Conjugate points along spherical harmonics

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 Y-lm with 1 <=|m|<= l, except for Y-1 +/- 1 and Y-2 +/- 1, conjugate points can be determined along the solution generated by the velocity field elm=del Y-perpendicular to(lm). 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 del(perpendicular to)Y(l1)0, we demonstrate that proper rotation rate can lead to the appearance of conjugate points along the corresponding solution, where l(1)=2k+1.is an element of 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.