大气科学  2018, Vol. 42 Issue (3): 634-639 PDF

Global Atmospheric Motion Should Follow Topological Theorem
LIU Shida
Department of Atmospheric and Oceanic Sciences, School of Physics, Peking University, Beijing 100871
Abstract: 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.
Key words: Synoptic chart      Topology theorem      Euler characteristic
1 引言

2 压力曲面及其临界点的典型形式、天气图的廓线

 $f(x, y, z) = 0$ (1)

 $p = p(x, y)$ (2)

 图 1 一维地形曲线 Figure 1 One-dimensional topographic curve

 图 2 三种压力曲面及其临界点 Figure 2 Three types of pressure surfaces and their critical points
 $\left\{ \begin{array}{l} p(x, y) = - {x^2} - {y^2}, \\ p(x, y) = {x^2} + {y^2}, \\ p(x, y) = - {x^2} + {y^2}, \end{array} \right.$ (3)

 $\partial p/\partial x = 0;{\rm{ }}\partial p/\partial y = 0.$ (4)

 $χ=(源点数目)+(汇点数目)-(鞍点数目),$ (6)

 $（高压中心点数目）+（低压中心的数目）-\\~~~~~~~~~~~~~（鞍点数目）=2.$ (7)

 $\chi = \mathop \sum \limits_{i = 1}^n {I_n}$ (8)

4 全球大气流场的实例

4.1 从北极沿经圈流向南极的经向流

 图 5 沿经圈的经向流 Figure 5 Meridional flow along the longitudes
4.2 沿纬圈的西风纬向流

 图 6 沿纬圈的西风环流 Figure 6 Westerly flow along the latitudes
4.3 全球四个大涡旋

 图 7 （a）北极—南极经圈和赤道将地球分成四部分；（b）每个部分在两个鞍点S1和S2间形成涡旋的剖面 Figure 7 (a) Four parts of the earth surface divided by the equator and a longitude circle; (b) cross section along two saddle points (S1, S2) where the vortex

 图 8 单圈环流地表面流场 Figure 8 Surface flow corresponding to single-cell circulation

4.5 三圈环流（Djuric，1994

 图 9 三圈环流中的地表东西风带 Figure 9 Surface westerlies and easterlies corresponding to the three-cell circulations

4.6 偶极子流场

 图 10 球只有一个奇点的偶极子流场 Figure 10 Global dipole flow with only one singular point
5 关于高空天气图如何使用庞加莱-霍普夫定理的问题

 图 11 500 hPa天气图上等高线波动等于平直等高线加上高压或低压 Figure 11 Geopotential perturbations at 500-hPa synoptic weather chart are equal to straight isohypses plus high or low pressure

 图 12 一个高压脊线和一个低压槽线的交点便是鞍点 Figure 12 The point of intersection between a high pressure ridge line and a low pressure trough line is the saddle point
6 结论

 Bluestein H B. 1992. Synoptic-Dynamic Meteorology in Midlatitudes:Volume Ⅰ. Principles of Kinematics and Dynamics [M]. New York, Oxford: Oxford University Press: 431pp. Bluestein H B. 1993. Synoptic-Dynamic Meteorology in Midlatitudes. Volume Ⅱ. Observations and Theory of Weather Systems [M]. Oxford: Oxford University Press: 594pp. Djuric D. 1994. Weather Analysis [M]. Prentice-Hall Inc.: 314pp. Flegg H G. 1974. From Geometry to Topology [M]. New York: Crane Russak and Co.: 275pp. Hatcher A. 2002. Algebraic Topology [M]. Cambridge: Cambridge University Press: 345pp. Hopf H. 1926. über die curvatura integra geschlossener hyperflächen [J]. Math. Ann., 95(1): 340-367. DOI:10.1007/BF01206615 刘式达, 刘式适, 傅遵涛. 2016. 大气运动的几何和拓扑[M]. 北京: 高等教育出版社. Liu Shida, Liu Shishi, Fu Zuntao. 2016. Geometry and Topology of Atmosphere Motions (in Chinese)[M]. Beijing: Higher Education Press. Morse M. 1929. Singular points of vector fields under general boundary conditions [J]. Amer. J. Math., 51(2): 165-178. DOI:10.2307/2370703 Poincaré H. 1902. Sur certaines surfaces algébriques, Troisième complément à l'Analysis sitûs [J]. Bull Soc. Math. France, 30: 49-70. DOI:10.24033/bsmf.657 Richeson D S. 2008. Euler's Gem:The Polyhedron Formula and the Birth of Topology [M]. Princeton, NJ: Princeton University Press: 368pp.