{"id":"b1eaa6b6-0f1f-4189-a0e5-e22763c68831","arxiv_id":"2507.07314","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"A new budget equation for the uncertainty field shows a self-similar, compression-driven inverse cascade of decorrelation, and DNS with identical forcing favor a t^{2/3} growth of uncertainty energy.","lead":"This paper derives a new scale-by-scale budget equation for the uncertainty between two initially close solutions of the Navier-Stokes equations, and uses it to show that uncertainty spreads through a predominantly inverse cascade of decorrelation. It also reports DNS evidence that, when both flows share the same forcing, uncertainty energy grows as t^{2/3} rather than the classic linear law.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Alternative scaling rests on an explicitly 'bold assumption' about reference-field correlations at r~lΔ; if it fails, the t^{2/3}/lΔ~t predictions reduce to the classical t/lΔ~t^{3/2} laws.","rationale":"In good faith, the paper contributes a careful derivation of a scale-by-scale uncertainty energy budget and identifies four interscale transfer mechanisms, with DNS showing a predominantly inverse, compression-driven cascade and a plausible tendency toward equilibrium. The weakest point is indeed the 'bold assumption' in §3 that reference-field correlations at r∼lΔ elevate the linear transfer scaling from (ε lΔ)^{1/3}⟨EΔ⟩/lΔ to U^(1)⟨EΔ⟩/lΔ. This assumption is load-bearing because it is the only step that distinguishes the novel t^{2/3}, lΔ∼t predictions from the classical t, lΔ∼t^{3/2} laws; without it, Eqs. (3.6a,b) are not obtained and the central claim's quantitative predictive content evaporates. The DNS fits in §5.1 favour t^{2/3} and lΔ∼t, and the §5.4 scatter plots support the production/dissipation scaling, but these do not directly validate the transfer-rate scaling at r≈lΔ. A direct regression of the transfer sum against the two candidate scalings would settle the matter. The reader's CONDITIONAL verdict is appropriate; this stress test does not change it, so verdict_should_be is UNCHANGED.","tokens_in":37807,"tokens_out":2387,"duration_ms":24693,"concrete_test":"Using the same F2 DNS fields, compute S(r,t) = ⟨Πδ,ref⟩ + ⟨Iδ↑⟩ + ⟨Iδ↓⟩ evaluated at r = lΔ(t) (or averaged over lλ^(1) < r < lΔ) and regress S against U^(1)⟨EΔ⟩/lΔ and against (ε lΔ)^{1/3}⟨EΔ⟩/lΔ across the stochastic range τ ∈ [5,10]. If the R² for the U^(1)-based scaling is not clearly better than for the (ε lΔ)^{1/3}-based scaling, the bold assumption of §3 is unsupported and the t^{2/3}/lΔ∼t prediction must be regarded as unconfirmed.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim's new power laws (Eqs. 3.6a-b → ⟨EΔ⟩∼(εU^(1)t)^{2/3}, lΔ∼U^(1)t) are not consequences of the derived budget alone; they require the assumption stated just before Eqs. (3.6a,b) that the linear reference-field contributions to ⟨Πδ,ref⟩, ⟨Iδ↑⟩, ⟨Iδ↓⟩ at scales near lΔ scale as U^(1)⟨EΔ⟩/lΔ rather than (ε lΔ)^{1/3}⟨EΔ⟩/lΔ. The authors call this 'a simple naive attempt to qualitatively account for significant correlations' and defer its derivation to future work. If the correct scaling is (ε lΔ)^{1/3}-based, the transfer rates collapse to the classical form, Eq. (3.4) reproduces the standard predictions ⟨EΔ⟩∼εt and lΔ∼ε^{1/2}t^{3/2}, and the DNS preference for t^{2/3} loses its theoretical basis. The DNS evidence in §5.4 shows linear correlations of ⟨PΔ⟩ and ⟨εΔ⟩ with U^(1)⟨EΔ⟩/lΔ, but that tests the production/dissipation scalings, not the transfer-rate scaling itself; the sum of the reference-field transfer terms at r≈lΔ is not directly regressed against the two candidate scalings. The equilibrium approximation ∂⟨Aδ⟩/∂t≈0 is only a tendency in Figs. 9(c,d), but even accepting it, the derivation of the novel scalings remains conditional on the unproven correlation scaling.