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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\; d | \langle\Phi\rangle = \lim_{\delta \mathcal{V} \to 0}\frac{1}{\delta \mathcal{V}}\int_{\delta \mathcal{V}} \Phi\; d\mathcal{V} | ||
</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>d | Introducing the differential volume element <math>d\mathcal{V} = \sqrt{g} d\psi d\theta d\phi</math> | ||
:<math> | :<math> | ||
\langle\Phi\rangle | \langle\Phi\rangle | ||
= \lim_{\delta V \to 0} \frac{1}{\delta V}\int_{\delta V} \Phi\; \sqrt{g} d\psi d\theta d\phi | = \lim_{\delta \mathcal{V} \to 0} \frac{1}{\delta \mathcal{V}}\int_{\delta \mathcal{V}} \Phi\; \sqrt{g} d\psi d\theta d\phi | ||
= \frac{d\psi}{d V}\int_0^{2\pi}\int_0^{2\pi}\Phi\; \sqrt{g} d\theta d\phi | = \frac{d\psi}{d V}\int_0^{2\pi}\int_0^{2\pi}\Phi\; \sqrt{g} d\theta d\phi | ||
</math> | </math> | ||
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</math> | </math> | ||
Note that <math> | Note that <math>d\mathcal{S} = |\nabla\psi|\sqrt{g}d\theta d\phi</math>, so the FSA is a surface integral ''weighted by'' <math>|\nabla V|^{-1}</math> : | ||
:<math> | :<math> | ||
\langle\Phi\rangle | \langle\Phi\rangle | ||
= \frac{d\psi}{d V}\int_0^{2\pi}\int_0^{2\pi}\Phi\; \sqrt{g} d\theta d\phi | = \frac{d\psi}{d V}\int_0^{2\pi}\int_0^{2\pi}\Phi\; \sqrt{g} d\theta d\phi | ||
= \frac{d\psi}{d V}\int_{S(\psi)}\frac{\Phi}{|\nabla\psi|}\; | = \frac{d\psi}{d V}\int_{S(\psi)}\frac{\Phi}{|\nabla\psi|}\; d\mathcal{S} | ||
= \int_{S(\psi)}\frac{\Phi}{|\nabla V|}\; | = \int_{S(\psi)}\frac{\Phi}{|\nabla V|}\; d\mathcal{S} | ||
</math> | </math> | ||
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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\; d | = \lim_{\delta \mathcal{V} \to 0}\frac{1}{\delta \mathcal{V}}\int_{\delta \mathcal{V}} \nabla\cdot\Gamma\; d\mathcal{V} | ||
= \lim_{\delta V \to 0}\frac{1}{\delta V}\int_{S(\delta V)} \Gamma\cdot \frac{\nabla V}{|\nabla V|} | = \lim_{\delta \mathcal{V} \to 0}\frac{1}{\delta \mathcal{V}}\int_{S(\delta \mathcal{V})} \Gamma\cdot \frac{\nabla V}{|\nabla V|}d\mathcal{S} | ||
= \lim_{\delta V \to 0}\left(\langle\Gamma\cdot\nabla V\rangle_{ | = \lim_{\delta \mathcal{V} \to 0}\frac{1}{\delta \mathcal{V}}\left(\langle\Gamma\cdot\nabla V\rangle_{(V+\delta \mathcal{V})} - \langle\Gamma\cdot\nabla V\rangle_{(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\; d | *<math> \int_{\mathcal{V}}\nabla\cdot\Gamma\; d\mathcal{V} = \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> | ||