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Toroidal coordinates: Difference between revisions
→Flux Surface Average
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\langle\Phi\rangle = \lim_{\delta \mathcal{V} \to 0}\frac{1}{\delta \mathcal{V}}\int_{\delta \mathcal{V}} \Phi\; d\mathcal{V} | \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. To avoid confusion, we denote volume elements or domains with the calligraphic <math>\mathcal{V}</math>. Capital <math>V</math> is reserved for the flux surface coordinate defined as the volume within a flux surface. | ||
Introducing the differential volume element <math>d\mathcal{V} = \sqrt{g} d\psi d\theta d\phi</math> | Introducing the differential volume element <math>d\mathcal{V} = \sqrt{g} d\psi d\theta d\phi</math> | ||
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</math> | </math> | ||
Note that <math> | Note that <math>dS = |\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|}\; dS | ||
= \int_{S(\psi)}\frac{\Phi}{|\nabla V|}\; | = \int_{S(\psi)}\frac{\Phi}{|\nabla V|}\; dS | ||
</math> | </math> | ||
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\langle\nabla\cdot\Gamma\rangle | \langle\nabla\cdot\Gamma\rangle | ||
= \lim_{\delta \mathcal{V} \to 0}\frac{1}{\delta \mathcal{V}}\int_{\delta \mathcal{V}} \nabla\cdot\Gamma\; d\mathcal{V} | = \lim_{\delta \mathcal{V} \to 0}\frac{1}{\delta \mathcal{V}}\int_{\delta \mathcal{V}} \nabla\cdot\Gamma\; d\mathcal{V} | ||
= \lim_{\delta \mathcal{V} \to 0}\frac{1}{\delta \mathcal{V}}\int_{S(\delta \mathcal{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|}dS | ||
= \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) | = \lim_{\delta \mathcal{V} \to 0}\frac{1}{\delta \mathcal{V}}\left(\langle\Gamma\cdot\nabla V\rangle_{S(V+\delta \mathcal{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> | ||