Paper: Global sonde datasets do not support a mesoscale transition in the turbulent energy cascade (Submitted; preprint): Using global radiosonde (IGRA, 2010–2025) + two dropsonde programs (ACTIVATE: 683 profiles; NOAA hurricane recon: 2325) to compute second-order structure functions of horizontal wind; vertical separations 0.2–8 km; horizontal 200–20,000 km; rigorous pairing (≤50 m vertical mis-match, ≤2 h launch offset), smoothing-aware thresholds (Δzmin ≈200 m from sonde inertia). Key result: no spectral “transition” regime; instead near-constant anisotropic scaling. Vertical Hurst exponent Hv ≈ 0.6 across troposphere (ACTIVATE 0.71±0.01, hurricanes 0.513±0.008, IGRA 0.62±0.02); inconsistent with gravity-wave/quasi-geostrophic (Hv=1) or 3D turbulence (1/3). Page 15 Fig. 4 shows robust power-law in 0.2–8 km; Fig. 5 (p.16) layered 2-km slabs: troposphere ~3/5, drop to ~0.4–0.5 above ~18–20 km. Horizontally, 200–1800 km: Hh ≈0.50±0.02; >~1800 km flattening (Hh→0); contradicts quasi-geostrophic (Hh=1). Two-D fit (Eq. 9) to Δv(Δx,Δz) (inset, Fig. 7 p.18): Hh=0.37±0.01, Hv=0.63±0.01; φh≈0.006 m^{2−2Hh}s{−2}, φv≈0.009 m^{2−2Hv}s{−2}; spheroscale ls = ε^{5/4}/φ{3/4} ≈ O(1 m) → circulation aspect ratios vary systematically with scale, not trivially anisotropic. Isoheight 2-D (Δx,Δy) (Fig. 8 p.19): Hx=0.35±0.05, Hy=0.33±0.03 (≈ horizontally isotropic), but φx≠φy (trivial anisotropy in amplitudes). Mechanism for elevated Hh (~0.5 vs 1/3): vertical smoothing biases horizontal statistics; explicit SAM LES test (100-m grid, 2–10 km layer; Fig. 9 p.22): applying 200-m vertical Gaussian smooth raises Hh from 0.305±0.008→0.42±0.01 and Hv from 0.69±0.02→0.79±0.03; implies apparent “Nastrom–Gage transition” can arise from sloping/smoothed sampling, not a mesoscale physics switch. Boundary-layer implication: even <2 km altitude does not show 3D-Kolmogorov Hv=1/3 down to 200 m separations; any isotropy likely <200 m. Data-processing caveat: strong country-by-country Hv bimodality (Appendix A1) indicates processing artifacts; uncertainties dominated by methodology. Overall: supports Lovejoy–Schertzer single anisotropic cascade (Hv≈3/5, Hh≈1/3) as unifying framework across scales.

Paper (Highlighted in ACP): Climatologically invariant scale invariance seen in distributions of cloud horizontal sizes: Mixing-engine thermodynamic theory (steady-state exchange across neutrally buoyant cloud edges) predicts perimeter distribution n(p)∝p^{−(1+β)} with β=1 on moist-isentropic layers; tested against 210-level SAM cloud-resolving output (204.8×204.8 km, 100 m Δx, GATE III forcing, hours 12–24 quasi-equilibrium): β=0.98±0.03, theory confirmed. Satellite view (GOES, Himawari, MSG, EPIC, VIIRS, MODIS, POLDER; Table 1) shows robust β=1.26±0.06 across seasons, latitude bands, and land–ocean (monthly spread ~1.21–1.32 only), implying climatology-invariant scaling controlled by stability rather than Coriolis, surface temperature, or aerosol regime; discrepancy with theory traced to perspective: vertical overlap “compression” in top-down imagery. Synthetic “compressed” SAM (vertically summed optical depth τ) reproduces β>1; β→1 only for optically thick thresholds (τ≳10), whereas MODIS reflectance masks R∈[0.1,0.7] exhibit near-constant β, indicating overlap/threshold interplay not trivially mappable between τ and R (Fig. 8). Methodological correction: removal of perimeter/area bins where ≥50% of clouds are truncated by domain boundaries eliminates spurious scale breaks; with this fix, area distributions power-law with α≈0.95±0.08 over ≥5 orders of magnitude, no observed cutoff up to a≈3×10^5 km² (effective diameter ≈600 km); prior smaller a_max likely an artifact of truncation handling (Figs. 2–3). Joint α,β imply perimeter–area exponent D≈1.5±0.1 (holes counted), consistent with slightly super-4/3 fractality. Additional rigor/mission design note: EPIC onboard 2×2 averaging + interpolation distorts small-scale perimeters, requiring elevated truncation thresholds; warns against compression that smooths edges in future missions. Practical implication: given stable, universal exponents, simulating counts of largest clouds suffices to constrain full distribution via scale invariance (layers vs satellite perspective matters for β).