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper derives a scale-by-scale (KHMH-type) budget equation for the uncertainty energy of the velocity-difference field between two nearby Navier-Stokes realisations in statistically stationary homogeneous turbulence. The budget identifies four interscale transfer rates, expresses them through the eigenvalues and eigenvectors of the reference field's relative deformation tensor, and attributes the dominant inverse transfer to compressions of the reference flow aligned with the uncertainty half-sum field. In the stochastic power-law regime the authors propose a self-similar equilibrium cascade, Eq. (3.4), in which the sum of the transfer rates is independent of scale and equal to the production-dissipation difference. From this equilibrium they obtain two alternative sets of scaling laws: the classical ones (⟨EΔ⟩∼εt, lΔ∼ε^{1/2}t^{3/2}) and new ones (⟨EΔ⟩∼(εU^(1)t)^{2/3}, lΔ∼U^(1)t), the latter resting on a stated 'bold assumption' about the scaling of reference-field correlations. DNS with identical forcing (F2) and with independent forcing (F1) are used to argue for the new laws, including finite-size Lyapunov exponent analysis and scatter plots of production and dissipation against the two candidate scalings.","tokens_in":38209,"tokens_out":8511,"duration_ms":98021,"significance":"If the central claims hold, the paper makes a substantive contribution: the two-point uncertainty budget of Section 2 is a parameter-free derivation from the Navier-Stokes equations, and the identification of reference-field relative-deformation compressions as the driver of inverse uncertainty transfer is a concrete physical mechanism that goes beyond previous single-point analyses. The paper also provides falsifiable predictions for the stochastic regime under identical forcing, a clean F1/F2 contrast, and useful exact checks at r=0 (⟨Dδ⟩=⟨εΔ⟩ and ⟨Iδ↑⟩=−⟨PΔ⟩) that anchor the DNS diagnostics. The main value of the contribution is therefore the new budget and the precise formulation of the competing scalings; the empirical support for the new power laws is suggestive but, as argued in the major comments, not yet conclusive.","major_comments":[{"comment":"The new growth laws (3.6a,b) are not consequences of the derived budget alone: they depend on the 'bold assumption' stated just before Eqs. (3.6a,b) that the linear reference-field contributions to ⟨Πδ,ref⟩, ⟨Iδ↑⟩ and ⟨Iδ↓⟩ at r≈lΔ scale as U(1)⟨EΔ⟩/lΔ rather than (εlΔ)^{1/3}⟨EΔ⟩/lΔ. The authors explicitly describe this as 'a simple naive attempt' and defer its derivation to future work, and if the alternative scaling is correct the derivation reproduces the classical laws (⟨EΔ⟩∼εt, lΔ∼ε^{1/2}t^{3/2}) instead. The DNS in §5.4 does not close this gap: Figures 16(c,d) regress ⟨PΔ⟩ and ⟨εΔ⟩ against U(1)⟨EΔ⟩/lΔ, which tests the consequences of the assumption through the balance (3.4), but the assumed object is the scaling of the transfer-rate sum itself at r≈lΔ, and that sum is not directly regressed against the two candidate quantities. I recommend adding a direct regression of (⟨Πδ,ref⟩+⟨Πδ,err⟩+⟨Iδ↑⟩+⟨Iδ↓⟩) at r≈lΔ, and ideally of each term, against U(1)⟨EΔ⟩/lΔ and (εlΔ)^{1/3}⟨EΔ⟩/lΔ, or a derivation of the correlation scaling from the statistics of 𝚵(1).","section":"§3, before Eqs. (3.6a,b)"},{"comment":"The equilibrium step ∂⟨Aδ⟩/∂t≈0 for r<lΔ is the foundation of Eq. (3.4), but it is introduced via the idealized complete-decorrelation cartoon (3.2), and the DNS evidence in Figs. 9(c,d) is presented only as a 'tendency' rather than a converged balance. The size of the residual time-derivative relative to the transfer terms is not quantified, and because the self-similar cascade and all subsequent scalings inherit this approximation, the manuscript should report, for example, the ratio |∂⟨Aδ⟩/∂t| to the dominant right-hand-side term in Eq. (3.1) as a function of τ and r/lΔ, and estimate the resulting correction to the predicted power laws.","section":"§3, Eq. (3.3)"},{"comment":"The empirical discrimination between the two candidate scalings is weaker than the text suggests. The power-law fits in Figs. 3 and 6 use the interval τ∈[4,10], roughly a factor of two in time, with three free parameters (a, b, c), so the difference between exponent 1 and 2/3 in ⟨EΔ⟩, and between 3/2 and 1 in lΔ, corresponds to