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DNS configuration, physical regime, and time/length units are insufficiently specified, limiting reproducibility and physical interpretation of reported exponents and correlation times (Sec. 2.1; affects Secs. 3.1–3.4). The manuscript does not clearly state the governing equations (compressible vs incompressible; what “isothermal” means operationally), forcing scheme and whether it is isotropic, numerical method, grid size and domain size, viscosity and resulting Reynolds number (e.g., Taylor-scale), Mach number (if compressible), snapshot spacing $\Delta t$, total simulated time in turnover units, and evidence of statistical stationarity. Without these, values like $\tau_{1/e}\approx 0.19$ and the fitted MSD range cannot be related to turbulence scales, and anisotropy cannot be contextualized.
Recommendation: Expand Sec. 2.1 substantially (ideally add a concise ‘simulation/analysis parameters’ table) to include: equations solved and equation of state; definition of ‘isothermal’; domain size and grid resolution; viscosity (and/or dissipation) and Reynolds number; Mach number (if relevant); forcing type, injection scale, and whether forcing introduces a preferred direction; numerical scheme and timestep; snapshot cadence $\Delta t$; total duration in large-eddy turnover times $T_L$; and a brief stationarity check (time series of kinetic energy/dissipation). Express key times (VACF correlation time, MSD fitting window) in units of $T_L$ and/or Kolmogorov time to enable physical comparison.
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MSD scaling claim ($\alpha\approx 1.82$) lacks essential methodological detail and robustness evidence against crossover/finite-size/periodic-box artifacts (Secs. 2.3, 3.1). In a periodic cube, MSD requires careful ‘unwrapping’ of trajectories across boundaries; the manuscript mentions minimum-image conventions for linking but does not clarify whether positions are unwrapped for MSD computation. Additionally, $\alpha\approx 1.82$ is close to ballistic ($\alpha=2$), so an apparent power law can arise from fitting across a ballistic-to-diffusive crossover rather than a true scale-invariant regime. The exact fitting lag range $\tau\in[\tau_{\min},\tau_{\max}]$, exclusion of short-time tracking jitter and long-time saturation, and sensitivity to the fit window are not clearly documented.
Recommendation: In Sec. 2.3 and Sec. 3.1: (i) explicitly state whether and how trajectories are unwrapped across periodic boundaries for displacement/MSD (and how ‘minimum-image’ is applied for multi-step lags); (ii) report the exact $\tau$-range used to fit $\alpha$ and justify it physically (between short-time ballistic/jitter and long-time finite-size effects); (iii) add a local-slope plot $\alpha(\tau)=d\log\mathrm{MSD}/d\log\tau$ or compensated MSD plots (MSD/$\tau^2$, MSD/$\tau$) to demonstrate a genuine scaling regime; (iv) show sensitivity of $\alpha$ to reasonable variations of the fitting window and to maximum lag used.
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Vortex identification and tracking are under-specified and not validated against common failure modes (splits/merges, segmentation jitter, crowded-field mis-association), risking biased MSD/VACF statistics (Sec. 2.2; impacts all results). The greedy nearest-neighbor linking within a fixed 3-cell radius can fail when structures move faster than the radius, when multiple candidates exist, or when segmentation changes shift centroids even without physical motion. The manuscript does not specify: exact centroid weighting, tie-breaking rules with multiple matches, how merges/splits are handled, trajectory initiation/termination rules, fraction of unlinked vortices, distribution of link distances, or diagnostics for unphysical jumps. The role of any pre-processing (e.g., smoothing/filtering before gradient/Q computation) and the rationale for the minimum-volume cutoff are also not adequately justified.
Recommendation: In Sec. 2.2: provide explicit definitions and algorithmic details: (i) centroid formula with the exact weight (e.g., $|\omega|$, $|\omega|^2$) and normalization over voxels in each connected component; (ii) tracking rules when multiple candidates fall within the search radius, and how merges/splits are detected/handled (terminate, branch, or assign based on overlap/volume continuity); (iii) justify the 3-cell radius via the observed one-step displacement distribution and test sensitivity to radius; (iv) justify minimum volume cutoff (e.g., relative to core size / grid resolution) and test sensitivity. Add validation diagnostics (Sec. 3 or appendix): histogram of link distances, fraction of ambiguous matches/unlinked objects, examples of trajectories overlaid on Q/$|\omega|$ fields, and outlier/jump filtering criteria if used. If feasible, consider a more robust assignment (e.g., Hungarian matching with cost + continuity constraints) or overlap-based tracking to reduce mis-associations.
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Statistical estimation and uncertainty quantification for $\alpha$, VACF metrics, and correlation analyses are under-described and likely over-precise (Secs. 2.3, 3.1–3.3). Reported uncertainties (e.g., $\pm 0.009$) and very high $R^2$ values appear to reflect fit residuals rather than sampling variability across trajectories and temporal correlations. VACF correlation time extraction (e.g., $\tau_{1/e}$) is not defined procedurally, and effective sample sizes for Pearson correlations are not addressed despite autocorrelated data.
