Coherent structures, instabilities, and the arrow of time

In fluid dynamics, viscosity sets the arrow of time: if viscosity is nonnegative, the Second Law of Thermodynamics is satisfied. But time-asymmetry can still be small, as in the famous demonstration by G. I. Taylor, if the nonlinear term of the Navier-Stokes equation is negligible. Nonlinearities matter most, in the sense of sensitive dependence on initial conditions, in regions where finite-time Lyapunov exponents are largest. Those same regions are home to Lagrangian Coherent Structures (LCS), the flow's most important barriers to mixing. We present experiments and simulations showing that the motion of forward-time LCS differs from that of backward-time LCS, as in prior work. Varying flow speed, we show that time-asymmetry of LCS motion is nearly zero for steady flow and increases with Reynolds number. In fact, time-asymmetry jumps discontinuously at the Reynolds number where an instability initiates periodic flow, and jumps again where periodicity gives way to chaos. Our results suggest that time-asymmetry is often driven by interactions between nonlinearities and viscosity, and are relevant to attempts to make LCS predictive, not merely descriptive, since attracting LCS are less predictable than repelling LCS.