{"id":"6937a56d-f124-4165-bc3a-78e7b4698c02","arxiv_id":"1909.01177","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":12,"one_line_summary":"Finite-time Lyapunov exponents from DNS are robust, quickly converging measures of chaos in homogeneous isotropic turbulence, and a Reynolds-dependent dissipation correction resolves the prior alpha discrepancy.","lead":"Direct numerical simulations show that finite-time Lyapunov exponents in forced turbulence are stable across lattice sizes and measurement steptimes, and their statistics converge faster than energy or dissipation. The paper also reconciles earlier conflicting scalings of the Lyapunov exponent with Reynolds number and tests the method in magnetohydrodynamic turbulence.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The FTLE mean is assumed to be the true maximal Lyapunov exponent without a δ0 or tangent-linear convergence test; the paper's own caveats indicate the Gaussianity and fast decorrelation may be reset artifacts, so the fitted exponents and stability claims are only conditionally supported.","rationale":"The paper's main claims are that the Eulerian maximal Lyapunov exponent measured by the FTLE method is robust, converges faster than Reynolds number or energy, and resolves the α=0.53 versus α=0.64 discrepancy through TE0=E/ε and the Reynolds-dependent dissipation rate. All of these claims require that the FTLE estimator returns the true maximal Lyapunov exponent of the underlying flow. The estimator uses a fixed finite perturbation δ0=10^-3 and finite steptime Δt=0.1 (Section II.A). The Oseledets theorem cited by the authors addresses the infinite-time, infinitesimal-perturbation limit; the finite-time and finite-perturbation caveat is acknowledged via reference [56]. The only scanned parameter is Δt (Section III.E), which does not test the δ0→0 limit and, if anything, moves away from the t→∞ limit at small Δt. The direct-method comparison is described only qualitatively, so it does not quantitatively bound the possible bias. A systematic bias would propagate directly into α, αE, γ, σλ/λ, and the MHD exponent α=0.14. The fast-statistics claim is similarly fragile: the authors themselves suggest in Sections III.A and IV that the near-Gaussian distribution and the short decorrelation time may be consequences of the perturbation procedure. If that is true, the robustness and faster-convergence claims describe the estimator rather than the turbulence itself. The paper still has real supporting evidence: steptime stability is demonstrated over a wide range, the lattice-size dependence is reported, the decaying-turbulence test favors Eq. (21), and the mean value agrees with the earlier direct-method result of [14]. These elements reduce but do not remove the concern, so the reader's conditional verdict is appropriate. A targeted δ0 and tangent-linear test would settle whether the concern actually lands, and I agree with the reader that this is the weakest assumption in the paper.","tokens_in":20943,"tokens_out":12591,"duration_ms":138929,"concrete_test":"Rerun the highest-Re hydrodynamic case (Table I, N=512, ν=0.0006) with δ0 = 10^-2, 10^-3, 10^-4, and 10^-5 at fixed Δt=0.1, and in parallel evolve the perturbation using the tangent-linearized NSE over the same intervals. If the mean λ or σλ varies by more than the statistical errors quoted in Table I, or if the nonlinear-reset FTLE mean disagrees systematically with the tangent-linear estimate, then the FTLE mean is biased and the fitted scaling exponents and MHD comparison are method-dependent.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central quantitative results—α=0.53±0.02, αE=0.658±0.006, σλ/λ≈0.2, the MHD α=0.14, and the claim that λ statistics converge in roughly 10TE0—all come from the FTLE estimator of Section II.A (Eqs. 9–12) with δ0=10^-3 fixed and Δt=0.1. Oseledets' theorem guarantees independence of the initial infinitesimal perturbation only in the δ0→0, t→∞ limit, and the paper explicitly concedes finite-perturbation and finite-time dependence [56]. No δ0 convergence study is reported; the steptime scan in Section III.E (Fig. 12) varies Δt, not δ0, and decreasing Δt moves further from the t→∞ limit. The one indirect check, the direct method of [14], is mentioned qualitatively but not quantified in this paper. If the mean FTLE carries a δ0-dependent bias, all fitted exponents and the hydrodynamic/MHD comparison shift together, and the claimed robustness becomes a property of the estimator rather than of the flow's maximal Lyapunov exponent. The paper itself notes in Sections III.A and IV that the near-Gaussianity and short decorrelation time 'could well be a consequence of the perturbation method,' so the fast-statistics claim lacks independent support.