content_image/346005.jpg content_image/346007.jpg content_image/346006.jpg content_image/346008.jpg content_image/346009.jpg content_image/346010.jpg
to be unity. Hence the metric tensor g$_{ij}$ satisfies det g$_{ij}$ ≡ g = 1. The function χ is found by the constraint that θ is cyclic on the unit interval 

<img src='content_image/346005.jpg'>

where η is a simple cyclic coordinate on the flux surface in the poloidal plane (e.g., path length on the curve, or the physical angle position about the magnetic axis with respect to any branch cut). Then q is found 

where the angle brackets give the flux surface average defined as 

<img src='content_image/346007.jpg'>

<img src='content_image/346006.jpg'>

Now, ψ$_{,V}$ = qχ$_{,V}$ is also defined and the toroidal angle coordinate is given by 

which completes the prescription. 

<img src='content_image/346008.jpg'>

The field aligning is a one-to-one and onto coordinate transform from { V, θ, ζ } to { x, y, s } given by 

<img src='content_image/346009.jpg'>

where a is a reference minor radius and α$_{k}$ is an arbitrary function of x which is chosen to make the off diagonal perpendicular metric element g xy k vanish at a particular location. This is called the shifted metric procedure [38]. The point is that at any position in s where perpendicular drifts or ∇ 2 ⊥ is evaluated, the coordinate elements are rectangular, and the information contained in magnetic shear enters only in derivatives in the parallel coordinate. The magnetic field components now satisfy 

<img src='content_image/346010.jpg'>

at all locations in space for any choice of α$_{k}$ . Hence although the perpendicular coordinate elements are rectangular only for one particular location in s the field aligning is exactly satisfied everywhere. 

