Bifurcation of Tetrahedral Non-Zonal Flows in the 2D Euler Equations on a Rotating Sphere
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arXiv
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866908954735411200 |
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| author | Cacchiò, Yuri |
| author_facet | Cacchiò, Yuri |
| contents | We investigate the emergence of finite-amplitude non-zonal flows on the sphere $\mathbb{S}^2$ arising from stationary solutions to the 2D Euler equations. By restricting the Laplace-Beltrami eigenspace to the invariant subspace of the tetrahedral symmetry group $\mathbf{T}$, we bypass the $(2l+1)$-dimensional kernel degeneracy, obtaining a scalar Liapunov-Schmidt reduction. We analyze four distinct physical non-linearities: a polynomial model, the sine-Gordon and sinh-Gordon models, and the exponential (Liouville) model. We explicitly derive the bifurcation parameter via spectral projections, proving that the bifurcation topology (subcritical or supercritical) is not a geometric invariant, but is governed by the parity of the nonlinearity and the mass conservation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_10092 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Bifurcation of Tetrahedral Non-Zonal Flows in the 2D Euler Equations on a Rotating Sphere Cacchiò, Yuri Analysis of PDEs 35Q31, 35B32, 37G40, 76B47 We investigate the emergence of finite-amplitude non-zonal flows on the sphere $\mathbb{S}^2$ arising from stationary solutions to the 2D Euler equations. By restricting the Laplace-Beltrami eigenspace to the invariant subspace of the tetrahedral symmetry group $\mathbf{T}$, we bypass the $(2l+1)$-dimensional kernel degeneracy, obtaining a scalar Liapunov-Schmidt reduction. We analyze four distinct physical non-linearities: a polynomial model, the sine-Gordon and sinh-Gordon models, and the exponential (Liouville) model. We explicitly derive the bifurcation parameter via spectral projections, proving that the bifurcation topology (subcritical or supercritical) is not a geometric invariant, but is governed by the parity of the nonlinearity and the mass conservation. |
| title | Bifurcation of Tetrahedral Non-Zonal Flows in the 2D Euler Equations on a Rotating Sphere |
| topic | Analysis of PDEs 35Q31, 35B32, 37G40, 76B47 |
| url | https://arxiv.org/abs/2604.10092 |