查看“︁极向–环向分解”︁的源代码

{{NoteTA
|G1=Physics
|G2=Math
}}
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[[不可壓縮流|不可压缩流体]]等。<ref>{{Cite book|url=http://cdsads.u-strasbg.fr/abs/1961hhs..book.....C|title=Hydrodynamic and hydromagnetic stability|publisher=Oxford: Clarendon|year=1961|series=International Series of Monographs on Physics|at=See discussion on page 622|author=Subrahmanyan Chandrasekhar|access-date=2016-04-29|archive-date=2012-02-12|archive-url=https://web.archive.org/web/20120212213109/http://cdsads.u-strasbg.fr/abs/1961hhs..book.....C}}</ref>考虑一个三维向量场'''F'''满足
: <math> \nabla \cdot \mathbf{F} = 0, </math>
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：
: <math>\mathbf{F} = \mathbf{T} + \mathbf{P} = \nabla \times \Psi \mathbf{r} + \nabla \times (\nabla \times \Phi \mathbf{r}), </math>
其中<math> \mathbf{r} </math>是球坐标<math> (r,\theta,\phi) </math>中的径向矢量，纵场<math> \mathbf{T} </math>为
: <math> \mathbf{T} = \nabla \times \Psi \mathbf{r} </math> 
<math> \Psi (r,\theta,\phi)</math>为一标量场，<span class="mw-ref" id="cxcite_ref-FOOTNOTEBackus198687_2-0" rel="dc:references" contenteditable="false" data-sourceid="cite_ref-FOOTNOTEBackus198687_2-0">[[#cite_note-FOOTNOTEBackus198687-2|<span class="mw-reflink-text"><nowiki>[2]</nowiki></span>]]</span><span class="mw-ref" id="cxcite_ref-FOOTNOTEBackus198687_2-0" rel="dc:references" contenteditable="false" data-sourceid="cite_ref-FOOTNOTEBackus198687_2-0"></span>横场<math> \mathbf{P} </math>为 
: <math> \mathbf{P} = \nabla \times \nabla \times \Phi \mathbf{r} </math>
<math> \Phi (r,\theta,\phi)</math>为一标量场。{{Sfn|Backus|1986|p=88}}<span class="mw-ref" id="cxcite_ref-FOOTNOTEBackus198688_3-0" rel="dc:references" contenteditable="false" data-sourceid="cite_ref-FOOTNOTEBackus198688_3-0"></span>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。{{Sfn|Backus|Parker|Constable|1996|p=178}}<span class="mw-ref" id="cxcite_ref-FOOTNOTEBackusParkerConstable1996178_4-0" rel="dc:references" contenteditable="false" data-sourceid="cite_ref-FOOTNOTEBackusParkerConstable1996178_4-0"></span>纵场与球心在原点的球面相切
: <math> \mathbf{r} \cdot \mathbf{T} = 0 </math>,{{Sfn|Backus|Parker|Constable|1996|p=178}}<span class="mw-ref" id="cite_ref-FOOTNOTEBackusParkerConstable1996178_4-1" rel="dc:references" contenteditable="false"></span>
而横场的旋度同样地与这些球面相切
: <math> \mathbf{r} \cdot (\nabla \times \mathbf{P}) = 0 </math>.{{Sfn|Backus|Parker|Constable|1996|p=179}}<span class="mw-ref" id="cite_ref-FOOTNOTEBackusParkerConstable1996179_5-0" rel="dc:references" contenteditable="false"></span>
若标量场<math> \Psi </math>和<math> \Phi </math>的平均值在任意半径为<math> r </math>的球面上都等于零，则这一分解方式是唯一的。{{Sfn|Backus|1986|p=88}}<span class="mw-ref" id="cite_ref-FOOTNOTEBackus198688_3-1" rel="dc:references" contenteditable="false"></span>

== 另见 ==
* {{tsl|en|Toroidal and poloidal|}}

== 脚注 ==
{{Reflist}}

== 参考资料 ==
* [http://cdsads.u-strasbg.fr/abs/1961hhs..book.....C ''Hydrodynamic and hydromagnetic stability''] {{Wayback|url=http://cdsads.u-strasbg.fr/abs/1961hhs..book.....C |date=20120212213109 }}, Chandrasekhar, Subrahmanyan; International Series of Monographs on Physics, Oxford: Clarendon, 1961, p.&nbsp;622.
* [http://www.springerlink.com/content/h584m7h23v1k5428/ Decomposition of solenoidal fields into poloidal fields, toroidal fields and the mean flow.]{{Dead link|date=2020年2月 |bot=InternetArchiveBot |fix-attempted=yes }} [http://www.springerlink.com/content/h584m7h23v1k5428/ Applications to the boussinesq-equations]{{Dead link|date=2020年2月 |bot=InternetArchiveBot |fix-attempted=yes }}, Schmitt, B. J. and von Wahl, W; in ''The Navier-Stokes Equations II — Theory and Numerical Methods'', pp.&nbsp;291&#x2013;305; Lecture Notes in Mathematics, Springer Berlin/ Heidelberg, Vol. 1530/ 1992.
* [http://cdsads.u-strasbg.fr/abs/1999ApJS..121..247L Anelastic Magnetohydrodynamic Equations for Modeling Solar and Stellar Convection Zones] {{Wayback|url=http://cdsads.u-strasbg.fr/abs/1999ApJS..121..247L |date=20120212213114 }}, Lantz, S. R. and Fan, Y.; The Astrophysical Journal Supplement Series, Volume 121, Issue 1, Mar. 1999, pp.&nbsp;247&#x2013;264.
* Plane poloidal-toroidal decomposition of doubly periodic vector fields: [http://www.austms.org.au/Publ/Jamsb/V47P1/2148.html Part 1.] {{Wayback|url=http://www.austms.org.au/Publ/Jamsb/V47P1/2148.html |date=20200720135029 }} [http://www.austms.org.au/Publ/Jamsb/V47P1/2148.html Fields with divergence] {{Wayback|url=http://www.austms.org.au/Publ/Jamsb/V47P1/2148.html |date=20200720135029 }} and [http://www.austms.org.au/Publ/Jamsb/V47P1/2203.html Part 2.] {{Wayback|url=http://www.austms.org.au/Publ/Jamsb/V47P1/2203.html |date=20200722224633 }} [http://www.austms.org.au/Publ/Jamsb/V47P1/2203.html Stokes equations] {{Wayback|url=http://www.austms.org.au/Publ/Jamsb/V47P1/2203.html |date=20200722224633 }}. G. D. McBain. [http://www.austms.org.au/Publ/ANZIAM/index.shtml ANZIAM  J.] {{Wayback|url=http://www.austms.org.au/Publ/ANZIAM/index.shtml |date=20080103180107 }} [http://www.austms.org.au/Publ/ANZIAM/V47P1/contents.html 47 (2005)] {{Wayback|url=http://www.austms.org.au/Publ/ANZIAM/V47P1/contents.html |date=20200711052442 }}
* {{Citation|first=George|last=Backus|title=Poloidal and toroidal fields in geomagnetic field modeling|journal=Reviews in Geophysics|volume=24|year=1986|pages=75–109|doi=10.1029/RG024i001p00075}}.
* {{Citation|first1=George|last1=Backus|first2=Robert|last2=Parker|first3=Catherine|last3=Constable|title=Foundations of Geomagnetism|publisher=Cambridge University Press|year=1996|isbn=0-521-41006-1}}.
[[Category:向量分析]]