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 ($N\approx40$). 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.