Paper: Finite domains cause bias in measured and modeled distributions of cloud sizes: primary driver of interstudy disagreements in cloud-size exponents/cutoffs = finite-domain truncation (not estimator choice). Linear regression on log-binned histograms is as accurate as MLE if bins with <~24 counts are excluded; failure rate <5% across sample/bin settings; justification via CLT on counts → Gaussian errors in log n for n≳24; correction to common misinterpretation in Clauset et al. (Appendix A). Evidence chain: synthetic truncated power laws (α=1, amin=10, amax=1000) → Fig. 2/3; real GOES-West cloud masks (10 noon images; ~4000×4000 km domain; subsampled) → Fig. 4–7; idealized percolation lattices (10k×10k) → Fig. 6–9. Mechanism: removing truncated clouds induces spurious “exponential tail” & apparent scale break at areas ~0.1–1% of subdomain area; including them shifts mass to smaller bins and creates nonphysical local maximum near domain area (domain-size–locked). Quantified bias (100×100 km subdomains vs full 4000×4000 km): α underestimated by 36% (LR) / 19% (MLE) when including truncated; overestimated by 24% (LR) / 20% (MLE) when excluding—plots still look power-law (Table 1; Fig. 7). Recommended procedure (domain-agnostic): per-bin truncated-fraction filter n_trunc/ n_total < 0.5 + bin-count ≥24; yields α nearly invariant across domain sizes (GOES LR/MLE ~0.90–1.02; percolation LR ~1.02–1.07 vs exact 1.055; Table 2). Practical design rule: for square domains with amin=10 ξ², need L/ξ ≈300 to retain ≥2 decades after filtering. Periodic boundaries do not fix finite-size bias—periodic lattices still show domain-scale peak (Fig. 8). Non-power-law cases also biased: exponential tails (percolation P=0.5) undersampled when truncation dominates; same 50% rule recovers stability (Fig. 9/Appendix C). Caution against “corrections” requiring cloud shape/anisotropy assumptions; fractal geometry + dependence on domain edge + non-independence challenge MLE and any deconvolution (Appendix B). Generalizable to geometric size distributions beyond clouds; code released.

Preprint: Toward less subjective metrics for quantifying the shape and organization of clouds: proposes two distinct fractal metrics with clear physics links, exposes pitfalls in decades of cloud “fractal” work, and supplies empirically tested fixes; key claims: strict, topological argument that perimeter–resolution definition for an individual-cloud fractal dimension Di is not interchangeable with the common perimeter–area method unless cloud area is resolution-invariant; holes make a(ξ) vary, so unfilled perimeter–area fits do not measure a true dimension; fill holes to restore scale definition (Fig. 5 p.13: unfilled Di rises with size 1.419→~1.8; filled Di stable ≈1.36–1.38 across 4 decades; Fig. 6 p.14). Introduces ensemble dimension De via total perimeter P(ξ)∝ξ^(1−De), better suited to fields + observational validation; analytical derivation (Appendix A p.28–33) shows De = β·Di where β is the perimeter distribution exponent n(p)∝p^(−1−β); orthogonal controls: shape roughness (Di) and relative abundance of small vs large clouds (β) (Fig. 7 p.17). Details: 72 MODIS granules (Jan 2021, 60°S–60°N), ocean-only, sun-glint/zenith filters; thresholds R=0.05–0.35 (more in App. C); Di computed after hole-filling; per-cloud a,p cutoffs to avoid unresolved edges; β from logarithmically binned, domain-truncation-corrected histograms; nested perimeters counted (App. B Table B1 p.34: nested β ≈ +0.1 vs summed). Empirics (Table 1 p.24; Figs. 10–11 p.22–23): Di ≈1.362–1.374 (weak R sensitivity); Dc (correlation dimension ≈De) ≈1.67–1.74 decreasing with R (multifractality); box De lower and more R-sensitive; product β·Di ≈1.72–1.81 matches Dc magnitude/trends imperfectly but supports De=βDi framing. Methodological best-practice: prefer correlation integral (careful r≥3ξ, r≤min(L,W)/10, no boundary-crossing circles) over box-counting; avoid perimeter–area on unfilled clouds. Perspective: stop forcing continuous reflectance into subjective object classes; treat cloud fields as continuous, scaling systems; De diagnostics offer objective, physics-tied validation of GCRM cloud organization and reveal turbulence anisotropy across scales (context: Rees et al. 2024).

Software: scaleinvariance: Unified Hurst-estimation + multifractal simulation toolkit; 1D/2D fBm + FIF generators; PyTorch-parallel (GPU/CPU) for large fields; causal/acausal; three Hurst estimators (Haar fluctuation, structure function, spectral) under one API; multidimensional arrays with axis-averaging; optional fit returns; examples/tests; docs + pip. Advantage vs typical single-method or NumPy-only stacks: end-to-end simulate→analyze, multidimensional support, faster simulation, reproducible validation.

Software: objscale: Object-based scaling analysis at scale; unbiased power-law exponents via finite-domain correction; individual + ensemble fractal dimensions (correlation integral preferred; also box-counting, coarsening); handles NaN/arbitrary boundaries, interior holes, resolution effects; metrics: area, perimeter, widths; Numba-accelerated (billions of objects feasible); utilities (border removal, largest object, coarsening). Replaces subjective heuristics with physics-linked, objective metrics—implements (Above: Toward less subjective metrics… ) and (Above: Finite domains cause bias… ). Advantage vs common workflows: corrects edge-truncation bias, reduces subjectivity, supports ensemble methods, scales to massive datasets.