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 $S^2$. We demonstrate that for any spherical harmonics $Y_{lm}$ with $1\le \left|m\right|\le l$, except for $Y_{1\pm 1}$ and $Y_{2\pm 1}$, conjugate points can be determined along the solution generated by the velocity field $e_{lm}={\nabla }^{\perp }{Y}_{lm}$. 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 ${\nabla }^{\perp }{Y}_{l_{1}0}$, we demonstrate that proper rotation rate can lead to the appearance of conjugate points along the corresponding solution, where $l_1=2k+1.\in N$ Additionally, we prove the existence of conjugate points along (complex) Rossby-Haurwitz waves and explore the effect of the Coriolis force on their stability.