全球大气运动应遵循的拓扑定理

Global Atmospheric Motion Should Follow Topological Theorem

地面天气图上的等压线斑图是空间压力曲面的廓线。全球压力曲面是凸凹不平的球面。压力曲面的峰、谷和通道(pass)对应于天气图上的高压中心、低压中心和鞍点(saddle;两个高压或两个低压间的通道)。尽管空间压力曲面的凸凹不平的位置随时间变化,相应的天气图上的高低压位置也不断变化,天气也随之变化,但是全球压力曲面的欧拉示性数(Euler characteristic)却是一个拓扑不变数,这个不变量就是球面的欧拉示性数为2。拓扑学的莫尔斯(Morse)定理,用大气学科的语言讲就是天气图上的(高压数目)+(低压数目)-(鞍点数目)=2。若将其推广到任何闭合曲面、任何奇点,则广义上称为庞加莱(Poincare)-霍普夫(Hopf)定理。显然,这个定理对天气预报有重大意义。本文列出了经向流、纬向流、单圈环流和三圈环流等例子。广大气象学工作者不但要知道大气运动应遵守流体力学的纳维-司托克斯(Navier-Stokes)方程,还要知道全球大气运动要遵循拓扑上的庞加莱-霍普夫定理。

The isobaric pattern in surface synoptic chart shows contours of space pressure surface. The global pressure surface is a spherical surface with concaves and convexes. The peaks, valleys and passes in the space pressure surface correspond to high pressure centers, low pressure centers and saddle points in surface synoptic chart. Although the locations of concaves and convexes in the spherical surface change with time, and the corresponding locations of high and low-pressure centers also change with time, the Euler characteristic of the spherical surface is a topology invariant, whose number is 2. Topologically, the Morse theorem states that if a gradient vector field on the spherical surface synoptic chart has many zeros, then（number of high pressure center） ＋（number of low pressure center）-（number of saddle point）=2. For any vector field, the extended theorem is called Poincare-Hopf theorem. This theorem is very important for weather prediction. The present paper shows application of this theorem in longitudinal flow, latitudinal flow, Hadley circulation, and three-cell circulation etc. Atmosphere scientists know for sure that global atmosphere motion follows not only the Navier-Stokes equation, but also the topological theorem.