only a modest factor of growth. The FSLE plots in Figs. 4(b) and 7(b) are more convincing, but they still cover a limited range, only the ε0=0.1 case is ensemble-averaged, and the F1/F2 comparison rests on a single F1 realisation. Reporting confidence intervals for the fitted exponents, or bootstrapping over the six F2 realisations, would substantially strengthen the central claim that the data 'coherently favour' the new laws.","section":"§5.1, Figs. 3, 4, 6, 7"},{"comment":"The scatter regressions in Fig. 16 pool serially correlated samples from the same trajectories and plot quantities that share the same growing factors ⟨EΔ⟩ and lΔ, so the reported R² values overstate the significance of the difference between the two candidate scalings. The authors should report log-log slopes with confidence intervals, account for autocorrelation in the time series, or use well-separated independent snapshots from the stochastic regime before concluding that the data favour scalings (3.6) over (3.5).","section":"§5.4, Fig. 16"}],"minor_comments":[{"comment":"The manuscript contains several typographical errors, including 'stange attractors', 'tems', 'exponental', 'uncerainty', 'aligments', 'baring', and 'the the'; these should be corrected during revision.","section":"Throughout"},{"comment":"The caption refers to 'all six cases', but the paper presents three F2 cases (one of them ensemble-averaged over six realisations); the wording should be corrected to describe the actual data sets being pooled.","section":"Fig. 16 caption"},{"comment":"Please check the index structure of the final term in Eq. (2.8); as written, the last term mixes the summation indices in a way that appears inconsistent with the preceding derivation, even though the term is later dropped by homogeneity.","section":"Eq. (2.8)"},{"comment":"The typesetting of 'kΔ /uni223C 1/lΔ' contains raw LaTeX/Unicode artifacts and should be rendered as kΔ ∼ 1/lΔ.","section":"Fig. 1"},{"comment":"The time ranges in Table 2 are stated to be the same for all three F2 cases; please specify whether these boundaries were determined from the ε0=0.1 ensemble alone or from each case separately, and give the associated uncertainties.","section":"Table 2"},{"comment":"The journal-format markers 'Focus on Fluids articles must not exceed this page length' and 'Rapids articles must not exceed this page length' should be removed before submission.","section":"Manuscript formatting"}],"recommendation":"major_revision","confidential_remarks":"The authors are unusually candid: they explicitly label the U(1)-scaling assumption as 'bold' and 'a simple naive attempt' and defer its derivation to future work. In my assessment, this is the correct locus for a major revision: the DNS validates the production/dissipation scalings and the equilibrium balance, but it does not directly test the transfer-rate scaling that distinguishes the new laws from the classical ones. The paper should not be rejected because the Section 2 budget is a genuine parameter-free contribution, but the central claim is currently conditional on an unproven correlation scaling, and the revision needed is identifiable and testable."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThe budget equation is the real contribution here; the new scaling is explicitly speculative. The authors derive a KHMH-style scale-by-scale budget for the uncertainty field w, and decompose the interscale transfer into four mechanisms tied to eigenvalues and alignments of the reference relative deformation tensor. That derivation is careful, parameter-free, and anchored by the r→0 checks (⟨Dδ⟩=⟨εΔ⟩, ⟨Iδ↑⟩=−⟨PΔ⟩). The DNS evidence for an inverse, compression-driven cascade is solid, and the four-way decomposition appears genuinely new.