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 $|\omega|>a$ 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 $|\omega|=a$. 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 $Z=|\omega|^{-1/4}\omega$. 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.