Flux coordinates: Difference between revisions

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Taking the curl of the covariant form of <math>\mathbf{B}</math> the equilibrium current density <math>\mathbf{j}</math> can be written as
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>
: <math>
\mathbf{j} = \nabla\psi\times\nabla\eta~.
\mu_0\mathbf{j} = \nabla\psi\times\nabla\eta~.
</math>
</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
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
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</math>
</math>
where the integral is performed along the magnetic axis.
where the integral is performed along the magnetic axis.
According to our definitions, <math>\beta</math> in the covariant form of the magnetic field relates to the current stream function <math>\eta</math> as <math>\beta = -\mu_0\eta</math>


== Magnetic coordinates ==
== Magnetic coordinates ==
204

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