Confined vortex surface and irreversibility. 2. Hyperbolic sheets and turbulent statistics
Alexander Migdal
Abstract
We continue the study of Confined Vortex Surfaces (CVS) that we introduced in the previous paper. We classify the solutions of the CVS equation and find the analytical formula for the velocity field for arbitrary background strain eigenvalues in the stable region. The vortex surface cross-section has the form of four symmetric hyperbolic sheets with a simple equation [Formula: see text] in each quadrant of the tube cross-section ([Formula: see text] plane). We use the dilute gas approximation for the vorticity structures in a turbulent flow, assuming their size is much smaller than the mean distance between them. We vindicate this assumption by the scaling laws for the surface shrinking to zero in the extreme turbulent limit. We introduce the Gaussian random background strain for each vortex surface as an accumulation of a large number of small random contributions coming from other surfaces far away. We compute this self-consistent background strain, relating the variance of the strain to the energy dissipation rate. We find a universal asymmetric distribution for energy dissipation. A new phenomenon is a probability distribution of the shape of the profile of the vortex tube in the [Formula: see text] plane. This phenomenon naturally leads to the “multifractal” scaling of the moments of velocity difference [Formula: see text]. More precisely, these moments have a nontrivial dependence of [Formula: see text], [Formula: see text], approximating power laws with effective index [Formula: see text]. We derive some general formulas for the moments containing multidimensional integrals. The rough estimate of resulting moments shows the log–log derivative [Formula: see text] which is approximately linear in [Formula: see text] and slowly depends on [Formula: see text]. However, the value of effective index is wrong, which leads us to conclude that some other solution of the CVS equations must be found. We argue that the approximate phenomenological relations for these moments suggested in a recent paper by Sreenivasan and Yakhot are consistent with the CVS theory. We reinterpret their renormalization parameter [Formula: see text] in the Bernoulli law [Formula: see text] as a probability to find no vortex surface at a random point in space.