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Flux coordinates: Difference between revisions
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→Contravariant Form of the current density
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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 == | ||