# Existence of a conjugate point in the incompressible Euler flow on an ellipsoid

## TL;DR

This paper investigates the existence of conjugate points in incompressible Euler flows on spheres and ellipsoids, revealing that certain flows on ellipsoids satisfy geometric criteria for conjugate points, unlike on spheres.

## Contribution

It demonstrates that zonal flows on ellipsoids can satisfy the M-criterion for conjugate points, extending geometric understanding beyond spherical cases.

## Key findings

- No zonal flow on a sphere satisfies the M-criterion.
- Some zonal flows on ellipsoids do satisfy the M-criterion.
- Conjugate points arise from nonlinear effects in inviscid flows.

## Abstract

Existence of a conjugate point in the incompressible Euler flow on a sphere and an ellipsoid is considered. Misiolek (1996) formulated a differential-geometric criterion (we call M-criterion) for the existence of a conjugate point in a fluid flow. In this paper, it is shown that no zonal flow (stationary Euler flow) satisfies M-criterion if the background manifold is a sphere, on the other hand, there are zonal flows satisfy M-criterion if the background manifold is an ellipsoid (even it is sufficiently close to the sphere). The conjugate point is created by the fully nonlinear effect of the inviscid fluid flow with differential geometric mechanism.