A Tensorial Extension of Fractional Dissipation at Inertial Scales
Abstract
We formulate a scale dependent tensorial extension of a fractional dissipation operator motivated by a macroscopic Navier Stokes correspondence. The starting point is a fractional model in which the order of dissipation decreases continuously from the Laplacian value toward the Kolmogorov value as a local Reynolds number increases. At the critical value of the local Reynolds number, the model reduces to an isotropic operator. We then ask whether the same macroscopic description can be continued into a regime in which velocity fields have Onsager, critical or subcritical regularity without introducing an unrelated closure or an additional isotropic dissipation scale. In addition, we extend the original AFNS free, energy, construction by one nonnegative concentration order parameter c. The new term is quadratic in c and linearly coupled to the defect beta function, so that minimization leaves the original AFNS free energy exactly unchanged when the concentration branch is inactive. The resulting three regime mosaic is regular, inertial, and concentrated/anomalous. The construction uses the coarse-grained stress and its critical normalization. The normalized anisotropy tensor provides the lowest order objective tensorial information available at the critical scale, while the scale derivative of the Duchon Robert energy defect, supplies a scalar measure of departure from the Onsager threshold. We propose the minimal constitutive deformation of the critical operator: in Weyl quantization, the isotropic fractional symbol is dressed by this exponential tensorial factor, giving a pseudodifferential operator that reduces to the scalar critical operator. The exponential parametrization guarantees positive eigenvalues and unit determinant for symmetric traceless, so the deformation redistributes dissipation among directions without introducing a new isotropic amplitude.
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BibTeX
@misc{lopez2026tensorial,
title = {{A Tensorial Extension of Fractional Dissipation at Inertial Scales}},
author = {José I. H. López},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.35905},
eprint = {2609.35905},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.35905},
}
RIS
TY - UNPB
TI - A Tensorial Extension of Fractional Dissipation at Inertial Scales
AU - José I. H. López
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.35905
UR - https://arxiv.org/abs/2609.35905
AB - We formulate a scale dependent tensorial extension of a fractional dissipation operator motivated by a macroscopic Navier Stokes correspondence. The starting point is a fractional model in which the order of dissipation decreases continuously from the Laplacian value toward the Kolmogorov value as a local Reynolds number increases. At the critical value of the local Reynolds number, the model reduces to an isotropic operator. We then ask whether the same macroscopic description can be continued into a regime in which velocity fields have Onsager, critical or subcritical regularity without introducing an unrelated closure or an additional isotropic dissipation scale. In addition, we extend the original AFNS free, energy, construction by one nonnegative concentration order parameter c. The new term is quadratic in c and linearly coupled to the defect beta function, so that minimization leaves the original AFNS free energy exactly unchanged when the concentration branch is inactive. The resulting three regime mosaic is regular, inertial, and concentrated/anomalous. The construction uses the coarse-grained stress and its critical normalization. The normalized anisotropy tensor provides the lowest order objective tensorial information available at the critical scale, while the scale derivative of the Duchon Robert energy defect, supplies a scalar measure of departure from the Onsager threshold. We propose the minimal constitutive deformation of the critical operator: in Weyl quantization, the isotropic fractional symbol is dressed by this exponential tensorial factor, giving a pseudodifferential operator that reduces to the scalar critical operator. The exponential parametrization guarantees positive eigenvalues and unit determinant for symmetric traceless, so the deformation redistributes dissipation among directions without introducing a new isotropic amplitude.
ER -