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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Alfvén's Theorem ： ウィキペディア英語版 Alfvén's Theorem In magnetohydrodynamics, the Alfvén's theorem, also known as Alfvén's frozen in theorem, states that in a fluid with infinite electric conductivity, magnetic field lines are frozen into the fluid and have to move along with it. Hannes Alfvén put the idea forward for the first time in 1942. In his own words: "In view of the infinite conductivity, every motion (perpendicular to the field) of the liquid in relation to the lines of force is forbidden because it would give infinite eddy currents. Thus the matter of the liquid is “fastened” to the lines of force...". In most astrophysical environments, as well as laboratory plasmas, the electric conductivity is not infinite, so the magnetic field lines are not ideally frozen into the fluid. However, with a high electric conductivity, or equivalently a small resistivity, the frozen in theorem can be approximately applied. This is called the frozen flux approximation which is widely used in dynamo theory. ==Mathematical statement== In a fluid with infinite electric conductivity, the change of magnetic flux over time can be written as $=\; \backslash int\_S\; \backslash cdot\; d\backslash vec\; +\; \backslash oint\_C\; \backslash vec\; \backslash cdot\; \backslash vec\; \backslash times\; d\; \backslash vec,$ where $\backslash vec$ and $\backslash vec$ are the magnetic and velocity fields respectively. Here, $\backslash vec$ is the surface enclosed by the curve $C$ with differential line element $d\backslash vec$. Using the induction equation $=\; \backslash vec\; \backslash times\; (\backslash vec\; \backslash times\; \backslash vec\; ),$ leads to $=\; \backslash int\_S\; \backslash vec\; \backslash times\; (\backslash vec\; \backslash times\; \backslash vec\; )\; \backslash cdot\; d\backslash vec\; +\; \backslash oint\_C\; \backslash vec\; \backslash cdot\; \backslash vec\; \backslash times\; d\; \backslash vec.$ These two integrals can be re-written using Stokes' theorem for the first one, and the vector identity $(\backslash vec\; \backslash times\; \backslash vec)\; \backslash cdot\; \backslash vec\; =\; -\backslash vec\; \backslash cdot\; (\backslash vec\; \backslash times\; \backslash vec)$ for the second one. The result is $\backslash int\_S\; \backslash vec\; \backslash cdot\; d\backslash vec\; =\; const.$ This is the mathematical form of the ''Alfvén's theorem'': the magnetic flux passing through a surface moving along with the fluid is conserved. This means that the plasma can move along with the local field lines. For the perpendicular motions of the fluid, the field lines will push the fluid or otherwise they will be dragged with the fluid. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Alfvén's Theorem」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.