\n\nThe weaker part is the alternative stochastic-regime scaling: ⟨EΔ⟩∼(ε U t)^{2/3}, lΔ∼U t under identical forcing. It rests on a 'bold assumption' (their words) that reference-field contributions to the transfer rates scale with U^(1) rather than (ε lΔ)^{1/3}. The paper is transparent that this is a naive first guess and defers a derivation to future work. If that assumption fails, the scalings collapse to the classical linear growth and lΔ∼t^{3/2}. The DNS in §5.4 tests production/dissipation scalings, but it does not directly regress the summed transfer rates at r≈lΔ against the two candidates. So the empirical preference for t^{2/3} is suggestive but not a direct confirmation of the mechanism.\n\nAlso minor: the equilibrium ∂⟨Aδ⟩/∂t≈0 is only approached, and the F1 comparison is based on a single realization. These are addressable.\n\nBottom line: the budget and the cascade phenomenology deserve serious referee time. The scaling law is a hypothesis for further testing, not a proven result. The authors frame it honestly, which raises confidence in the rest of the paper. I would bring it to a reading group and cite the budget equation in my own work. Recommend sending to peer review.\n\nBest,\n[Your name]","headline":"The budget equation is the substantive contribution; the new t^{2/3} scaling is clearly labeled speculation that should not be taken as established.","tokens_in":38721,"tokens_out":1792,"would_cite":true,"duration_ms":20247,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["47.27.-i"],"model":"deepseek-v4-flash","headline":"Uncertainty in three-dimensional turbulence grows through a self-similar equilibrium cascade of decorrelation, predominantly inverse and driven by compressions of the reference flow's relative deformation tensor, predicting $\\langle…","keywords":["uncertainty","stochasticity","predictability","interscale cascade","relative deformation tensor","compressions","homogeneous turbulence","Navier-Stokes"],"falsifier":"Measure the sphere-averaged transfer rates in (3.4) in a higher-Reynolds identical-forcing DNS and test whether their sum is independent of $r$ in $l_\\lambda^{(1)}\\ll r\\ll l_\\Delta$ and whether $\\langle P_\\Delta\\rangle$ and $\\langle\\varepsilon_\\Delta\\rangle$ scale as $U^{(1)}\\langle E_\\Delta\\rangle/l_\\Delta$ rather than $\\langle E_\\Delta\\rangle^{3/2}/l_\\Delta$; a failure of the $r$-independence or a collapse onto the $\\langle E_\\Delta\\rangle^{3/2}/l_\\Delta$ scaling would falsify the self-similar equilibrium cascade's specific prediction.","tokens_in":37566,"feed_emoji":"🌀","tokens_out":7390,"duration_ms":74393,"temperature":0.7,"pith_summary":"This paper asks how a tiny initial difference between two solutions of the Navier-Stokes equations grows into large-scale unpredictability in statistically stationary turbulence. It derives a scale-by-scale budget for the uncertainty field, the difference between the two velocity fields, and identifies four interscale transfer mechanisms, dominated by compressions of the reference flow's relative deformation tensor. In the stochastic, post-chaotic regime the budget becomes a self-similar equilibrium cascade of decorrelation: over an inertial range of scales the sum of the transfer rates is independent of scale and balances production minus dissipation. The authors use this to predict that, when the two flows share the same forcing, average uncertainty energy grows as $t^{2/3}$ and its integral scale as $t$, and their direct numerical simulations support these power laws over the classical linear growth.","feed_headline":"Uncertainty spreads by an inverse cascade of decorrelation","feed_subtitle":"Budget equation plus simulations show compressions move uncertainty upward, giving t^{2/3} energy growth.","key_machinery":"The carrying object is the scale-by-scale uncertainty energy budget, a K\\'arm\\'an-Howarth-Monin-Hill-type equation for $|\\delta\\boldsymbol{w}|^2$ with a spherical average over scales. Its four interscale transfer rates are $\\Pi_{\\delta,\\mathrm{ref}}$ and $\\Pi_{\\delta,\\mathrm{err}}$ (reference-field and self-induced two-point transfers) and $I_\\delta^{\\uparrow}$, $I_\\delta^{\\downarrow}$ (production-related transfers coupling the uncertainty half-sum and half-difference). All three linear rates are governed by the reference field's relative deformation tensor $\\Xi^{(1)}_{ij}=\\tfrac12(\\delta u^{(1)}_i e_j/r + \\delta u^{(1)}_j