Conjugate points along spherical harmonics
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866911162312949760 |
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| author | Suri, Ali |
| author_facet | Suri, Ali |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_10578 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Conjugate points along spherical harmonics Suri, Ali Differential Geometry 58D05, 35Q35, 53C22, 53C80 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. |
| title | Conjugate points along spherical harmonics |
| topic | Differential Geometry 58D05, 35Q35, 53C22, 53C80 |
| url | https://arxiv.org/abs/2402.10578 |