204
edits
Toroidal coordinates: Difference between revisions
→General Curvilinear Coordinates
| Line 106: | Line 106: | ||
</math></center> | </math></center> | ||
It can be seen that<ref>W.D. D'haeseleer, ''Flux coordinates and magnetic field structure: a guide to a fundamental tool of plasma theory'', Springer series in computational physics, Springer-Verlag (1991) ISBN 3540524193</ref> <math>g \equiv \det(g_{ij}) = J^2 \Rightarrow J = \sqrt{g}</math> | It can be seen that<ref>W.D. D'haeseleer, ''Flux coordinates and magnetic field structure: a guide to a fundamental tool of plasma theory'', Springer series in computational physics, Springer-Verlag (1991) ISBN 3540524193</ref> <math>g \equiv \det(g_{ij}) = J^2 \Rightarrow J = \sqrt{g}</math> | ||
== Flux Coordinates == | |||
A flux coordinate set is one that includes a flux surface label as a coordinate. A flux surface label is a function that is constant and single valued on each flux surface. In our naming of the general curvilinear coordinates we have already adopted the usual flux coordinate convention for toroidal equilibrium with nested flux surfaces with <math>\psi</math> being the flux surface label and <math>\theta, \phi</math> are <math>2\pi</math>-periodic poloidal and toroidal-like angles. | |||
=== Flux Surface Average === | |||
The flux surface average of a function <math>\Phi</math> is defined as the limit | |||
<center><math> | |||
\langle\Phi\rangle = \lim_{\delta V \to 0}\frac{1}{\delta V}\int_{\delta V} \Phi\; dV | |||
</math></center> | |||
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> | |||
<center><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 | |||
</math></center> | |||
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 | |||
<center><math> | |||
\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} | |||
</math></center> | |||
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> : | |||
<center><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 | |||
</math></center> | |||
Applying Gauss' theorem to the definition of FSA we get to the identity | |||
<center><math> | |||
\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~. | |||
</math></center> | |||
Some useful properties of the FSA are | |||
== Magnetic == | == Magnetic == | ||