e_i/r)$, whose eigenvalues are one compressive, one zero, and one stretching; the sign of each transfer is set by whether compressions or stretchings dominate. The equilibrium approximation $\\partial\\langle A_\\delta\\rangle/\\partial t\\approx 0$ for $r<l_\\Delta$ converts the budget into equation (3.4), the self-similar equilibrium cascade, and the bold assumption that correlations make the linear transfer rates scale as $U^{(1)}\\langle E_\\Delta\\rangle/l_\\Delta$ yields the power laws.","core_discovery":"The central claim is that uncertainty in statistically stationary homogeneous turbulence does not merely grow chaotically; once its integral scale exceeds the Taylor scale, it sustains a self-similar equilibrium cascade of decorrelation. Equation (3.4) expresses this balance: the difference between average dissipation and production of uncertainty equals the sum of four interscale transfer rates, and that sum is independent of scale $r$ in the range $l_\\lambda^{(1)}\\ll r\\ll l_\\Delta$. The cascade is predominantly inverse: uncertainty produced at small scales by one-point compressions of the reference strain rate is carried to larger scales by two-point compressions of the reference field's relative deformation tensor, aligned with the uncertainty half-sum. Two forward linear transfers and one weak nonlinear inverse transfer coexist. From the equilibrium scalings, with the assumption that reference-field correlations make the linear transfers scale as $U^{(1)}\\langle E_\\Delta\\rangle/l_\\Delta$, the paper derives $\\langle E_\\Delta\\rangle \\sim (\\varepsilon U^{(1)} t)^{2/3}$ and $l_\\Delta\\sim U^{(1)} t$, and the DNS of identical-forcing flows supports these over the classical $\\varepsilon t$ and $t^{3/2}$ laws.","pith_inferences":["If the $t^{2/3}$ law holds at higher Reynolds number, earlier linear-growth measurements may have been contaminated by forcing-generated uncertainty, meaning that the intrinsic stochastic cascade is distinguishable only when the forcing is shared between realisations.","The same relative-deformation-tensor decomposition could be applied to passive scalar uncertainty or to Lagrangian predictability, where alignment statistics with compressive eigenframes may again set the error-growth rate.","The equilibrium assumption $\\partial\\langle A_\\delta\\rangle/\\partial t\\approx 0$ is only approached as time grows within the power-law window; if the window is too short, the fitted exponents may mix transient and equilibrium behaviour, a testable concern for future simulations."],"forward_implications":["If the equilibrium cascade is correct, the classical linear growth $\\langle E_\\Delta\\rangle\\sim\\varepsilon t$ and $l_\\Delta\\sim t^{3/2}$ applies only when the two flows are forced independently; with identical forcing, contamination reaches larger scales faster, as $l_\\Delta\\sim U^{(1)} t$.","The scale-by-scale budget (2.9) provides a direct diagnostic: in the stochastic regime, the sum of the four interscale transfer rates is independent of $r$ between the Taylor scale and $l_\\Delta$, so simulations and experiments can identify the regime by checking that plateau.","Since the three linear transfer rates are all controlled by compressions of the reference field's relative deformation tensor, compression statistics and alignment of the uncertainty field with the compressive eigenvector become measurable predictors of predictability loss.","The necessary condition for a dual-cascade decomposition, $\\langle P_\\Delta\\rangle/\\langle\\varepsilon_\\Delta\\rangle \\approx -(\\langle\\Pi_{\\delta,\\mathrm{err}}\\rangle+\\langle I_\\delta^{\\uparrow}\\rangle)/(\\langle\\Pi_{\\delta,\\mathrm{ref}}\\rangle+\\langle I_\\delta^{\\downarrow}\\rangle)$, holds in the DNS; the full dual cascade may require higher Reynolds numbers to verify."],"supporting_citations":[{"why":"Establishes the one-point uncertainty production by strain-rate compressions and the chaotic/exponential growth regimes that this paper extends.","marker":"Ge et al. (2023)"},{"why":"Supplies the compression/stretching framework