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Toroidal coordinates: Difference between revisions
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→Flux Coordinates
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== Flux Coordinates == | == 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. | 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. | ||
Different flux surface labels can be chosen like toroidal or poloidal magnetic fluxes or the volume contained within the flux surface. | |||
=== Flux Surface Average === | === Flux Surface Average === | ||
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</math></center> | </math></center> | ||
==== Useful properties of FSA ==== | |||
Some useful properties of the FSA are | Some useful properties of the FSA are | ||
:<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\; dV = \langle\Gamma\cdot\nabla V\rangle = V'\langle\Gamma\cdot\nabla \psi\rangle</math> | |||
:<math> \langle \mathbf{B}\cdot\nabla f \rangle = 0~,\qquad \forall~ \mathrm{single~valued~} f, ~ \mathrm{if}~ \nabla\cdot\mathbf{B} = 0 ~\mathrm{and}~ \nabla V\cdot\mathbf{B} = 0 </math> | |||
== Magnetic == | == Magnetic == | ||