Functions/Subroutines
subroutine	curent (lvol, mn, Nt, Nz, iflag, ldItGp)
	Computes the plasma current, \(I \equiv \int B_\theta \, d\theta\), and the "linking" current, \(G \equiv \int B_\zeta \, d\zeta\). More...

Detailed Description

Function/Subroutine Documentation

Computes the plasma current, \(I \equiv \int B_\theta \, d\theta\), and the "linking" current, \(G \equiv \int B_\zeta \, d\zeta\).

enclosed currents

In the vacuum region, the enclosed currents are given by either surface integrals of the current density or line integrals of the magnetic field,
\begin{eqnarray} \int_{\cal S} {\bf j}\cdot d{\bf s} = \int_{\partial \cal S} {\bf B}\cdot d{\bf l}, \end{eqnarray}
and line integrals are usually easier to compute than surface integrals.
The magnetic field is given by the curl of the magnetic vector potential, as described in e.g. bfield() .
The toroidal, plasma current is obtained by taking a "poloidal" loop, \(d{\bf l}={\bf e}_\theta \, d\theta\), on the plasma boundary, where \(B^s=0\), to obtain
\begin{eqnarray} I \equiv \int_{0}^{2\pi} \! {\bf B} \cdot {\bf e}_\theta \, d\theta = \int_{0}^{2\pi} \! \left( - \partial_s A_\zeta \; \bar g_{\theta\theta} + \partial_s A_\theta \; \bar g_{\theta\zeta} \right) \, d\theta, \end{eqnarray}
where \(\bar g_{\mu\nu} \equiv g_{\mu\nu} / \sqrt g\).
The poloidal, "linking" current through the torus is obtained by taking a "toroidal" loop, \(d{\bf l}={\bf e}_\zeta \, d\zeta\), on the plasma boundary to obtain
\begin{eqnarray} G \equiv \int_{0}^{2\pi} \! {\bf B}\cdot {\bf e}_\zeta \, d\zeta = \int_{0}^{2\pi} \! \left( - \partial_s A_\zeta \; \bar g_{\theta\zeta} + \partial_s A_\theta \; \bar g_{\zeta\zeta} \right) \, d\zeta. \end{eqnarray}

Fourier integration

Using \(f \equiv - \partial_s A_\zeta \; \bar g_{\theta\theta} + \partial_s A_\theta \; \bar g_{\theta\zeta}\), the integral for the plasma current is
\begin{eqnarray} I = {\sum_i}^\prime f_i \cos(n_i \zeta) 2\pi, \label{eq:plasmacurrent_curent} \end{eqnarray}
where \(\sum^\prime\) includes only the \(m_i=0\) harmonics.
Using \(g \equiv - \partial_s A_\zeta \; \bar g_{\theta\zeta} + \partial_s A_\theta \; \bar g_{\zeta\zeta}\), the integral for the linking current is
\begin{eqnarray} G = {\sum_i}^\prime g_i \cos(m_i \zeta) 2\pi, \label{eq:linkingcurrent_curent} \end{eqnarray}
where \(\sum^\prime\) includes only the \(n_i=0\) harmonics.
The plasma current, Eqn. \((\ref{eq:plasmacurrent_curent})\), should be independent of \(\zeta\), and the linking current, Eqn. \((\ref{eq:linkingcurrent_curent})\), should be independent of \(\theta\).
Todo:
Perhaps this can be proved analytically; in any case it should be confirmed numerically.

Parameters

[in]	lvol	index of volume
[in]	mn	number of Fourier harmonics
[in]	Nt	number of grid points along \(\theta\)
[in]	Nz	number of grid points along \(\zeta\)
[in]	iflag	some integer flag
[out]	ldItGp	plasma and linking current

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