准大气共轨的球面几何推导与结构保持仿真

我们对球面上准大气共轨方程进行几何推导， starting from rotating shallow water equations。我们利用在涡度和散度变量中的扰动级数方法。该推导采用渐近分析技术，导致在不近似 Coriolis 系数的情况下，在球面上得到全局准大气共轨势涡动模型。 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。