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The theoretical status, assumptions, and even the intended observable behind the central mapping $\alpha = 2/\xi$ are not stated precisely, making it hard to interpret deviations as (i) genuine counterexamples, (ii) pre-asymptotic limitations, or (iii) regime mismatch (Abstract; Sec. 1; Sec. 2.4; Sec. 3.3–3.5; Sec. 4). In particular: (a) the definition of $\xi$ is ambiguous due to mixed conventions (Sec. 2.1–2.2; Eq. (2); the statement $\xi=1+\zeta_2$ and the Kraichnan spectrum parameterization $E(k)\propto k^{-(1+\zeta_2)}$); (b) it is not clearly justified that single-particle displacement in the specific 3D Kraichnan setting should converge to a space-fractional $\alpha$-stable law (many classic Kraichnan anomalous results concern pair dispersion/scalars and depend on compressibility/dimension); and (c) the paper does not explicitly list the hypotheses under which the RG fixed point is expected (e.g., dimensionality, incompressibility/compressibility, isotropy, white-in-time vs finite correlation, absence of strong trapping, intermittency relevance).
Recommendation: Add a dedicated theory/conventions subsection early (e.g., new Sec. 1.1 or Sec. 2.0) that: (i) defines $\xi$ unambiguously in the paper’s convention (e.g., $\langle |\delta u(r)|^2\rangle\sim r^{\xi}$ or via $E(k)$), and explicitly derives/quotes the relation between the spectrum exponent and structure-function exponent in the dimensions used (disambiguate symbols, e.g., $\zeta_{2,{\rm SF}}$ vs a spectral slope parameter); (ii) states what Lagrangian quantity is predicted to be $\alpha$-stable (single-particle displacement component? radial displacement? increments?) and under what model class (Gaussian, homogeneous, isotropic, incompressible?, white-in-time); (iii) clarifies whether the mapping pertains to single-particle or pair dispersion and why the chosen observable is the correct test in your settings; (iv) lists assumptions/known limitations (dimension, compressibility, trapping/topology) and, for each model in Sec. 3.1–3.5, explicitly states which assumptions hold/violate; (v) adds primary references where $\alpha=2/\xi$ (and any accompanying relation such as $H=\xi/2$) is derived/discussed for turbulent transport, distinguishing rigorous vs heuristic arguments. Then adjust the language in Sec. 4 from unconditional “universal fixed point” to conditional statements tied to these hypotheses.
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There is an internal mismatch between the Lévy/space-fractional picture ($\alpha<2$) and the heavy reliance on MSD/Hurst exponents as a primary validation target (Sec. 2.3; Sec. 3.3; the MSD-based benchmark $H=\xi/2$; also noted around p.6–7). For ideal symmetric $\alpha$-stable Lévy flights with $\alpha<2$, $\langle \Delta x^2\rangle$ diverges, so MSD-based scaling is not a mathematically consistent asymptotic diagnostic unless a specific regularization is part of the model (finite domain, truncation/tempering, finite-resolution cutoffs). As written, MSD is used both to assess “diffusive” behavior and to compare against a fractional model whose second moment need not exist, which can bias conclusions (e.g., toward $\alpha\approx 2$ under finite-variance/finite-sample regimes).
Recommendation: Make the modeling choice explicit and consistent: either (a) pivot the primary Lagrangian scaling tests to observables compatible with $\alpha<2$, e.g., fractional moments $\langle |\Delta x|^q\rangle$ for $q<\alpha$, inter-quantile scaling, or scaling of the typical displacement (median/quantiles); or (b) explicitly define a regularized/tempered/truncated stable model induced by finite inertial range, periodic domain, or numerical cutoff, and derive what MSD scaling should be expected under that regularization (including how it depends on resolution and time). Then rewrite Sec. 3.3–3.5 so that “agreement/disagreement” is judged using observables whose asymptotics match the stated model class.
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The Lagrangian inference pipeline for $\alpha$ and especially $\alpha(\tau)$ is under-specified and lacks calibration/robustness checks, so central conclusions (“$\alpha(\tau)$ pinned near 2” in 3D; extremely small $\alpha$ in 1D) cannot be independently assessed (Sec. 2.3–2.4; Sec. 3.3–3.5; Fig. 6–8). Key missing details include: number of tracers/realizations and trajectory lengths; sampling interval; how $\phi(k,\tau)=\langle e^{ik\Delta x(\tau)}\rangle$ is computed (k-grid, averaging, overlapping windows); the fit strategy for $\exp(-D(\tau)|k|^{\alpha(\tau)})$ (joint fit vs sequential, weights, goodness-of-fit criteria, treatment of large-k noise/aliasing); and uncertainty quantification. Moreover, without careful k-window choice, finite-variance/truncated tails can make the small-k behavior look quadratic, biasing $\alpha$ toward 2 even for non-Gaussian laws. The use of Hill plots/tail estimators also needs justification for symmetric stable/tempered-stable settings.
