Conjugate points along spherical harmonics

2024/02/16 by Suri, Ali · 1 citation
#35Q35 #53C22 #53C80 #58D05 #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2402.10578

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 \Sph². 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 e_lm=∇^⊥ 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 ∇^⊥ Y_l1~0, we demonstrate that varying the 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.

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• Ricci curvature for hydrodynamics on the sphere

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