Sparse M\"untz--Sz\'asz Recovery for Boundary-Anchored Velocity Profiles: A Short-Record Roughness Diagnostic in Turbulence

TL;DR

This paper introduces a sparse convex-relaxation method for estimating local roughness exponents from short velocity profiles in turbulence, providing a finite-scale diagnostic of directional roughness and anisotropy.

Contribution

The authors develop a novel $ ext{l}_1$-regularized regression framework using a M"untz--Szász/Jacobi dictionary for turbulence roughness diagnostics from short data records.

Findings

Achieves high self-consistency with $F_1\approx0.93$ at $N=40$

Detects directional vorticity-related roughness contrasts with statistical significance

Shows finite-range persistence of roughness features across scales

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 M\"untz--Sz\'asz/Jacobi dictionary and is interpreted throughout as a finite-scale, directional roughness diagnostic rather than a pointwise H\"older 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 $Re_\lambda\approx433$--$1300$, 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 $\hat{\alpha}$ 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 $\Pi_\alpha$ (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 $\Pi_\alpha$ 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.