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Toroidal coordinates: Difference between revisions
→Flux Surface Average
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The flux surface average of a function <math>\Phi</math> is defined as the limit | The flux surface average of a function <math>\Phi</math> is defined as the limit | ||
:<math> | :<math> | ||
\langle\Phi\rangle = \lim_{\delta V \to 0}\frac{1}{\delta V}\int_{\delta V} \Phi\; | \langle\Phi\rangle = \lim_{\delta V \to 0}\frac{1}{\delta V}\int_{\delta V} \Phi\; d^3x | ||
</math> | </math> | ||
where <math>\delta V</math> is the volume confined between two flux surfaces. It is therefore a ''volume average'' over an infinitesimal spatial region rather than a surface average. | where <math>\delta V</math> is the volume confined between two flux surfaces. It is therefore a ''volume average'' over an infinitesimal spatial region rather than a surface average. | ||
Introducing the differential volume element <math> | Introducing the differential volume element <math>d^3x = \sqrt{g} d\psi d\theta d\phi</math> | ||
:<math> | :<math> | ||
\langle\Phi\rangle | \langle\Phi\rangle | ||
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:<math> | :<math> | ||
\langle\nabla\cdot\Gamma\rangle | \langle\nabla\cdot\Gamma\rangle | ||
= \lim_{\delta V \to 0}\frac{1}{\delta V}\int_{\delta V} \nabla\cdot\Gamma\; | = \lim_{\delta V \to 0}\frac{1}{\delta V}\int_{\delta V} \nabla\cdot\Gamma\; d^3x | ||
= \lim_{\delta V \to 0}\frac{1}{\delta V}\int_{S(\delta V)} \Gamma\cdot \frac{\nabla V}{|\nabla V|}dS | = \lim_{\delta V \to 0}\frac{1}{\delta V}\int_{S(\delta V)} \Gamma\cdot \frac{\nabla V}{|\nabla V|}dS | ||
= \lim_{\delta V \to 0}\left(\langle\Gamma\cdot\nabla V\rangle_{S(V+\delta V)} - \langle\Gamma\cdot\nabla V\rangle_{S(V)} \right) | |||
= \frac{d}{dV}\langle\Gamma\cdot\nabla V\rangle~. | = \frac{d}{dV}\langle\Gamma\cdot\nabla V\rangle~. | ||
</math> | </math> | ||
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*<math> \langle\nabla\cdot\Gamma\rangle = \frac{d}{dV}\langle\Gamma\cdot\nabla V\rangle = \frac{1}{V'}\frac{d}{d\psi}V'\langle\Gamma\cdot\nabla \psi\rangle</math> | *<math> \langle\nabla\cdot\Gamma\rangle = \frac{d}{dV}\langle\Gamma\cdot\nabla V\rangle = \frac{1}{V'}\frac{d}{d\psi}V'\langle\Gamma\cdot\nabla \psi\rangle</math> | ||
*<math> \int_{V}\nabla\cdot\Gamma\; | *<math> \int_{V}\nabla\cdot\Gamma\; d^3x = \langle\Gamma\cdot\nabla V\rangle = V'\langle\Gamma\cdot\nabla \psi\rangle</math> | ||
*<math> \langle \sqrt{g}^{-1}\rangle = \frac{4\pi^2}{V'} | *<math> \langle \sqrt{g}^{-1}\rangle = \frac{4\pi^2}{V'} | ||
</math> | </math> | ||