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript reports direct numerical simulations of forced homogeneous isotropic turbulence, plus a set of MHD runs, using the finite-time Lyapunov exponent (FTLE) method of Boffetta and Musacchio with perturbation amplitude δ0 = 10^-3 and steptime Δt = 0.1. It measures the mean Lyapunov exponent λ and its fluctuation σλ, fits λT0 = D Re^α (α = 0.53 ± 0.02) and λTE0 = DE Re^αE (αE = 0.658 ± 0.006), and attributes the difference between these exponents to the Reynolds-number-dependent dimensionless dissipation rate Cε(Re) of McComb et al. It further claims that FTLE histograms are approximately Gaussian while Reynolds-number histograms are not, that λ and σλ are insensitive to lattice size and steptime, that the Lyapunov signal decorrelates in about TE0/10 so that roughly 10TE0 of data suffice for converged statistics, that a decaying-turbulence test favors TE0 = E/ε as the relevant large-eddy timescale, and that MHD turbulence exhibits a much weaker scaling, α = 0.14 ± 0.02.","tokens_in":21379,"tokens_out":7845,"duration_ms":76899,"significance":"If the results hold, the paper provides a useful diagnostic: the Eulerian maximal Lyapunov exponent appears to be a comparatively cheap and stable statistical observable in both hydrodynamic and MHD turbulence. The explicit reconciliation of the earlier α = 0.53 versus 0.64 discrepancy through Cε(Re) is a concrete, checkable contribution, and the decaying-turbulence experiment is a genuine out-of-sample test because its constants were obtained from forced runs and then used to predict the growth of the difference field in an unforced flow. The autocorrelation analysis and the run-time rule Tr ≈ 100Td are practically valuable. The main weakness is that the FTLE estimator is not validated against the δ0→0 and t→∞ limits required by Oseledets theory, leaving the quantitative claims conditional.","major_comments":[{"comment":"The central quantitative claims rest on the assumption that the mean FTLE, λ = ⟨λ̃⟩, is the true maximal Lyapunov exponent, but this is never validated. Oseledets' theorem guarantees independence of the initial perturbation only in the δ0→0 and t→∞ limits, and the paper explicitly concedes finite-perturbation and finite-time dependence [56]. Figure 12 varies the steptime Δt, not the perturbation amplitude δ0, which is fixed at 10^-3; decreasing Δt moves further from the t→∞ limit, and the agreement with the direct method is described only qualitatively in Section III.E. A δ0 scan (for example 10^-4 to 10^-5, ideally including a tangent-linear check) is needed to show that α, αE, σλ/λ ≈ 0.2, and the MHD comparison are properties of the flow rather than of the estimator. As written, a δ0-dependent bias would shift all fitted exponents and the hydrodynamic/MHD comparison together.","section":"II.A (Eqs. 11–12) and III.E (Fig. 12)"},{"comment":"The claims that FTLEs are approximately Gaussian and that λ statistics decorrelate on a time Td ≈ 0.5 ≈ TE0/10 are flagged by the authors themselves as possibly being reset artifacts: Section III.A states that the near-Gaussianity 'could well be a consequence of the perturbation method', and Section IV states that the short self-correlation time 'might indicate that the perturbations introduced in the FTLE method have a relevant effect' on the measured signal. Because the fast-statistics result (run time of about 10TE0 rather than 100TE0) is one of the paper's main conclusions, it needs independent support. For example, the autocorrelation of a direct-method or tangent-linear signal could be compared, or Td could be shown to be insensitive to both Δt and δ0 over a wide range, rather than only noting that Td is larger than the steptime.","section":"III.A and IV (Figs. 14–17)"},{"comment":"The reconciliation of α = 0.53 ± 0.02 with αE = 0.658 ± 0.006 is algebraically transparent, but the paper states 'strong agreement' between its simulation data and Eq. (19) without showing a plot or giving a quantitative comparison. Since Eq. (23) uses the constants Cε,∞ = 0.486 and C = 18.9 from [50] to predict ⟨α⟩exp ≈ 0.52, the reader needs to see the measured Cε(Re) values in the same figure or table to confirm that this correction is actually operating as claimed in the present runs.","section":"III.D (Eqs. 19–24)"},{"comment":"The weighted least-squares fits in Sections III.C and III.D use