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Toroidal coordinates: Difference between revisions
→Magnetic field representation in flux coordinates
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=== Magnetic field representation in flux coordinates === | === Magnetic field representation in flux coordinates === | ||
Any magnetic field <math>\mathbf{B}</math> can be written as | |||
:<math> | <math> \mathbf{B} = \nabla\alpha\times\nabla\nu </math> | ||
\mathbf{B} = \nabla\psi\times\nabla\nu | called its Clebsch representation. For a magnetic field with flux surfaces <math>(\psi = \mathrm{const}\; , \; \nabla\psi\cdot\mathbf{B} = 0)</math> we can choose, say, <math>\alpha</math> to be the flux surface label <math>\psi</math> | ||
</math> | :<math> | ||
\mathbf{B} = \nabla\psi\times\nabla\nu | |||
</math> | |||
Field lines are then given as the intersection of the constant-<math>\psi</math> and constant-<math>\nu</math> surfaces. This form provides a general expression for <math>\mathbf{B}</math> in terms of the covariant basis vectors of a flux coordinate system | |||
:<math> | :<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~. | \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> | </math> | ||
in terms of the function <math>\nu</math>, sometimes referred to as the magnetic field's ''stream function''. | |||
== Magnetic == | == Magnetic == | ||