirical DNS study, not a derivation whose output is equivalent to its input. The central results are measured power-law exponents for kinetic energy decay, obtained from the simulations and then compared with external theoretical predictions (Saffman/BS, Kolmogorov–Batchelor/LKB, and Migdal’s Euler ensemble). The initial spectra are deliberately prescribed with k^2 or k^4 low-wavenumber forms, but the observed exponents are not defined by those inputs; they emerge from the nonlinear dynamics and are fitted from the data. The LKB case actually exposes a contradiction with the 'permanence of large eddies' assumption—the k^4 slope erodes while n≈10/7 persists—which the authors explicitly acknowledge as strange. This is an unresolved physical inconsistency, not a circular step, because the theoretical prediction is not being manufactured from the simulation’s own fitted parameters. Self-citations (John et al. 2022; Khurshid, Donzis & Sreenivasan; Buaria & Sreenivasan) are used for background, previous simulation durations, and dissipation-range fitting forms; they are not load-bearing for the paper’s main claim of non-universality. The bulk sensitivity analysis removes low-wavenumber shells and reports the resulting exponent changes; while it is unsurprising that truncation affects decay, the quantitative outcomes are data-driven and are not presented as a prediction derived from the same data. No fitted parameter is renamed as a prediction, and no uniqueness theorem is imported from the authors’ prior work. The paper is therefore self-contained in its empirical comparisons and shows no significant circularity.

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

The paper's conclusions depend primarily on fitted exponents from DNS and on several domain assumptions about the fidelity of the simulations and the interpretation of finite-domain effects. The main unresolved tension is the LKB erosion of the k^4 spectrum despite persistence of the 10/7 exponent.

free parameters (6)
  • n_BS (energy decay exponent, BS) = 1.31 (Reλ=30), 1.28 (45), 1.26 (70), 1.26 (93), 1.25 (105), 1.25 (145)
    Measured from local slope of log En vs log t over the plateau; used to claim BS decay consistent with n≈5/4 but non-universal.
  • n_LKB (energy decay exponent, LKB) = ≈10/7 ≈ 1.43 for all LKB cases
    Measured from plateau of n(t); central to the claim that LKB decay obeys Kolmogorov's prediction.
  • α_LM (growth exponent of Migdal length) = 0.53 (both BS and LKB)
    Power-law fit to L_M(t) over the plateau; compared against the theoretical 0.5.
  • Ω decay exponents = -2.25 (BS), -2.42 (LKB)
    Enstrophy vs time power-law fits, used to support the enstrophy universality discussion.
  • Virtual-origin parabola coefficients = BS: t/T = 1.65 + 832.2 L_M + 237791.8 L_M^2; LKB: t/T = -14.6 + 2020.0 L_M + 221317.1 L_M^2
    Least-squares fits used to claim t ~ L_M^2.
  • Dissipation-range spectral fit parameters (α, β, γ) = See Table 4: α≈0.327-(-0.343), β≈4.1-8.3, γ≈0.82-1.02
    Fitted to Eq. (4.7) to describe the viscous cutoff shape; used for the claim that γ→1 at late times.
assumptions (6)
  • domain assumption The incompressible Navier-Stokes equations with periodic boundary conditions, solved pseudospectrally, faithfully represent decaying homogeneous turbulence.
    Basis of all DNS results; not proven in the paper.
  • domain assumption A random Gaussian initial velocity field with a prescribed Pope-model spectrum effectively realizes the BS (k^2) and LKB (k^4) low-wavenumber conditions.
    Central to controlling initial large-scale structure; the paper notes low-k mode discreteness causes fluctuations.
  • domain assumption Ensemble averages over O(3-10) realizations with different random seeds yield statistically converged decay exponents and spectra.
    The paper uses up to 10 runs per case but does not quantify convergence or sample-to-sample spread.
  • domain assumption Dynamic grid modification with a threshold kmaxη=6 does not alter the physical decay dynamics.
    Validated for a single case (Reλ=93) against unmodified DNS; assumed transferable to all cases.
  • ad hoc to paper The persistence of the low-wavenumber initial spectrum controls the decay exponent (permanence of large eddies).
    The paper relies on this to connect initial spectrum to decay exponent, yet observes that for LKB the k^4 region erodes while the exponent persists—contradicting this axiom.
  • ad hoc to paper Finite-domain 'boundary effects' are equivalent to the effects of truncating low wavenumbers at k0 in the spectrum integrals of Sec. 5.
    Used to extrapolate from k0 sensitivity to the role of finite box size; direct equivalence is not demonstrated.