for interscale energy transfers that the paper adapts to the uncertainty field.","marker":"Chen & Vassilicos (2022)"},{"why":"Proposed the inverse cascade of uncertainty in EDQNM and the cartoon model of decorrelation that underpins the equilibrium scalings.","marker":"Leith & Kraichnan (1972)"},{"why":"DNS of chaos and predictability in homogeneous turbulence that reported linear uncertainty-energy growth, the baseline the F2 forcing deviates from.","marker":"Boffetta & Musacchio (2017)"},{"why":"Provides the finite-size Lyapunov exponent method used to distinguish the competing power laws.","marker":"Boffetta et al. (2002)"},{"why":"Introduced the multi-scale predictability problem of backward error contamination that the paper formalises as the inverse cascade.","marker":"Lorenz (1969)"},{"why":"The pseudo-spectral code strategy used for the direct numerical simulations.","marker":"Vincent & Meneguzzi (1991)"}],"fun_headline_variants":["Inverse cascade of decorrelation drives uncertainty growth","Uncertainty spreads via inverse cascade in turbulence","Compressions drive inverse cascade of decorrelation in turbulence","Inverse cascade of uncertainty: compressions matter"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The power-law predictions rest on the assumption that correlations between the two velocity fields make the reference-field transfer rates at scale $l_\\Delta$ scale with the large-scale r.m.s. velocity $U^{(1)}$ instead of the Kolmogorov-scale velocity $(\\varepsilon l_\\Delta)^{1/3}$; if that scaling is wrong, the predicted laws revert to the classical ones.","fun_headline_variants_meta":{"raw":{"variants":["Inverse cascade of decorrelation drives uncertainty growth","Uncertainty spreads via inverse cascade in turbulence","Compressions drive inverse cascade of decorrelation in turbulence","Inverse cascade of uncertainty: compressions matter"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000989,"raw_usage":{"total_tokens":4191,"prompt_tokens":942,"completion_tokens":3249,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":558,"completion_tokens_details":{"reasoning_tokens":3189}},"tokens_in":558,"tokens_out":3249,"duration_ms":22082,"temperature":1.0,"reasoning_tokens":3189,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T18:43:26.539388+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Measure the sphere-averaged transfer rates in (3.4) in a higher-Reynolds identical-forcing DNS and test whether their sum is independent of $r$ in $l_\\lambda^{(1)}\\ll r\\ll l_\\Delta$ and whether $\\langle P_\\Delta\\rangle$ and $\\langle\\varepsilon_\\Delta\\rangle$ scale as $U^{(1)}\\langle E_\\Delta\\rangle/l_\\Delta$ rather than $\\langle E_\\Delta\\rangle^{3/2}/l_\\Delta$; a failure of the $r$-independence or a collapse onto the $\\langle E_\\Delta\\rangle^{3/2}/l_\\Delta$ scaling would falsify the self-similar equilibrium cascade's specific prediction.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the compression/stretching framework for interscale energy transfers that the paper adapts to the uncertainty field."},{"cited_title":"& Kraichnan, R.H","cited_arxiv_id":null,"evidence_quote":"Proposed the inverse cascade of uncertainty in EDQNM and the cartoon model of decorrelation that underpins the equilibrium scalings."},{"cited_title":"& Musacchio, S","cited_arxiv_id":null,"evidence_quote":"DNS of chaos and predictability in homogeneous turbulence that reported linear uncertainty-energy growth, the baseline the F2 forcing deviates from."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the finite-size Lyapunov exponent method used to distinguish the competing power laws."},{"cited_title":"1969 The predictability of a flow which possesses many scales of motion","cited_arxiv_id":null,"evidence_quote":"Introduced the multi-scale predictability problem of backward error contamination that the paper formalises as the inverse cascade."},{"cited_title":"& Meneguzzi, M","cited_arxiv_id":null,"evidence_quote":"The pseudo-spectral code strategy used for the direct numerical simulations."}],"review_version":1}