翻訳と辞書 Words near each other ・ Graeme Fowler ・ Graeme Friedman ・ Graeme Frost ・ Graeme G. Cook ・ Graeme Gahan ・ Graeme Garden ・ Graeme Geddes ・ Gradyville, Kentucky ・ Gradyville, Pennsylvania ・ Gradzanowo Kościelne ・ Gradzanowo Włościańskie ・ Gradzanowo Zbęskie ・ Gradzanowo Zbęskie-Kolonia ・ Gradów ・ Gradówek ・ Grad–Shafranov equation ・ Grae Worster ・ Graea ・ Graeae ・ Graeae Theatre Company ・ Graeagle, California ・ Graebner ・ Graeco-Armenian ・ Graeco-Aryan ・ Graeco-Latin square ・ Graeco-Phrygian ・ Graeco-Roman Museum ・ Graeco-Roman paganism ・ Graecoanatolica ・ Graecoanatolica anatolica Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Grad–Shafranov equation ： ウィキペディア英語版 Grad–Shafranov equation The Grad–Shafranov equation (H. Grad and H. Rubin (1958); Vitalii Dmitrievich Shafranov (1966)) is the equilibrium equation in ideal magnetohydrodynamics (MHD) for a two dimensional plasma, for example the axisymmetric toroidal plasma in a tokamak. This equation is a two-dimensional, nonlinear, elliptic partial differential equation obtained from the reduction of the ideal MHD equations to two dimensions, often for the case of toroidal axisymmetry (the case relevant in a tokamak). The flux function $\backslash psi$ is both a dependent and an independent variable in this equation: : $\backslash Delta^\backslash psi\; =\; -\backslash mu\_R^\backslash frac-\backslash frac\backslash frac,$ where $\backslash mu\_0$ is the magnetic permeability, $p(\backslash psi)$ is the pressure, $F(\backslash psi)=RB\_$ and the magnetic field and current are, respectively, given by : $\backslash vec=\backslash frac\backslash nabla\backslash psi\backslash times\; \backslash hat\_+\backslash frac\backslash hat\_,$ : $\backslash mu\_0\backslash vec=\backslash frac\backslash frac\backslash nabla\backslash psi\backslash times\; \backslash hat\_-\backslash frac\backslash Delta^\backslash psi\; \backslash hat\_.$ The elliptic operator $\backslash Delta^$ is :$\backslash Delta^\backslash psi\; \backslash equiv\; R^\; \backslash vec\; \backslash cdot\; \backslash left(\; \backslash frac\; \backslash psi\; \backslash right)\; =\; R\backslash frac\backslash left(\backslash frac\backslash frac\backslash right)+\backslash frac$. The nature of the equilibrium, whether it be a tokamak, reversed field pinch, etc. is largely determined by the choices of the two functions $F(\backslash psi)$ and $p(\backslash psi)$ as well as the boundary conditions. == Derivation (in slab coordinates) == In the following, it is assumed that the system is 2-dimensional with $z$ as the invariant axis, i.e. $\backslash partial\; /\backslash partial\; z\; =\; 0$ for all quantities. Then the magnetic field can be written in cartesian coordinates as :$\backslash bold\; =\; (\backslash partial\; A/\backslash partial\; y,-\backslash partial\; A\; /\backslash partial\; x,B\_z(x,y)),$ or more compactly, :$\backslash bold\; =\backslash nabla\; A\; \backslash times\; \backslash hat\},$ where $A(x,y)\backslash hat\; \backslash times\; \backslash bold,$ where ''p'' is the plasma pressure and j is the electric current. It is known that ''p'' is a constant along any field line, (again since $\backslash nabla\; p$ is everywhere perpendicular to B). Additionally, the two-dimensional assumption ($\backslash partial\; /\; \backslash partial\; z\; =\; 0$) means that the z- component of the left hand side must be zero, so the z-component of the magnetic force on the right hand side must also be zero. This means that $\backslash bold\_\backslash perp\; \backslash times\; \backslash bold\_\backslash perp\; =\; 0$, i.e. $\backslash bold\_\backslash perp$ is parallel to $\backslash bold\_\backslash perp$. The right hand side of the previous equation can be considered in two parts: :$\backslash bold\; \backslash times\; \backslash bold\; =\; j\_z\; (\backslash hat)\; +\backslash bold\; \backslash times\; \backslash hat\_\backslash perp\; =\; \backslash nabla\; A\; \backslash times\; \backslash hat\_\backslash perp\; =\; (1/\backslash mu\_0)\backslash nabla\; B\_z\; \backslash times\; \backslash hat\_\backslash perp$ as required, the vector $\backslash nabla\; B\_z$ must be perpendicular to $\backslash bold\_\backslash perp$, and $B\_z$ must therefore, like $p$, be a field-line invariant. Rearranging the cross products above leads to :$\backslash hat\_\backslash perp\; =\; \backslash nabla\; A\; -(\backslash bold\; \backslash cdot\backslash nabla\; A)\; \backslash bold\; =\; \backslash nabla\; A$, and :$\backslash bold\_\backslash perp\; \backslash times\; B\_z\backslash bold\backslash cdot\backslash nabla\; B\_z)\backslash bold=-(1/\backslash mu\_0)\; B\_z\backslash nabla\; B\_z.$ These results can be substituted into the expression for $\backslash nabla\; p$ to yield: :$\backslash nabla\; p\; =\; -\backslash left(\backslash nabla^2\; A\backslash right)\backslash nabla\; A-(1/\backslash mu\_0)B\_z\backslash nabla\; B\_z.$ Since $p$ and $B\_z$ are constants along a field line, and functions only of $A$, hence $\backslash nabla\; p\; =\; (d\; p\; /dA)\backslash nabla\; A$ and $\backslash nabla\; B\_z\; =\; (d\; B\_z/dA)\backslash nabla\; A$. Thus, factoring out $\backslash nabla\; A$ and rearraging terms yields the Grad–Shafranov equation: : $\backslash nabla^2\; A\; =\; -\backslash mu\_0\; \backslash frac\backslash left(p\; +\; \backslash frac\backslash right).$ 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「Grad–Shafranov equation」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.