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Flux coordinates: Difference between revisions
→Useful properties of FSA
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==== Useful properties of FSA ==== | ==== 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> \langle \mathbf{B}\cdot\nabla f \rangle = \langle \nabla\cdot(\mathbf{B} f) \rangle = 0~,\qquad \forall~ \mathrm{single~valued~} f(\mathbf{x}), ~ \mathrm{if}~ \nabla\cdot\mathbf{B} = 0 ~\mathrm{and}~ \nabla \psi\cdot\mathbf{B} = 0 </math> | |||
The two identities above are the basic simplifying properties of the FSA: The first cancels the contribution of 'conservative forces' like the pressure gradient or electrostatic electric fields. The second reduces the number of spatial variables to only the radial one. Further, it is possible to show that, if <math>\nabla\cdot\Gamma = 0</math> then <math>\langle\Gamma\cdot\nabla V\rangle = 0</math> and not simply constant as the above suggest. This can be seen by simply using Gauss' theorem | |||
*<math> \int_{\mathcal{V}}\nabla\cdot\Gamma\; d\mathcal{V} = \langle\Gamma\cdot\nabla V\rangle = V'\langle\Gamma\cdot\nabla \psi\rangle \qquad \mathrm{where~} \mathcal{V} \mathrm{~is~the~volume~enclosed~by~a~flux~surface.} | *<math> \int_{\mathcal{V}}\nabla\cdot\Gamma\; d\mathcal{V} = \langle\Gamma\cdot\nabla V\rangle = V'\langle\Gamma\cdot\nabla \psi\rangle \qquad \mathrm{where~} \mathcal{V} \mathrm{~is~the~volume~enclosed~by~a~flux~surface.} | ||
</math> | |||
The FSA relates to the conventional volume integral as | |||
*<math> \int_{\mathcal{V}(V_1<V<V_2)} s\; d\mathcal{V} = \int_{V_1}^{V_2} \langle s \rangle\; dV | |||
</math> | |||
whereas the conventional surface integral over a <math>\psi = constant</math> is | |||
*<math> \int_{\psi = \psi_0} s\; d\mathcal{S} = \langle s |\nabla V| \rangle | |||
</math> | </math> | ||
Other useful properties are | |||
*<math> \langle \nabla \psi\cdot\nabla\times \mathbf{A} \rangle = -\langle \nabla\cdot( \nabla\psi\times\mathbf{A}) \rangle = 0~. | *<math> \langle \nabla \psi\cdot\nabla\times \mathbf{A} \rangle = -\langle \nabla\cdot( \nabla\psi\times\mathbf{A}) \rangle = 0~. | ||
</math> | </math> | ||