标题： 屏蔽泊松方程在球面上的格林函数 显示英文标题

标题： Green's function of the screened Poisson's equation on the sphere

摘要： 在地球物理流体动力学中，筛选泊松方程出现在浅水、准地转方程中。 最近，已经尝试使用不同的数值方法在球面上求解这些方程。 这些方法包括涡旋方法，它们通过求解一个泊松方程从（相对）涡度计算流函数。 或者，可以直接从位势涡度（PV）计算流函数，这在无粘情况下沿物质轨迹守恒的条件下，可能提供构建更优涡旋方法的可能性。 然而，在球壳上，筛选泊松方程不具有已知的格林函数，这限制了此类方法在球面情况下的扩展。 在本文中，我们以级数形式和积分形式推导了球壳上筛选泊松方程的格林函数表达式。 然后给出了级数表示的收敛性证明。 由于级数收敛缓慢，使用一种分离形式获得了一种稳健且高效的近似，该形式分离了奇异行为。 对于不同值的筛选常数，对解进行了说明和分析。 显示英文摘要

摘要： In geophysical fluid dynamics, the screened Poisson equation appears in the shallow-water, quasi geostrophic equations. Recently, many attempts have been made to solve those equations on the sphere using different numerical methods. These include vortex methods, which solve a Poisson equation to compute the stream-function from the (relative) vorticity. Alternatively, the stream-function can be computed directly from potential vorticity (PV), which would offer the possibility of constructing more attractive vortex methods because PV is conserved along material trajectories in the inviscid case. On the spherical shell, however, the screened Poisson equation does not admit a known Green's function, which limits the extension of such approaches to the case of a sphere. In this paper, we derive an expression of Green's function for the screened Poisson equation on the spherical shell in series form and in integral form. A proof of convergence of the series representation is then given. As the series is slowly convergent, a robust and efficient approximation is obtained using a split form which isolates the singular behavior. The solutions are illustrated and analyzed for different values of the screening constant.