Conjugate points along spherical harmonics

Abstract

Utilizing structure constants, we present a version of the Misiolek criterion for identifying conjugate points. We propose an approach that enables us to locate these points along solutions of the quasi-geostrophic equations on the sphere S2. We demonstrate that for any spherical harmonics Ylm with 1≤|m|≤l, except for Y1±1 and Y2±1, conjugate points can be determined along the solution generated by the velocity field elm=∇⊥Ylm. Subsequently, we investigate the impact of the Coriolis force on the occurrence of conjugate points. Moreover, for any zonal flow generated by the velocity field ∇⊥Yl10, we demonstrate that proper rotation rate can lead to the appearance of conjugate points along the corresponding solution, where l1=2k+1.∈N Additionally, we prove the existence of conjugate points along (complex) Rossby-Haurwitz waves and explore the effect of the Coriolis force on their stability.