Toroidal coordinates: Difference between revisions

Toroidal coordinates (view source)

Revision as of 15:53, 18 August 2010

783 bytes added , 18 August 2010

→‎Covariant Form

Arturo

204

edits

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 ==== Covariant Form ====
-If we consider an equilibrium magnetic field such that <math> \mathbf{j}\times\mathbf{B} \propto \nabla\psi</math>, then both <math> \mathbf{B}\cdot\nabla\psi = 0</math> and <math> \nabla\times\mathbf{B}\cdot\nabla\psi = 0</math> and the magnetic field can be written as
+If we consider an equilibrium magnetic field such that <math> \mathbf{j}\times\mathbf{B} \propto \nabla\psi</math>, where <math> \mathbf{j}</math> is the current density , then both <math> \mathbf{B}\cdot\nabla\psi = 0</math> and <math> \nabla\times\mathbf{B}\cdot\nabla\psi = 0</math> and the magnetic field can be written as
 :<math>
 \mathbf{B} = \beta\nabla\psi + \nabla\chi
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 I_{tor} = \int_S \mu_0\mathbf{j}\cdot d\mathbf{S}\; ; ~\mathrm{with}~ \partial S ~\mathrm{such~that}~ (\Delta\theta = 2\pi, \Delta\phi = 0)~.
 </math>
+===== Contravariant Form of the current density =====
+Taking the curl of the covariant form of <math>\mathbf{B}</math> the equilibrium current density <math>\mathbf{j}</math> can be written as
+: <math>
+\mathbf{j} = \nabla\psi\times\nabla\eta~.
+</math>
+By very similar arguments as those used for <math>\mathbf{B}</math> (note that both <math>\mathbf{B}</math> and <math>\mathbf{j}</math> are solenoidal fields tangent to the flux surfaces) it can be shown that the general expression for <math>\eta</math> is
+:<math>
+\eta(\psi,\theta,\phi) = \frac{1}{2\pi}(\dot{I}_{tor}\theta
+		- \dot{I}_{pol}\phi)
+		+ \tilde{\eta}(\psi,\theta,\phi)~.
+</math>
+Note that the poloidal current is now defined through a ribbon and not a disc.
 == Magnetic coordinates ==