σλ as a scatter measure, but the effective number of independent samples is not reported and the standard errors on the individual λ values are not given in Table I. If the fit weights are computed as σλ/√n without accounting for the autocorrelation (Td ≈ 0.5 and Δt = 0.1 imply several correlated samples per decorrelation time), the reported uncertainties on α and αE could be underestimated. Please state the weights used and the effective number of independent FTLE samples, or otherwise justify the error bars.","section":"III.C and IV (Eq. 30, Table I)"}],"minor_comments":[{"comment":"The autocorrelation lag is also denoted Δt, the same symbol as the FTLE steptime; although the text notes the distinction, using a separate symbol such as τlag would avoid confusion.","section":"IV (Figs. 14–17)"},{"comment":"The text reports n = 11863 FTLE samples from a run of 200T0, but with Δt = 0.1 and T0 = 2.43 from Table I this gives roughly 4860 samples; please clarify the run length or the steptime used for this figure.","section":"III.A (Fig. 4)"},{"comment":"Equation (14) mixes ensemble-average and sample notations; it should be written as an explicit sample variance, for example σ²λ = (1/(n−1)) Σ (λ̃i − λ)², to make the estimator unambiguous.","section":"II.A (Eq. 14)"},{"comment":"Equation (17) adds the two propagated errors linearly; for independent fluctuations the quadrature sum is the standard choice, and the linear sum should be justified as a conservative upper bound.","section":"III.C (Eq. 17)"},{"comment":"The fit giving α = 0.14 ± 0.02 is not shown in Figure 20; please include the fitted line or state the fit procedure explicitly.","section":"V (Fig. 20)"}],"recommendation":"major_revision","confidential_remarks":"The paper would be considerably strengthened by depositing the FTLE time series, the simulation parameters, and a short δ0-scan study; given the central role of the FTLE estimator, I would not recommend acceptance without that validation. The MHD section is exploratory but fits the journal's scope."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague, here's my read of Ho et al. It's a solid, squarely empirical contribution to the Eulerian-chaos literature, not a breakthrough. What's genuinely new: a systematic demonstration that the mean and variance of FTLEs are insensitive to lattice size and to FTLE steptime, an approximate Gaussianity result for FTLE distributions, a run-time rule based on signal decorrelation, and a compact resolution of the alpha = 0.53 vs 0.64 discrepancy. The resolution is the best part: by using TE0 = E/epsilon and the Reynolds-dependent C_epsilon(Re) of McComb et al., the paper turns two apparently inconsistent scalings into one relation, and the predicted alpha ~ 0.52 matches the direct measurement. The decaying-turbulence test is a nice out-of-sample check, and it favors Eq. (21) without having been fit to that data, though the authors do allow a small post-hoc shift within error bars.\n\nThe main soft spot is real and the paper half-admits it: every quantitative claim—the exponents, the fluctuation scaling, the MHD comparison—passes through the FTLE estimator with delta0 fixed at 1e-3 and dt = 0.1, and there is no delta0 convergence study. Oseledets guarantees independence of the initial perturbation only in the delta0 -> 0, t -> infinity limits, and the authors cite Goldhirsch et al. for the finite-time, finite-perturbation dependence but never probe it. If the estimator carries a delta0-dependent bias, all fitted exponents shift together and the MHD contrast is suspect. One mitigating fact is that the hydrodynamic alpha = 0.53 matches the direct-method value from the same group [14], which is an independent estimator; that makes me think the mean is probably fine. But the fast-decorrelation and near-Gaussianity claims are explicitly conjectured by the authors to be possible artifacts of the perturbation method, so I would not treat those as flow physics until shown otherwise.\n\nOther gaps are minor: steptime stability is shown for one simulation per fluid case; no code or data is shipped; the smallest lattice sizes are marginally resolved. None of that sinks the paper. The central scaling argument holds up; the missing delta0 test is a required revision, not a fatal flaw.