Geometric derivation and structure-preserving simulation of quasi-geostrophy on the sphere
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| Main Authors: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866915577778405376 |
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| author | Luesink, Erwin Franken, Arnout Ephrati, Sagy Geurts, Bernard |
| author_facet | Luesink, Erwin Franken, Arnout Ephrati, Sagy Geurts, Bernard |
| contents | We present a geometric derivation of the quasi-geostrophic equations on the sphere, starting from the rotating shallow water equations. We utilise perturbation series methods in vorticity and divergence variables. The derivation employs asymptotic analysis techniques, leading to a global quasi-geostrophic potential vorticity model on the sphere without approximation of the Coriolis parameter. The resulting model forms a closed system for the evolution of potential vorticity with a rich mathematical structure, including Lagrangian and Hamiltonian descriptions. Formulated using the Lie-Poisson bracket reveals the geometric invariants of the quasi-geostrophic model. Motivated by these geometric results, simulations of quasi-geostrophic flow on the sphere are presented based on structure-preserving Lie-Poisson time-integration. We explicitly demonstrate the preservation of Casimir invariants and show that the hyperbolic quasi-geostrophic equations can be simulated in a stable manner over long time. We show the emergence of longitudonal jets, wrapped around the circumference of the sphere in a general direction that is perpendicular to the axis of rotation. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2402_13707 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Geometric derivation and structure-preserving simulation of quasi-geostrophy on the sphere Luesink, Erwin Franken, Arnout Ephrati, Sagy Geurts, Bernard Fluid Dynamics Mathematical Physics We present a geometric derivation of the quasi-geostrophic equations on the sphere, starting from the rotating shallow water equations. We utilise perturbation series methods in vorticity and divergence variables. The derivation employs asymptotic analysis techniques, leading to a global quasi-geostrophic potential vorticity model on the sphere without approximation of the Coriolis parameter. The resulting model forms a closed system for the evolution of potential vorticity with a rich mathematical structure, including Lagrangian and Hamiltonian descriptions. Formulated using the Lie-Poisson bracket reveals the geometric invariants of the quasi-geostrophic model. Motivated by these geometric results, simulations of quasi-geostrophic flow on the sphere are presented based on structure-preserving Lie-Poisson time-integration. We explicitly demonstrate the preservation of Casimir invariants and show that the hyperbolic quasi-geostrophic equations can be simulated in a stable manner over long time. We show the emergence of longitudonal jets, wrapped around the circumference of the sphere in a general direction that is perpendicular to the axis of rotation. |
| title | Geometric derivation and structure-preserving simulation of quasi-geostrophy on the sphere |
| topic | Fluid Dynamics Mathematical Physics |
| url | https://arxiv.org/abs/2402.13707 |