# TJ-II:Magnetic coordinates

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The diagnostics perform measurements in real space. The location of points is given in one of these coordinate systems (units according to the S.I. system, m and rad):

1. Cartesian $(X, Y, Z)$ with its origin at the centre of the TJ-II device, the X-axis pointing due North, the Y-axis due West, and the Z-axis up (see TJ-II:Sectors).
2. Cylindrical $(R, \phi, Z)$, where $R^2 = X^2 + Y^2$ and $\tan \phi = Y/X$.

In order to make comparisons between diagnostics, it is useful to convert these real-space coordinates to flux coordinates. This coordinate transform depends on the particular magnetic configuration used in a given experiment. Two tools are available to do so (See the on-line documentation - only internal laboratory access):

First, vacuum equilibrium calculations from VMEC. These are then used to obtain magnetic flux coordinates $(\psi, \theta, \phi)$. A set of routines is available to perform the corresponding coordinate transforms.[1] The drawback of the VMEC calculations is (a) that magnetic islands are ignored, and (b) that only a limited number of configurations is available.

Second, magnetic field line calculations using the Biot-Savart Law. The approximate magnetic flux is recovered from an interpolation procedure. A set of routines is available to perform the corresponding coordinate transforms. More information can be found in files g3d_readme.doc and g3d_gridfile.doc. Since the latter procedure is more flexible and generally applicable than the VMEC-based calculations, the latter is preferred.

It should be noted that these coordinate transforms are approximate and not error-free. The errors in the vacuum field calculation are due to three sources:

1. The error in the placement of the coils.
2. The error in the value of the currents flowing through the coils.
3. The error due to the fact that the model uses a finite amount of filaments to model the current in the coils (which has a continuous distribution).

Additional errors are due to finite-pressure effects (estimated to be quite small in TJ-II) and net plasma currents.

## Field direction

The direction of the dominant toroidal field component is in the $+\phi$ direction (counterclockwise, seen from the top), to accommodate the Heavy Ion Beam Probe diagnostic.

## References

1. File:TJ2 Library V2.pdf: TJ-II Library Manual, Informe Técnico del CIEMAT Nº 963