\n\nThis one deserves a serious referee. It is useful for anyone running forced turbulence DNS who wants a practical run-time rule or a reconciliation of past Lyapunov-exponent scalings. I would send it to peer review with a request for a delta0 scan and, ideally, code/data release. I probably wouldn't cite it myself until the estimator sensitivity is closed.","headline":"Solid empirical contribution with a genuinely useful resolution of the alpha discrepancy, but the FTLE estimator's delta0 sensitivity is untested and should be fixed before the robustness claims are taken at face value.","tokens_in":21902,"tokens_out":2823,"would_cite":false,"duration_ms":31414,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"The maximum Lyapunov exponent is a fast, steptime-independent measure of chaos in isotropic turbulence.","keywords":["finite-time Lyapunov exponents","homogeneous isotropic turbulence","direct numerical simulation","Eulerian chaos","Reynolds-number scaling","dimensionless dissipation rate","decorrelation time","MHD turbulence"],"falsifier":"Repeat the same FTLE procedure at fixed Reynolds number, lattice size, and steptime while varying the renormalization amplitude $\\delta_0$ from $10^{-4}$ to $10^{-2}$; if the mean exponent or its standard deviation drifts systematically with $\\delta_0$, the ensemble average is not the true maximal Lyapunov exponent and the fitted scaling exponents are contaminated. A complementary check is to measure the unrenormalized exponential growth slope (the direct method) on identical runs and compare it with the FTLE average.","tokens_in":20740,"feed_emoji":"🌪️","tokens_out":17638,"duration_ms":148012,"temperature":0.7,"pith_summary":"The paper claims that in forced homogeneous isotropic turbulence, the maximum Lyapunov exponent — the rate at which two nearly identical velocity fields diverge — is a robust statistical measure of chaos: its mean and fluctuations are insensitive to numerical lattice size and to the renormalization steptime, even when that steptime is reduced to a few simulation timesteps. It reaches stable statistics much faster than energy, dissipation, or Reynolds number because its time signal decorrelates on a timescale of about a tenth of the large-eddy turnover time, so roughly ten turnover times suffice for converged Lyapunov averages. The paper also reconciles two previously conflicting measurements of how the exponent scales with Reynolds number by showing that the conflict came from using different definitions of the large-eddy turnover time, and it assembles evidence that the energy-transfer time $E/\\varepsilon$, not the integral-scale time $L/U$, is the timescale that governs Eulerian chaos in turbulence.","feed_headline":"Turbulence chaos metric stabilizes ten times faster than energy","feed_subtitle":"Finite-time Lyapunov exponents converge in a tenth of the run time and settle a Reynolds-scaling dispute.","key_machinery":"The load-bearing object is the finite-time Lyapunov exponent (FTLE) procedure: at each steptime $\\Delta t$, the difference between two velocity fields is measured through the energy in the difference spectrum, the perturbation is rescaled to a fixed small amplitude $\\delta_0=10^{-3}$, and one exponent sample is recorded as $\\tilde\\lambda = (1/\\Delta t)\\ln(\\delta_{\\Delta t}/\\delta_0)$. Repeating this produces an ensemble of exponents whose mean and variance can be studied statistically; the paper's robustness claims are statements about how this ensemble behaves when $\\Delta t$, lattice size, and Reynolds number are varied. The second load-bearing ingredient is the identity connecting the two large-eddy timescales, $T_{E0}=E/\\varepsilon = (3/(2C_\\varepsilon(\\mathrm{Re})))T_0$, where $C_\\varepsilon(\\mathrm{Re})=C_{\\varepsilon,\\infty}+C/\\mathrm{Re}$ comes from an asymptotic expansion of the structure-function equations; this identity converts the measured $T_{E0}$-based exponent $\\alpha_E$ into a slightly Reynolds-number-dependent $\\alpha$ for $T_0$-based scaling, reconciling the earlier $\\alpha\\approx0.53$ and $\\alpha\\approx0.64$ measurements. The decorrelation time $T_d$ of each signal, read off from the small-lag parabolic behavior of $1-\\rho_{X,\\Delta t}$, supplies the run-time rule $T_r\\approx100 T_d$.","core_discovery":"On the paper's own terms, the central discovery is that a single number — the Eulerian maximum Lyapunov exponent $\\lambda$, averaged over many finite-time measurements — is a stable and quickly convergent descriptor of turbulence. Using direct numerical simulations with a fixed dissipation-rate forcing, the authors measure $\\lambda$ by repeatedly perturbing a copy of the velocity field, letting the pair evolve for a short steptime $\\Delta t$, and recording the logarithmic growth before renormalizing the difference to a small fixed amplitude $10^{-3}$. They