HRT

Abstract

We do not understand the laws of motion of vortex and magnetic-field lines in high-Reynolds-number turbulent flows. The current lore is self-contradictory. On the one hand, vortex/magnetic-field lines are often assumed to wander and elongate nearly as material lines in the limit of small viscosity/resistivity, and thus also to intensify, as a consequence of the Kelvin/Alfvén theorems. On the other hand, the topology of the lines is assumed to be continuously altered by viscous/resistive reconnection, implying breakdown of those same theorems. We discuss experimental and numerical evidence that these laws are both sometimes observed and sometimes violated in high-Reynolds-number turbulence. Unfortunately, we have no rational criterion to say when the Kelvin/Alfvén theorems or the Helmholtz laws of ``frozen-in'' motion should hold and when they should not. The problem has grown more perplexing with the theoretical discovery of "spontaneous stochasticity" for Lagrangian particle evolution in a Kolmogorov inertial range. As a consequence of the forgetting of initial separations in Richardson pair-diffusion, Lagrangian trajectories are not unique and must be replaced with random distributions of trajectories in the limit of small viscosity. This result presents a major crisis to our understanding of the turbulent dynamics of vortex/magnetic-field lines. As a possible resolution, we discuss a conjectured generalization of the Kelvin/Alfvén theorems, namely, that they survive as "backward martingales" of the spontaneous stochastic flows at high Reynolds-number. This conjectured relation provides a precise mathematical framework for the theory of turbulent reconnection. We discuss current rigorous results related to the conjecture and also important questions for investigation by experiment and simulation.