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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2503.14191 |
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| _version_ | 1866912281456017408 |
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| author | Constantin, Adrian Germain, Pierre Lin, Zhiwu Zhu, Hao |
| author_facet | Constantin, Adrian Germain, Pierre Lin, Zhiwu Zhu, Hao |
| contents | We investigate some qualitative aspects of the dynamics of the Euler equation on a rotating sphere that are relevant or stratospheric flows. Zonal flow dominates the dynamics of the stratosphere and for most known planetary stratospheres the observed flow pattern is a small perturbation of an n-jet, which corresponds to choosing the Legendre polynomial of degree n as the stream function. Since the 1-jet and the 2-jet are stable, the main interest is the onset of instability for the 3-jet. We confirm long standing conjectures based on numerical simulations by proving that the 3-jet is linearly unstable if and only if the rotation rate belongs to a critical interval. Turning to the nonlinear problem, we prove that linear instability implies nonlinear instability and that, as the rotation rate goes to infinity, nearby traveling waves change gradually from a cat's eyes streamline pattern to a profile with no stagnation points. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_14191 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The onset of instability for zonal stratospheric flows Constantin, Adrian Germain, Pierre Lin, Zhiwu Zhu, Hao Analysis of PDEs 86A10, 35Q35 We investigate some qualitative aspects of the dynamics of the Euler equation on a rotating sphere that are relevant or stratospheric flows. Zonal flow dominates the dynamics of the stratosphere and for most known planetary stratospheres the observed flow pattern is a small perturbation of an n-jet, which corresponds to choosing the Legendre polynomial of degree n as the stream function. Since the 1-jet and the 2-jet are stable, the main interest is the onset of instability for the 3-jet. We confirm long standing conjectures based on numerical simulations by proving that the 3-jet is linearly unstable if and only if the rotation rate belongs to a critical interval. Turning to the nonlinear problem, we prove that linear instability implies nonlinear instability and that, as the rotation rate goes to infinity, nearby traveling waves change gradually from a cat's eyes streamline pattern to a profile with no stagnation points. |
| title | The onset of instability for zonal stratospheric flows |
| topic | Analysis of PDEs 86A10, 35Q35 |
| url | https://arxiv.org/abs/2503.14191 |