Commutative Coarse-graining on the Sphere
Abstract
Filtering by spatial convolution underlies coarse-graining and Large Eddy Simulation, the analysis of multiscale dynamics, and data processing across the sciences. Yet, as commonly practiced on the sphere, it violates a basic consistency requirement: commutation with differential operators, which is necessary for deriving coarse-grained dynamics and preserving the physics across lengthscales. We discuss the root of the problem, which is intrinsic to the curved geometry rather than being a result of metric terms, coordinate system, or numerical grid. A recently proposed generalized spherical convolution is, for a given scalar kernel, the unique linear operator commuting with the radial projection, divergence, and curl. Any consistent linear operator, be it spectral truncation, wavelet transform, diffusion, Green's-function inversion, or a machine-learned convolution, necessarily takes this form. From a simulation of an unstable jet, we demonstrate commutation to within numerical round-off, whereas common filtering that ignores curvature produces significant spurious divergence where none exists and gradient errors comparable to the gradients themselves.
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BibTeX
@misc{teeraratkul2026commutative,
title = {{Commutative Coarse-graining on the Sphere}},
author = {Chayut Teeraratkul and Hussein Aluie},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.26896},
eprint = {2610.26896},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.26896},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Commutative Coarse-graining on the Sphere
AU - Chayut Teeraratkul
AU - Hussein Aluie
PY - 2026
DA - 2026/10/16
PB - arXiv
DO - 10.48550/arxiv.2610.26896
UR - https://arxiv.org/abs/2610.26896
AB - Filtering by spatial convolution underlies coarse-graining and Large Eddy Simulation, the analysis of multiscale dynamics, and data processing across the sciences. Yet, as commonly practiced on the sphere, it violates a basic consistency requirement: commutation with differential operators, which is necessary for deriving coarse-grained dynamics and preserving the physics across lengthscales. We discuss the root of the problem, which is intrinsic to the curved geometry rather than being a result of metric terms, coordinate system, or numerical grid. A recently proposed generalized spherical convolution is, for a given scalar kernel, the unique linear operator commuting with the radial projection, divergence, and curl. Any consistent linear operator, be it spectral truncation, wavelet transform, diffusion, Green's-function inversion, or a machine-learned convolution, necessarily takes this form. From a simulation of an unstable jet, we demonstrate commutation to within numerical round-off, whereas common filtering that ignores curvature produces significant spurious divergence where none exists and gradient errors comparable to the gradients themselves.
ER -