极向–环向分解
在向量分析中,极向–环向分解(英文:poloidal–toroidal decomposition)是亥姆霍兹分解的一个受限制的形式,常用于螺线向量场在球坐标系下的分析,如磁场和不可压缩流体等。[1]考虑一个三维向量场F满足
可以被表示为一个轴矢量场(toroidal vector field)和一个极矢量场(poloidal vector field)的和:
其中是球坐标中的径向矢量,纵场为
为一标量场,[2]横场为
为一标量场。[2]这一向量分解法是对称的,因为纵场的旋度是横场,而横场的旋度是纵场。[3]纵场与球心在原点的球面相切
- ,[3]
而横场的旋度同样地与这些球面相切
- .[4]
若标量场和的平均值在任意半径为的球面上都等于零,则这一分解方式是唯一的。[2]
另见
编辑脚注
编辑- ^ Subrahmanyan Chandrasekhar. Hydrodynamic and hydromagnetic stability. International Series of Monographs on Physics. Oxford: Clarendon. 1961. See discussion on page 622 [2016-04-29]. (原始内容存档于2012-02-12).
- ^ 2.0 2.1 Backus 1986,第88頁.
- ^ 3.0 3.1 Backus, Parker & Constable 1996,第178頁.
- ^ Backus, Parker & Constable 1996,第179頁.
参考资料
编辑- Hydrodynamic and hydromagnetic stability (页面存档备份,存于互联网档案馆), Chandrasekhar, Subrahmanyan; International Series of Monographs on Physics, Oxford: Clarendon, 1961, p. 622.
- Decomposition of solenoidal fields into poloidal fields, toroidal fields and the mean flow.[永久失效連結] Applications to the boussinesq-equations[永久失效連結], Schmitt, B. J. and von Wahl, W; in The Navier-Stokes Equations II — Theory and Numerical Methods, pp. 291–305; Lecture Notes in Mathematics, Springer Berlin/ Heidelberg, Vol. 1530/ 1992.
- Anelastic Magnetohydrodynamic Equations for Modeling Solar and Stellar Convection Zones (页面存档备份,存于互联网档案馆), Lantz, S. R. and Fan, Y.; The Astrophysical Journal Supplement Series, Volume 121, Issue 1, Mar. 1999, pp. 247–264.
- Plane poloidal-toroidal decomposition of doubly periodic vector fields: Part 1. (页面存档备份,存于互联网档案馆) Fields with divergence (页面存档备份,存于互联网档案馆) and Part 2. (页面存档备份,存于互联网档案馆) Stokes equations (页面存档备份,存于互联网档案馆). G. D. McBain. ANZIAM J. (页面存档备份,存于互联网档案馆) 47 (2005) (页面存档备份,存于互联网档案馆)
- Backus, George, Poloidal and toroidal fields in geomagnetic field modeling, Reviews in Geophysics, 1986, 24: 75–109, doi:10.1029/RG024i001p00075.
- Backus, George; Parker, Robert; Constable, Catherine, Foundations of Geomagnetism, Cambridge University Press, 1996, ISBN 0-521-41006-1.