Curvature and stability of quasi-geostrophic motion
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866918143799066624 |
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| author | Suri, Ali |
| author_facet | Suri, Ali |
| contents | This paper outlines the study of the curvature of the quantomorphism group and its central extension, as well as the quasi-geostrophic equation. By utilizing spherical harmonics and structure constants, a formula for computing the curvature of the $L^2$ metric on the central extension $\hat{\cg}=\cg\ltimes_Ω\mathbb{R}$ is derived, where $\cg$ represents the Lie algebra of $\cD^s_μ(\mathbb{S}^2)$. The sectional curvatures of the planes containing $Y_{10}$ and the tradewind current are calculated as special cases. The impact of the Rossby and Froude numbers, as well as the Coriolis effect, on the (exponential) stability of these quasi-geostrophic motions is highlighted. Finally, a lower bound for weather prediction error in a simplified model governed by the tradewind current and the Coriolis effect on a rotating sphere is suggested. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_03403 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Curvature and stability of quasi-geostrophic motion Suri, Ali Differential Geometry 58D05, 35Q35, 53C22, 53C80 This paper outlines the study of the curvature of the quantomorphism group and its central extension, as well as the quasi-geostrophic equation. By utilizing spherical harmonics and structure constants, a formula for computing the curvature of the $L^2$ metric on the central extension $\hat{\cg}=\cg\ltimes_Ω\mathbb{R}$ is derived, where $\cg$ represents the Lie algebra of $\cD^s_μ(\mathbb{S}^2)$. The sectional curvatures of the planes containing $Y_{10}$ and the tradewind current are calculated as special cases. The impact of the Rossby and Froude numbers, as well as the Coriolis effect, on the (exponential) stability of these quasi-geostrophic motions is highlighted. Finally, a lower bound for weather prediction error in a simplified model governed by the tradewind current and the Coriolis effect on a rotating sphere is suggested. |
| title | Curvature and stability of quasi-geostrophic motion |
| topic | Differential Geometry 58D05, 35Q35, 53C22, 53C80 |
| url | https://arxiv.org/abs/2310.03403 |