极向–环向分解

Template:NoteTA 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。考虑一个三维向量场F满足

$\mathrm{\nabla }\cdot 𝐅=0,$

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：

$𝐅=𝐓+𝐏=\mathrm{\nabla }\times \mathrm{\Psi }𝐫+\mathrm{\nabla }\times (\mathrm{\nabla }\times \mathrm{\Phi }𝐫),$

其中$𝐫$是球坐标$(r,\theta ,\varphi )$中的径向矢量，纵场$𝐓$为

$𝐓=\mathrm{\nabla }\times \mathrm{\Psi }𝐫$

$\mathrm{\Psi }(r,\theta ,\varphi )$为一标量场，[2]横场$𝐏$为

$𝐏=\mathrm{\nabla }\times \mathrm{\nabla }\times \mathrm{\Phi }𝐫$

$\mathrm{\Phi }(r,\theta ,\varphi )$为一标量场。Template:Sfn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。Template:Sfn纵场与球心在原点的球面相切

$𝐫\cdot 𝐓=0$,Template:Sfn

而横场的旋度同样地与这些球面相切

$𝐫\cdot (\mathrm{\nabla }\times 𝐏)=0$.Template:Sfn

若标量场$\mathrm{\Psi }$和$\mathrm{\Phi }$的平均值在任意半径为$r$的球面上都等于零，则这一分解方式是唯一的。Template:Sfn

另见

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参考资料

• Hydrodynamic and hydromagnetic stability Template:Wayback, 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.Template:Dead link Applications to the boussinesq-equationsTemplate:Dead link, 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 Template:Wayback, 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. Template:Wayback Fields with divergence Template:Wayback and Part 2. Template:Wayback Stokes equations Template:Wayback. G. D. McBain. ANZIAM J. Template:Wayback 47 (2005) Template:Wayback
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