Dependence of the asymptotic energy dissipation on third-order velocity scaling

TL;DR

The paper addresses whether the zeroth-law of turbulence—nonzero dissipation in the inviscid limit—can be inferred from the scaling of third-order longitudinal velocity increments in three-dimensional turbulence, leveraging the Kolmogorov $\\frac{4}{5}$ law. Under a uniform-in-$\\nu$ bound on the longitudinal absolute third-order increment $A^\\parallel_3(\\ell)/\\langle|\\mathbf{u}^\\nu|^3\\rangle \\le 8 \\ell^{\\xi_{3,\\parallel}}$, the authors derive $\\epsilon^* \\le 10 \\lim_{\\ell\\to 0} \\ell^{(\\xi_{3,\\parallel}-1)}$, so that $\\xi_{3,\\parallel}>1$ implies $\\epsilon^* = 0$ and energy conservation in the inviscid limit. Because the longitudinal exponent typically satisfies $\\xi_{3,\\parallel} \\ge \\xi_3$ and often exceeds the total exponent due to transverse intermittency, this criterion is sharper than prior work linking dissipation to the total third-order scaling. If $\\epsilon^* = 0$, Kolmogorov's law implies $S^\\parallel_3(\\ell)/\\ell \\to 0$ as $\\ell\\to 0$, suggesting asymptotic symmetry of the longitudinal velocity increment and possible Lagrangian time-reversibility in the vanishing-viscosity limit; if $\\xi_{3,\\parallel} \\le 1$ and $\\epsilon^* > 0$, small-scale asymmetry and skewness persist.

Abstract

The asymptotic energy dissipation is connected to the third-order scaling of the longitudinal velocity increment magnitude in three-dimensional turbulence via the Kolmogorov $4/5$ law. It is shown that the third-order longitudinal absolute velocity increment scaling should not exceed unity for anomalous dissipation to occur, that is for non-vanishing average dissipation in the inviscid limit -- also known as the ``zeroth law" of turbulence. Conversely, if the third-order longitudinal absolute velocity increment scaling exceeds unity then the average dissipation must asymptotically vanish and the velocity increment field will becomes symmetric at least at the level of its skewness. This work highlights the importance of the third-order absolute velocity increment scaling in assessing the status of the ``zeroth-law" of turbulence.