find that $\\lambda$ and its standard deviation $\\sigma_\\lambda$ barely change when the lattice size is varied at fixed viscosity, and barely change when $\\Delta t$ is varied from values near the simulation timestep up to the direct-method limit. The exponent's autocorrelation time is about $0.5$ in simulation time units, roughly a tenth of the large-eddy turnover time, whereas energy decorrelates only after about two turnover times; hence a run of about ten turnover times gives converged Lyapunov statistics, while energy needs roughly a hundred. Fitting $\\lambda T_0 = D\\,\\mathrm{Re}^\\alpha$ with $T_0=L/U$ gives $\\alpha=0.53\\pm0.02$, agreeing with an earlier direct-method measurement; fitting with $T_{E0}=E/\\varepsilon$ gives $\\alpha_E=0.658\\pm0.006$. The apparent conflict between the two exponents is explained by the Reynolds-dependent dimensionless dissipation rate $C_\\varepsilon(\\mathrm{Re}) = C_{\\varepsilon,\\infty} + C/\\mathrm{Re}$, which converts $\\alpha_E$ into an effective $\\alpha\\approx0.52$, and decaying-turbulence growth curves favor the $T_{E0}$-based functional form. In magnetohydrodynamic runs the robustness of $\\lambda$ survives, but its Reynolds-number scaling is much shallower, $\\alpha=0.14\\pm0.02$.","pith_inferences":["If the near-Gaussianity of the FTLE distribution is partly produced by the rapid renormalization injecting a fresh perturbation every steptime, then varying the perturbation amplitude or changing the forcing scheme should alter the distribution's shape and variance; this is a testable consequence the paper does not develop.","The observed excess fluctuation level $\\sigma_\\lambda/\\lambda\\approx0.2$ over the value propagated from $\\mathrm{Re}$ and $T_0$ suggests either substantial finite-time and finite-perturbation effects or an attractor without a unique maximal exponent; distinguishing these would require measuring the full Lyapunov spectrum or varying $\\delta_0$ systematically.","Because the decorrelation time $T_d$ is much longer than the steptime, the raw count of FTLE samples overstates the number of independent measurements; the effective independent-sample count is about $T_r/T_d$, so confidence intervals computed from raw counts alone are likely too optimistic.","The MHD results suggest that the exponent's Reynolds-number scaling tracks the existence of a single governing dissipative timescale; varying the magnetic Prandtl number in future simulations would test whether the shallow $\\alpha=0.14$ scaling persists or shifts with the added magnetic timescales."],"forward_implications":["A simulation run of roughly ten large-eddy turnover times is enough for converged Lyapunov statistics, while energy or Reynolds-number averages need about a hundred, so the Lyapunov exponent is a practical fast-converging flow diagnostic.","Because $\\lambda$ and $\\sigma_\\lambda$ are nearly independent of the FTLE steptime down to very small values, short steptimes can be used to generate large samples at low computational cost.","Through the dimensional relation linking $\\lambda$, $\\mathrm{Re}$, and the large-eddy time, a measured $\\lambda$ can serve as a fast proxy for Reynolds number or dissipation in homogeneous isotropic turbulence once calibration constants are known.","The effective exponent $\\alpha$ in $\\lambda T_0 = D\\,\\mathrm{Re}^\\alpha$ is not a fixed universal number: it increases with Reynolds number and approaches $\\alpha_E\\approx0.66$ at high $\\mathrm{Re}$, so the $\\alpha=1/2$ dimensional prediction is not the fundamental scaling.","In MHD turbulence the Lyapunov exponent remains robust and steptime-stable, but its Reynolds-number power is much smaller ($\\alpha=0.14\\pm0.02$), so hydrodynamic calibrations should not be transferred to MHD flows."],"supporting_citations":[{"why":"Supplies the earlier direct-method result $\\alpha=0.53\\pm0.03$ that this paper's fitted $\\alpha$ agrees with.","marker":"[14]"},{"why":"Supplies the earlier FTLE result $\\alpha=0.64\\pm0.05$ and the large uncertainty in $\\gamma$, which this paper explains and revises.","marker":"[15]"},{"why":"Documents the large, fast fluctuations in Eulerian Lyapunov exponents that motivate the statistical analysis here.","marker":"[16]"},{"why":"Provides the magnetohydrodynamic DNS context and the previous $\\lambda$ versus inverse dissipative-time plot that the MHD section extends.","marker":"[17]"},{"why":"Gives the dimensional prediction that the maximal Lyapunov exponent scales as the inverse of the smallest dissipative timescale.","marker":"[48]"},{"why":"Supplies