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Toroidal coordinates: Difference between revisions
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→Covariant Form
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==== Covariant Form ==== | ==== 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> | :<math> | ||
\mathbf{B} = \beta\nabla\psi + \nabla\chi | \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)~. | 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> | </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 == | == Magnetic coordinates == | ||