Recommendation: Expand Sec. 2.3–2.4 into a reproducible protocol: (i) per model, report tracer count, realizations, total time, $\Delta t$, and the ratio of analyzed lags to correlation time $\tau_c$; (ii) specify whether $\Delta x$ is a 1D component or 3D vector magnitude, and update Eq. (4) accordingly (vector $\mathbf{k}$ vs scalar k); (iii) define the k-grid and the exact fit window $[k_{\min},k_{\max}]$ used for each $\tau$, with a rule for excluding noisy/aliased points; (iv) state the objective function and whether $D(\tau)$ and $\alpha(\tau)$ are jointly fit, plus goodness-of-fit diagnostics; (v) add bootstrap/subsampling confidence intervals for $\alpha$, $\alpha(\tau)$, and any reported plateaus; (vi) validate the estimator on synthetic data with known $\alpha$ (and with finite-sample sizes comparable to your simulations) to demonstrate identifiability and bias. For tail indices, either justify Hill under your assumptions or replace/augment with stable-law-oriented estimators (e.g., characteristic-function regression methods, stable MLE with diagnostics) and report estimator stability to threshold choices.
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The key negative result in the 3D Kraichnan model—near-Gaussian transport and lack of approach to the predicted $\alpha=2/\xi$ regime—is attributed to finite spectral resolution and finite time (“pre-asymptotic trapping”), but the numerical setup is not fully specified and the attribution is not supported by a systematic parameter/finite-size study (Sec. 2.1–2.3; Sec. 3.3–3.4). Crucial missing details include: box size and boundary conditions (periodicity and wrap-around effects on displacement statistics), wavenumber range $[k_{\min},k_{\max}]$, number of Fourier modes $N_k$ in each direction and incompressibility projection, how white-in-time is implemented, forcing/dissipation or spectral shaping, tracer integrator and time step, total integration time and number of statistically independent samples. With a single (or very limited) resolution/time horizon, it remains unclear whether the observed Gaussianity reflects genuine inapplicability of the mapping to the chosen observable, or merely insufficient scale separation.
Recommendation: In Sec. 2.1–2.3 add a complete Kraichnan-implementation description and summarize it in a parameter table (also including tracer counts, $\Delta t$, T, $k$-range, boundary conditions). Then, in Sec. 3.3–3.4 provide at least a minimal finite-size/time study: vary $N_k$ and/or inertial-range width $k_{\max}/k_{\min}$, and/or total time T, and show how (a) PDF non-Gaussianity (e.g., kurtosis/flatness), (b) $\alpha(\tau)$ plateaus/crossovers (with CI), and (c) any alternative consistent scaling observable (see Major Issue 2) change. If computational constraints prevent this, explicitly state the limitation and rephrase “pre-asymptotic trapping explains” into “consistent with a finite-size/time explanation; not demonstrated here.”
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The 1D synthetic turbulence section argues a “fundamental breakdown” of $\alpha=2/\xi$ due to “topological trapping,” but the mechanism is not quantitatively diagnosed and the appropriate effective model may not be space-fractional Lévy flight at all (Sec. 3.5; Sec. 4). Strong subdiffusion (e.g., very small H) is often associated with waiting-time mechanisms (CTRW/subordination; time-fractional diffusion), whereas the paper fits space-fractional $\alpha$-stable forms. In addition, it is not clarified whether the RG mapping was ever expected to hold in 1D (many results are for $d\ge 2$) and how specific modeling choices (boundary conditions, temporal correlations, stagnation-point statistics) control trapping.
Recommendation: First, in Sec. 2.1 precisely define the 1D synthetic velocity field (construction, spectrum, intermittency parameters, temporal correlation, domain/boundaries). Second, in Sec. 3.5 add quantitative trapping diagnostics: e.g., residence-time distributions in low-|u| regions, waiting-time statistics near stagnation points, distribution/density of velocity zeros, or displacement autocorrelation signatures; include representative trajectory visualizations tied to those statistics. Third, discuss model class: test whether a time-fractional/subordinated model better captures the observed subdiffusion (e.g., fit waiting-time tails and relate to a time-fractional order), and clearly separate “outside assumptions of $\alpha=2/\xi$” from “counterexample within assumptions.” Finally, soften “fundamental breakdown” unless you can cite theory predicting the mapping for your 1D setting.
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The Lorenz–96 component is currently underdeveloped relative to how it is motivated (Sec. 2.1–2.3 vs Sec. 3.1–3.2). It is presented as a deterministic-chaotic contrast, but the Results show mainly Eulerian structure-function exponents and a decorrelation time, without the corresponding Lagrangian transport analysis (MSD alternatives, PDFs, $\alpha$, $\alpha(\tau)$). This leaves an incomplete “bridge” in the narrative and weakens the manuscript’s claim to test robustness across stochastic vs deterministic dynamics.
Recommendation: Either (i) add a Lorenz–96 Lagrangian subsection (e.g., Sec. 3.6) that mirrors the Kraichnan diagnostics (using observables consistent with your chosen fractional model), including $\alpha(\tau)$ with uncertainty, and discuss how deterministic chaos affects (or fails to affect) approach to the predicted mapping; or (ii) explicitly reposition Lorenz–96 as an Eulerian-only benchmark and remove/temper any claims that it informs Lagrangian fractional transport. In both cases, document the Lorenz–96 configuration in Sec. 2.1 (N, forcing F, integration method/$\Delta t$, simulation length, and how the discrete index is treated as a spatial coordinate).