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Toroidal coordinates: Difference between revisions
→Flux Coordinates
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:<math> \int_{V}\nabla\cdot\Gamma\; dV = \langle\Gamma\cdot\nabla V\rangle = 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> | :<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 field representation in flux coordinates === | |||
The choice of a system of flux coordinates allows to express the (nested flux-surfaces forming) magnetic field <math>\mathbf{B}</math> as | |||
:<math> | |||
\mathbf{B} = \nabla\psi\times\nabla\nu | |||
</math> | |||
called the Clebsch representation. It is also an expression of <math>\mathbf{B}</math> in terms of the covariant basis vectors | |||
:<math> | |||
\mathbf{B} = \frac{\partial\nu}{\partial\theta}\nabla\psi\times\nabla\theta + \frac{\partial\nu}{\partial\phi}\nabla\psi\times\nabla\phi = \frac{1}{\sqrt{g}}\frac{\partial\nu}{\partial\theta}\mathbf{e}_\phi -\frac{1}{\sqrt{g}}\frac{\partial\nu}{\partial\phi}\mathbf{e}_\theta = B^\phi\mathbf{e}_\phi + B^\theta\mathbf{e}_\theta~. | |||
</math> | |||
Field lines are then given as the intersection of the constant-<math>\psi</math> and constant-<math>\nu</math> surfaces. The function <math>\nu</math> is sometimes referred to as the magnetic field's ''stream function''. | |||
== Magnetic == | == Magnetic == | ||