the Reynolds-dependent dimensionless dissipation rate $C_\\varepsilon(\\mathrm{Re})=C_{\\varepsilon,\\infty}+C/\\mathrm{Re}$ used to reconcile the two scaling exponents.","marker":"[50]"},{"why":"Provides the multiplicative ergodic theorem that underlies the claim that the maximal Lyapunov exponent is independent of the initial infinitesimal perturbation.","marker":"[55]"},{"why":"Establishes that finite-time and finite-perturbation effects introduce dependence on the FTLE parameters, the acknowledged source of extra fluctuations.","marker":"[56]"},{"why":"Gives the classical dissipation relation $\\varepsilon=C_\\varepsilon U^3/L$ that connects the two large-eddy turnover-time definitions.","marker":"[57]"}],"fun_headline_variants":["Lyapunov exponents in turbulence converge ten times faster than energy","Turbulence chaos metric stabilizes faster and resolves Reynolds-number dispute","Finite-time Lyapunov exponent robust to lattice size and step time in turbulence","MHD turbulence flattens Lyapunov Reynolds exponent to 0.14"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The paper assumes that averaging finite-time Lyapunov exponents, each obtained by renormalizing the perturbed field to a fixed tiny size at every short steptime, gives the true maximal Lyapunov exponent of the flow; it never tests whether changing that fixed size or extending the averaging time changes the result.","fun_headline_variants_meta":{"raw":{"variants":["Lyapunov exponents in turbulence converge ten times faster than energy","Turbulence chaos metric stabilizes faster and resolves Reynolds-number dispute","Finite-time Lyapunov exponent robust to lattice size and step time in turbulence","MHD turbulence flattens Lyapunov Reynolds exponent to 0.14"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000598,"raw_usage":{"total_tokens":2902,"prompt_tokens":1155,"completion_tokens":1747,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":771,"completion_tokens_details":{"reasoning_tokens":1667}},"tokens_in":771,"tokens_out":1747,"duration_ms":13325,"temperature":1.0,"reasoning_tokens":1667,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T05:25:29.374013+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Repeat the same FTLE procedure at fixed Reynolds number, lattice size, and steptime while varying the renormalization amplitude $\\delta_0$ from $10^{-4}$ to $10^{-2}$; if the mean exponent or its standard deviation drifts systematically with $\\delta_0$, the ensemble average is not the true maximal Lyapunov exponent and the fitted scaling exponents are contaminated. A complementary check is to measure the unrenormalized exponential growth slope (the direct method) on identical runs and compare it with the FTLE average.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the earlier direct-method result $\\alpha=0.53\\pm0.03$ that this paper's fitted $\\alpha$ agrees with."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the earlier FTLE result $\\alpha=0.64\\pm0.05$ and the large uncertainty in $\\gamma$, which this paper explains and revises."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Documents the large, fast fluctuations in Eulerian Lyapunov exponents that motivate the statistical analysis here."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the magnetohydrodynamic DNS context and the previous $\\lambda$ versus inverse dissipative-time plot that the MHD section extends."},{"cited_title":"de Divitiis, Adv.Math","cited_arxiv_id":null,"evidence_quote":"Gives the dimensional prediction that the maximal Lyapunov exponent scales as the inverse of the smallest dissipative timescale."},{"cited_title":"Biferale, G","cited_arxiv_id":null,"evidence_quote":"Supplies the Reynolds-dependent dimensionless dissipation rate $C_\\varepsilon(\\mathrm{Re})=C_{\\varepsilon,\\infty}+C/\\mathrm{Re}$ used to reconcile the two scaling exponents."},{"cited_title":"d’Ovidio, V","cited_arxiv_id":null,"evidence_quote":"Provides the multiplicative ergodic theorem that underlies the claim that the maximal Lyapunov exponent is independent of the initial infinitesimal perturbation."},{"cited_title":"Haller and T","cited_arxiv_id":null,"evidence_quote":"Establishes that finite-time and finite-perturbation effects introduce dependence on the FTLE parameters, the acknowledged source of extra fluctuations."},{"cited_title":"Haller, Annu","cited_arxiv_id":null,"evidence_quote":"Gives the classical dissipation relation $\\varepsilon=C_\\varepsilon U^3/L$ that connects the two large-eddy turnover-time definitions."}],"review_version":1}