Threshold Decomposition of Vorticity Stretching and Palinstrophy
Abstract
We give an exact finite-time decomposition of vortex stretching by resolving the vorticity equation according to the size of the vorticity. For almost every threshold a>0, the stretching occurring where |ω|>a equals the change of the corresponding superlevel mass plus two nonnegative viscous terms: one measuring changes in vorticity direction and one measuring dissipation across the level surface |ω|=a. Integrating this identity over all thresholds recovers the classical Lp vorticity balance and, for p=2, the usual spacetime palinstrophy. At the scale-critical exponent p=3/2, the dissipation can be written through the nonlinear variable Z=|ω|−1/4ω. The threshold formulation also gives a positive measure that can be used to assign overlapping high-vorticity episodes without double counting. Under additional compactness assumptions it leads, through empirical transition couplings supported on a closed physical transition relation, to a stationary Markov renormalization law; no deterministic return map is required. These are structural results; no unconditional three-dimensional regularity theorem is claimed.
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BibTeX
@misc{shahmurov2026threshold,
title = {{Threshold Decomposition of Vorticity Stretching and Palinstrophy}},
author = {Rishad Shahmurov},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2605.01873},
eprint = {2605.01873},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.01873},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Threshold Decomposition of Vorticity Stretching and Palinstrophy
AU - Rishad Shahmurov
PY - 2026
DA - 2026/09/10
PB - arXiv
DO - 10.48550/arxiv.2605.01873
UR - https://arxiv.org/abs/2605.01873
AB - We give an exact finite-time decomposition of vortex stretching by resolving the vorticity equation according to the size of the vorticity. For almost every threshold a>0, the stretching occurring where |ω|>a equals the change of the corresponding superlevel mass plus two nonnegative viscous terms: one measuring changes in vorticity direction and one measuring dissipation across the level surface |ω|=a. Integrating this identity over all thresholds recovers the classical L^p vorticity balance and, for p=2, the usual spacetime palinstrophy. At the scale-critical exponent p=3/2, the dissipation can be written through the nonlinear variable Z=|ω|^−1/4ω. The threshold formulation also gives a positive measure that can be used to assign overlapping high-vorticity episodes without double counting. Under additional compactness assumptions it leads, through empirical transition couplings supported on a closed physical transition relation, to a stationary Markov renormalization law; no deterministic return map is required. These are structural results; no unconditional three-dimensional regularity theorem is claimed.
ER -