Toroidal coordinates: Difference between revisions

Toroidal coordinates (view source)

825 bytes added , 18 August 2010

204

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 :<math>
 \mathbf{B} =  \nabla\psi\times \left( \frac{\Psi_{tor}'}{2\pi}\theta_f
-		- \frac{\Psi_{pol}'}{2\pi}\phi_f \right)
+		- \frac{\Psi_{pol}'}{2\pi}\phi_f \right)~.
 </math>
+Note that, in general, the contravariant components of the magnetic field in a magnetic coordinate system
+:<math>
+B^{\theta_f} = \frac{\Psi_{pol}'}{2\pi\sqrt{g}}\; ;\quad B^{\phi_f} = \frac{\Psi_{tor}'}{2\pi\sqrt{g}}
+</math>
+are not flux functions, but their quotient is
+:<math>
+\frac{B^{\theta_f}}{B^{\phi_f}} = \frac{\Psi_{pol}'}{\Psi_{tor}'} \equiv \frac{\iota}{2\pi}~,
+</math>
+<math>\iota</math> being the [[rotational transform]].
+Any transformation of the angles of the from
+:<math>
+		\theta_F = \theta_f +\Psi_{pol}' G(\psi, \theta_f, \phi_f)\; ;\quad \phi_F = \phi_f +\Psi_{tor}' G(\psi, \theta_f, \phi_f)
+</math>
+where <math>G</math> is periodic in the angles, preserves the straightness of the field lines. The spatial function <math>G(\psi, \theta_f, \phi_f)</math>, is called the ''generating function''.
 Magnetic coordinates adapt to the magnetic field, and therefore to the [[MHD equilibrium]] (also see [[Flux surface]]).
 Magnetic coordinates simplify the description of the magnetic field.