Conjugate and cut points in ideal fluid motion

TL;DR

This paper investigates the existence and properties of conjugate and cut points in the space of ideal fluid flows, revealing conditions for stability and instability in various geometries using geometric and analytical methods.

Contribution

It establishes the existence of conjugate points in Kolmogorov flows and non-existence in Arnold stable states, advancing understanding of fluid flow stability in geometric terms.

Findings

Conjugate points exist in Kolmogorov flows on rectangular tori.

No conjugate points in Arnold stable steady states on certain geometries.

Closest cut points are either conjugate points or midpoints of periodic flows.

Abstract

Two fluid configurations along a flow are conjugate if there is a one parameter family of geodesics (fluid flows) joining them to infinitesimal order. Geometrically, they can be seen as a consequence of the (infinite dimensional) group of volume preserving diffeomorphisms having sufficiently strong positive curvatures which `pull' nearby flows together. Physically, they indicate a form of (transient) stability in the configuration space of particle positions: a family of flows starting with the same configuration deviate initially and subsequently re-converge (resonate) with each other at some later moment in time. Here, we first establish existence of conjugate points in an infinite family of Kolmogorov flows - a class of stationary solutions of the Euler equations - on the rectangular flat torus of any aspect ratio. The analysis is facilitated by a general criterion for identifying conjugate points in the group of volume preserving diffeomorphisms. Next, we show non-existence of conjugate points along Arnold stable steady states on the annulus, disk and channel. Finally, we discuss cut points, their relation to non-injectivity of the exponential map (impossibility of determining a flow from a particle configuration at a given instant) and show that the closest cut point to the identity is either a conjugate point or the midpoint of a time periodic Lagrangian fluid flow.