Exponential growth in two-dimensional topological fluid dynamics

TL;DR

This paper explores how certain stirring protocols in two-dimensional fluids cause exponential growth in the complexity of material lines and vorticity gradients, with implications for Euler flow behavior.

Contribution

It applies Thurston-Nielsen theory to demonstrate exponential growth of vorticity gradients in Euler flows under specific periodic stirring protocols.

Findings

Exponential growth of material line length under certain protocols

Existence of non-periodic Euler solutions with exponential vorticity gradient growth

Application of topological methods to fluid dynamics problems

Abstract

This paper describes topological kinematics associated with the stirring by rods of a two-dimensional fluid. The main tool is the Thurston-Nielsen (TN) theory which implies that depending on the stirring protocol the essential topological length of material lines grows either exponentially or linearly. We give an application to the growth of the gradient of a passively advected scalar, the Helmholtz-Kelvin Theorem then yields applications to Euler flows. The main theorem shows that there are periodic stirring protocols for which generic initial vorticity yields a solution to Euler's equations which is not periodic and further, the $L^\infty$ and $L^1$-norms of the gradient of its vorticity grow exponentially in time.