Remarks on Fixed- and Variable-Order Fractional-Laplacian Closures for TurbulentDissipation

Abstract

Two independent attempts, separated by two decades, to encode turbulent dissipation with a fractional Laplacian both converge on the operator order as the signature of the fully developedinertial range: the additive, fixed order model, and the dynamically deforming operator of the Adaptive Fractional Navier Stokes. We show that, despite sharing this fixed point, the two constructions belongto distinct universality classes: Chen operator is a superposition of two fixed order kernels whose relative weight drifts with scale, while AFNS posits a single operator whose order itself flows continuously with the local Reynolds number. We derive a scale resolved observable, the effective spectral order, whose logarithmic anomaly term discriminates between these two mechanisms in principle, and then obtain a sharper, purely analytic verdict from the two dimensional, enstrophy conserving limit: the generative mechanism matching a stable process index to the empirical pair dispersion law has no solution once vortex stretching is kinematically switched off. Whether this dichotomy leaves a residual signature in real quasi two dimensional flows is left as an open experimental challenge.