标题： 球面上的热准地转模型：推导与结构保持模拟 显示英文标题

标题： Thermal quasi-geostrophic model on the sphere: derivation and structure-preserving simulation

摘要： 我们通过热旋转浅水方程的渐近展开推导了球面上的全局热准地转模型。该模型不依赖科里奥利力的渐近展开，并通过包括一个额外的输运浮力场来扩展球面上的准地转模型，该场作为位势涡度的源项。我们给出了其以半直积李-泊松括号形式的哈密顿描述。哈密顿公式揭示了存在无限多个守恒定律，即卡西米尔，这些由两个任意光滑函数参数化。基于泽特林为流体力学提出的自洽矩阵近似，提供了一种保持结构的离散化方法。采用保持卡西米尔的时间积分器，以数值上完全保留所得的有限维李-泊松结构。模拟结果揭示了涡度和浮力前沿的形成，以及由浮力-地形相互作用引起的浮力动力学中的大尺度结构。 显示英文摘要

摘要： We derive the global model of thermal quasi-geostrophy on the sphere via asymptotic expansion of the thermal rotating shallow water equations. The model does not rely on the asymptotic expansion of the Coriolis force and extends the quasi-geostrophic model on the sphere by including an additional transported buoyancy field acting as a source term for the potential vorticity. We give its Hamiltonian description in terms of semidirect product Lie--Poisson brackets. The Hamiltonian formulation reveals the existence of an infinite number of conservation laws, Casimirs, parameterized by two arbitrary smooth functions. A structure-preserving discretization is provided based on Zeitlin's self-consistent matrix approximation for hydrodynamics. A Casimir-preserving time integrator is employed to numerically fully preserve the resulting finite-dimensional Lie--Poisson structure. Simulations reveal the formation of vorticity and buoyancy fronts, and large-scale structures in the buoyancy dynamics induced by the buoyancy-bathymetry interaction.