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Sparse Müntz--Szász Recovery for Boundary-Anchored Velocity Profiles: A Short-Record Roughness Diagnostic in Turbulence

D Yang Eng

Abstract

We present a sparse convex-relaxation framework for estimating effective local scaling exponents from short boundary-anchored velocity-increment profiles ($N\approx40$). The detector solves an $\ell_1$-regularized regression in a mixed Müntz--Szász/Jacobi dictionary and is interpreted throughout as a finite-scale, directional roughness diagnostic rather than a pointwise Hölder exponent. On isotropic datasets from the Johns Hopkins Turbulence Database, an internal subsampling benchmark against $N=200$ detector labels gives $F_1\approx0.93$ across nine unweighted reruns, and a balanced synthetic control gives balanced accuracy $0.928$ at $N=40$, indicating useful short-record self-consistency without constituting an external calibration. Across $Re_λ\approx433$--$1300$, the fixed-window sharp fraction remains of order $30$--$50\%$, but a scale-normalized control does not isolate a clean Reynolds-number trend. The recovered $\hatα$ is only weakly associated with dissipation, whereas higher vorticity is consistently associated with smaller detected roughness exponents in conditioned samples. Directional controls on 60 high-vorticity centers further show a positive vorticity-aligned contrast $Π_α$ (mean $0.093$, bootstrap 95\% CI $[0.028,0.158]$), stronger on the true vorticity axis than on fake axes, together with a statistically detectable low-order quadrupolar component in a joint Legendre fit. A seeded scale-transfer scan shows positive $Π_α$ at both the smallest and largest tested radii, supporting finite-range persistence without a strong theorem-level nonlocal claim. The method is therefore best viewed as a finite-scale geometric diagnostic complementary to energetic observables, capable of resolving directional structure and low-order anisotropic organization in high-vorticity regions.

Sparse Müntz--Szász Recovery for Boundary-Anchored Velocity Profiles: A Short-Record Roughness Diagnostic in Turbulence

Abstract

We present a sparse convex-relaxation framework for estimating effective local scaling exponents from short boundary-anchored velocity-increment profiles (). The detector solves an -regularized regression in a mixed Müntz--Szász/Jacobi dictionary and is interpreted throughout as a finite-scale, directional roughness diagnostic rather than a pointwise Hölder exponent. On isotropic datasets from the Johns Hopkins Turbulence Database, an internal subsampling benchmark against detector labels gives across nine unweighted reruns, and a balanced synthetic control gives balanced accuracy at , indicating useful short-record self-consistency without constituting an external calibration. Across --, the fixed-window sharp fraction remains of order --, but a scale-normalized control does not isolate a clean Reynolds-number trend. The recovered is only weakly associated with dissipation, whereas higher vorticity is consistently associated with smaller detected roughness exponents in conditioned samples. Directional controls on 60 high-vorticity centers further show a positive vorticity-aligned contrast (mean , bootstrap 95\% CI ), stronger on the true vorticity axis than on fake axes, together with a statistically detectable low-order quadrupolar component in a joint Legendre fit. A seeded scale-transfer scan shows positive at both the smallest and largest tested radii, supporting finite-range persistence without a strong theorem-level nonlocal claim. The method is therefore best viewed as a finite-scale geometric diagnostic complementary to energetic observables, capable of resolving directional structure and low-order anisotropic organization in high-vorticity regions.

Paper Structure

This paper contains 46 sections, 9 equations, 8 figures, 5 tables.

Table of Contents

  1. Introduction: Beyond Global Smoothness
  2. The intermittency problem
  3. Limitations of classical diagnostics
  4. Geometric approaches to turbulence
  5. Compressed sensing and geometric separation
  6. Contributions
  7. Methodology
  8. Data Source: Johns Hopkins Turbulence Database
  9. Profile extraction.
  10. Preprocessing and normalization.
  11. Signal Model
  12. Dictionary Construction
  13. LASSO Detection
  14. Handling of null detections.
  15. Exponent Refinement
  16. ...and 31 more sections

Figures (8)

  • Figure 1: Analytic dictionary geometry. Bars show the computed $\kappa_{\mathrm{pop}}(\alpha)$ curve, and the line shows the singular-vs.-polynomial cross-coherence proxy $\mu_{\mathrm{cross}}(\alpha)$. These are continuous weighted-model diagnostics; they are not empirical detection frequencies.
  • Figure 2: Balanced synthetic control: estimated $\hat{\alpha}$ vs. true $\alpha_{\mathrm{true}}$ for Sparse LASSO at $N{=}40$, $\sigma{=}0.01$, with 200 sharp and 200 smooth synthetic profiles. At this operating point, the detector attains accuracy $=\,0.9275$, balanced accuracy $=\,0.9275$, precision $=\,0.873$, recall $=\,1.000$, specificity $=\,0.855$, and $F_1=0.932$, with no null detections. The visible spread in the smooth class reflects nontrivial false positives, so this figure supports numerical sanity rather than perfect synthetic classification.
  • Figure 3: Subsampling study on JHTDB profiles. The panel plots detection rates for three reference-$\alpha$ bands as a function of $N$. In a recent unweighted run, the same experiment yields an overall sharp/smooth $F_1$ score of $0.927$ at $N{=}40$ when $N{=}200$ detector outputs on the same profiles are used as reference labels. Across the nine saved unweighted reruns used in the manuscript summary, the corresponding $N{=}40$$F_1$ value spans $0.84$ to $1.00$; the two March 23 transition artifacts from the temporary quadrature-weighting test are excluded from that count. This figure should therefore be interpreted as an internal run-dependent self-consistency study. The vertical marker is a heuristic guide used in the plotting script.
  • Figure 4: Baseline comparison at $N=40$. The synthetic panel in this figure is a restricted sharp-only method-comparison benchmark, whereas the balanced synthetic control appears separately in Fig. \ref{['fig:synth_scatter']}. The JHTDB panel uses reference labels produced by the same detector at $N=200$ on the same profiles. Under those conventions, sparse methods outperform the simple ElasticNet and structure-function baselines tested here.
  • Figure 5: Targeted control experiments: (a) magnitude vs. longitudinal increments; (b) $r_{\max}/\eta_K$-normalized Reynolds control; (c) widened-margin sensitivity to $\sigma$ and coefficient threshold; (d) widened-margin Jacobi degree $K$ and weight $\beta$ ablation.
  • ...and 3 more figures

Theorems & Definitions (2)

  • Remark 1: Grid truncation
  • Remark 2: Coherence metrics: $\mu$ vs. $\kappa_{\mathrm{pop}}$