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Cite this review

Pith. "Pith review of The Asymptotic State of Decaying Turbulence." pith.science (2026). https://pith.science/paper/AFD4HFPP

@misc{pith2026260212501,
  author       = {Pith},
  title        = {Pith review of: The Asymptotic State of Decaying Turbulence},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AFD4HFPP}},
  note         = {Machine review of arXiv:2602.12501}
}
abstract

The long-time evolution of decaying homogeneous turbulence is a fundamental building block of the subject. We investigate the problem by using a comprehensive suite of Direct Numerical Simulations. The simulations cover initial Taylor microscale Reynolds numbers $Re_\lambda$ from $30 \text{ to } 145$, with multiple independent realizations obtained at each $Re_{\lambda}$ to ensure statistical robustness. The energy spectrum is initialized with the Birkhoff-Saffman (BS) form (with $E(k)\sim k^2$ for small $k$) in one case, and the Loitsianskii-Kolmogorov-Batchelor (LKB) form (with $E(k)\sim k^4$ for small $k$), in another. Simulations are performed for unprecedented durations, of the order of 200,000 initial eddy-turnover times in some instances. For both BS and LKB, the turbulent kinetic energy $En$ shows, after an initial transient, unambiguous power-law decay, $En\sim t^{-n}$, with nearly constant decay exponents $n$, whose values are consistent with past theoretical results (and thus not universal). We compute various length scales, second-order structure functions, and the spectral form at large wavenumbers; we note that an initially set $-5/3$ slope disappears quickly, while a perceptible $-1$ power region appears. In particular, we compare the present findings with predictions from the recent theory for decaying turbulence developed by Migdal 2026 Philos. Trans. R. Soc. A 384, 20250032. (doi:10.1098/rsta.2025.0032). The agreement for the BS case is excellent except for the large-wavenumber spectrum. A general discussion and assessment of results is provided in terms of the putative universality of energy decay. A main conclusion is that the energy decay is significantly influenced by ``boundary effects", and that universality likely manifests only when those effects are removed. Alternatively, it may be more useful to discuss the universality of enstrophy decay.

Figures

Figures reproduced from arXiv: 2602.12501 by the authors.

Figure 1
Figure 1. Initial energy spectra (t = 0) used for the simulations. The spectra are normalized by the initial integral length L and initial total energy En(t = 0). The wavenumber k is normalized by L. (a) BS spectra with the prescribed E(k) ∼ k 2 scaling at low wavenumbers. (b) LKB spectra with the prescribed E(k) ∼ k 4 scaling at low wavenumbers. Different colored curves correspond to different initial Taylor-microscale Reyno… view at source ↗
Figure 2
Figure 2. Validation of the grid modification method for the Reλ = 93 simulation. The plot compares the time evolution of the decay exponent n(t) for the standard simulation (solid black line) against simulations using grid modification with various thresholds (dashed lines). The curves for lower thresholds depart measurably, whereas the kmaxη = 6.0 case (red dashed line) overlaps perfectly with the unmodified simulation, dem… view at source ↗
Figure 3
Figure 3. Time evolution of (a) total kinetic energy [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (13 more)
Figure 4
Figure 4. Figure 4: Time evolution of characteristic length scales and the Taylor Reynolds number for [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]
Figure 5
Figure 5. Figure 5: Time evolution of (a) normalized total kinetic energy [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]
Figure 6
Figure 6. Figure 6: Time evolution of the integral length scale [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]
Figure 7
Figure 7. Figure 7: Time evolution of the energy spectrum E(k) from a single simulation, BS case. The different colored curves represent the spectrum at various normalized times t/T, as indicated in the legend. (a) The uncompensated energy spectrum, E(k), is plotted against the wavenumber…
Figure 8
Figure 8. Figure 8: Time evolution of the energy spectrum E(k) from a single simulation. The different colored curves in both panels represent the spectrum at various normalized times t/T, as indicated in the legend. (a) The uncompensated energy spectrum E(k) plotted against the wavenumbe…
Figure 9
Figure 9. Figure 9: Time evolution of the local logarithmic slope of the energy spectrum, [PITH_FULL_IMAGE:figures/full_fig_p012_9.png]
Figure 10
Figure 10. Figure 10: The Migdal length LM, defined in Eq. (4.4), for simulations with BS (E(k) ∼ k 2 for k → 0) and LKB (E(k) ∼ k 4 for k → 0) initial spectra. (a, c): Time evolution of LM normalized by the box length, Lbox, for (a) the BS and (c) the LKB cases. The best fit over the regi…
Figure 11
Figure 11. Figure 11: The relationship between total kinetic energy [PITH_FULL_IMAGE:figures/full_fig_p015_11.png]
Figure 12
Figure 12. Figure 12: The local scaling exponent of the second-order structure function, [PITH_FULL_IMAGE:figures/full_fig_p016_12.png]
Figure 13
Figure 13. Figure 13: The local scaling exponent ζ2(x, t)for simulations with an initial LKB spectrum (E(k) ∼ k 4 ). The exponent is defined as ζ2(x, t) = r∂r log(⟨∆v2 ⟩)(r), plotted against the normalized separation x. (a) The evolution of ζ2(x, t) at various normalized times t/T (dashed …
Figure 14
Figure 14. Figure 14: Spectral statistics in the dissipation range for the [PITH_FULL_IMAGE:figures/full_fig_p018_14.png]
Figure 15
Figure 15. Figure 15: Evolution of enstrophy for the Reλ = 145 simulations. The solid blue lines represent the BS case, and the solid orange lines represent the LKB case. (a) Time evolution of normalized enstrophy Ω(t)/Ω(0). The BS case decays with a slope of −2.25, consistent with n = 1.2…
Figure 16
Figure 16. Figure 16: Time evolution of bulk turbulence parameters for [PITH_FULL_IMAGE:figures/full_fig_p021_16.png]

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Forward citations

Cited by 2 Pith papers

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Pith tools

Reviewed August 2, 2026 · model on record in the stance chip above.