球调和函数和相应的大气环流拓扑模型OA

The Nodal sets(zero sets)of spherical harmonics divide the spherical surface into numerous small regions through their intersections,leading to the formation of vertices(V),edges(E),and faces(F).The relationship among these elements and the Euler characteristic(χ)of spherical topology is expressed as χ=V-E+F=2.If the flow field on the spherical surface is vortical,the Nodal sets physically correspond to locations where the vertical vorticity on the sphere is zero.These Nodal sets divide the sphere into alternating regions of positive and negative vorticity,which represent cyclonic and anticyclonic systems,respectively.For vortical fields consisting solely of the zonal flow,the Nodal sets coincide with the locations of zero vertical vorticity and the latitude-weighted maxima of the zonal flow.If the flow field on the spherical surface is a nonvortical gradient field,the zonal Nodal set represents lines of constant potential or isobars.The meridional circulation,such as the north-south meridional flow,is perpendicular to the Nodal set.In this case,the Nodal set corresponds to regions of zero horizontal divergence and divides the spherical surface into alternating areas of positive and negative horizontal divergence.Based on these considerations,qualitative models of atmospheric circulation,such as meridional and zonal flows,Hadley circulation,and the three-cell circulation system with its associated planetary wind belts,can be inferred from a topological perspective on the sphere.These conceptual models can be validated through analysis of the properties of critical points in the flow field.