Recommendation: Specify in Sec. 2.3/3.1–3.3: number of trajectories and points contributing at each lag; how trajectories of different lengths are weighted; and how uncertainties are computed. Prefer bootstrap/block-bootstrap over trajectories (or over time blocks) to obtain confidence intervals for $\alpha$, VACF, $\tau_{1/e}$, and Pearson r (with autocorrelation-adjusted effective N). Report $\alpha$ with realistically rounded uncertainty. For VACF, define velocity estimation (forward/backward/central differences) and how $\tau_{1/e}$ is obtained (interpolation vs fit).
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Connection between VACF and MSD is not tested, missing an opportunity to substantiate the ‘persistent random walk’ mechanism (Secs. 3.1–3.2). For correlated random walks, the Taylor–Kubo relation links MSD to the time integral of the VACF; without a consistency check, the narrative ‘slowly decaying VACF $\Rightarrow$ superdiffusive MSD’ remains suggestive rather than quantitatively supported.
Recommendation: Add a Taylor–Kubo consistency check (Secs. 3.1–3.2): compute MSD predicted from the measured VACF (or vice versa) and compare to the directly measured MSD over the same $\tau$-range. Even a qualitative agreement plot would strengthen the mechanistic claim and help diagnose whether tracking noise or finite-sample issues affect either statistic.
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Step-size distribution analysis contains a key statistical/model mismatch and insufficient tail diagnostics, weakening the argument against Lévy-flight-like mechanisms (Sec. 3.2; also noted in Sec. 2.3). The manuscript states it compares step magnitudes $|\Delta \mathbf{r}|$ to a Gaussian via a K–S test; however, the magnitude of a multivariate Gaussian increment is not Gaussian (it follows a Maxwell/chi distribution). As written, the goodness-of-fit inference is not meaningful, and kurtosis statements are ambiguous (raw vs excess; magnitude vs components).
Recommendation: Revise Sec. 3.2 to test appropriate quantities/distributions: (i) assess normality on component increments $\Delta x,\Delta y,\Delta z$ (QQ plots and a tail-sensitive test such as Anderson–Darling, or K–S with clearly stated parameter estimation); and/or (ii) compare $|\Delta \mathbf{r}|$ to the Maxwell distribution implied by the component variances. To address Lévy-flight alternatives, add explicit tail diagnostics (log–log tail plots; fit/upper-bound power-law tails using standard methods; report uncertainty and finite-size limitations). Rephrase conclusions to ‘consistent with Gaussian increments within statistical resolution’ rather than ‘Lévy flights ruled out.’
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Large-scale advective field subtraction is under-defined and its interpretation is currently ambiguous, yet it supports a key conclusion (Sec. 3.4). The manuscript does not specify the filtering/projection operator, scale cutoff, whether filtering is isotropic, whether it is applied per snapshot or time-averaged, how residual positions are constructed from a velocity subtraction, and how sensitive $\alpha_{\mathrm{residual}}\approx 1.93$ is to filter choice. The reported increase toward ballistic after subtraction could also be a filtering artifact (e.g., removing decorrelating components), so the physical conclusion ‘intrinsic vortex dynamics’ is not yet supported.
Recommendation: In Sec. 3.4: define the large-scale field mathematically (e.g., spectral low-pass at $k\le k_c$ or spatial convolution with scale $\ell$) and specify implementation details/parameters; clarify whether subtraction is performed on velocities or displacements and provide the discrete formula used to form residual trajectories consistent with Eq. (2). Add a sensitivity study of $\alpha$ and VACF to filter scale $\ell$ or cutoff $k_c$, and report the fraction of kinetic energy removed. Plot MSD/VACF before/after subtraction to support interpretation; if results depend strongly on filter choice, temper claims accordingly.
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Strong anisotropy findings (component-wise $\alpha$ and especially high y-direction centroid–fluid coupling) are not adequately contextualized relative to the underlying flow configuration and statistical significance (Secs. 2.1, 3.1, 3.3, 4). In nominally homogeneous/isotropic turbulence, a large directional asymmetry suggests anisotropic forcing, a mean drift, insufficient averaging, or analysis artifacts. The current text attributes anisotropy to ‘large-scale structure’ without showing basic Eulerian isotropy diagnostics or uncertainty bars on $\alpha_x,\alpha_y,\alpha_z$.
Recommendation: Clarify in Sec. 2.1 whether forcing or numerics introduce preferred directions and whether any mean flow/drift is removed. In Secs. 3.1 and 3.3, provide Eulerian anisotropy measures (e.g., $\langle u_i^2\rangle$, Reynolds stresses) and confidence intervals for $\alpha_x,\alpha_y,\alpha_z$ and correlations, accounting for autocorrelation. In Sec. 4, frame anisotropy as configuration-dependent unless supported by additional evidence.
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Physical interpretation of ‘centroid velocity’ vs ‘fluid velocity at centroid’ is potentially over-literal (Secs. 2.4, 3.3). A vortex centroid is not a material point; centroid motion can reflect advection plus deformation/threshold-induced segmentation changes. Interpreting high correlation as ‘vortex follows the flow’ requires caveats, and weak correlations may be partly methodological.
Recommendation: Add a clarifying discussion (Secs. 2.4/4) that centroid motion mixes advection and structural evolution. Where feasible, include a validation/contrast: e.g., compare centroid motion to advection of passive tracers seeded near the centroid, or compare against an alternative ‘vortex center’ definition (peak $|\omega|$ location, swirling-strength core) on a subset. At minimum, qualify interpretations of centroid–fluid coupling accordingly.