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Existence of a conjugate point in the incompressible Euler flow on a sphereand an ellipsoid is considered. Misiolek (1996) formulated adifferential-geometric criterion (we call M-criterion) for the existence of aconjugate 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 asphere, on the other hand, there are zonal flows satisfy M-criterion if thebackground manifold is an ellipsoid (even it is sufficiently close to thesphere). The conjugate point is created by the fully nonlinear effect of theinviscid fluid flow with differential geometric mechanism.

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