Toroidal coordinates: Difference between revisions

=== Flux Surface Average ===

The flux surface average of a function <math>\Phi</math> is defined as the limit

\langle\Phi\rangle = \lim_{\delta V \to 0}\frac{1}{\delta V}\int_{\delta V} \Phi\; dV  

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>dV = \sqrt{g} d\psi d\theta d\phi</math>

\langle\Phi\rangle  

= \lim_{\delta V \to 0} \frac{1}{\delta V}\int_{\delta 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

or, noting that <math>\langle 1\rangle = 1</math>, we have <math>\frac{dV}{d\psi} = \int_0^{2\pi}\int_0^{2\pi} \sqrt{g} d\theta d\phi</math> and  

we get to a more practical form of the Flux Surface Average

\langle\Phi\rangle  

= \frac{\int_0^{2\pi}\int_0^{2\pi}\Phi\; \sqrt{g} d\theta d\phi}

{\int_0^{2\pi}\int_0^{2\pi} \sqrt{g} d\theta d\phi}  

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> :

\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_{S(\psi)}\frac{\Phi}{|\nabla\psi|}\; dS

= \int_{S(\psi)}\frac{\Phi}{|\nabla V|}\; dS

Applying Gauss' theorem to the definition of FSA we get to the identity

\langle\nabla\cdot\Gamma\rangle  

= \lim_{\delta V \to 0}\frac{1}{\delta V}\int_{\delta V} \nabla\cdot\Gamma\; dV

= \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_0^{2\pi} \int_0^{2\pi} \Gamma\cdot \nabla V\; \sqrt{g} d\theta d\phi = \frac{d}{dV}\langle\Gamma\cdot\nabla V\rangle~.

==== Useful properties of FSA ====