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 $\Sph^2$. We demonstrate that for any spherical harmonics $Y_{lm}$ with $1 \leq |m| \leq l$, except for $Y_{1\pm1}$ 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 varying the rotation rate can lead to the appearance of conjugate points along the corresponding solution, where $l_1 = 2k+1. \in \mathbb{N}$ Additionally, we prove the existence of conjugate points along (complex) Rossby-Haurwitz waves and explore the effect of the Coriolis force on their stability.