6.2. Radial grid

After the EquilibriumRegion objects are created, the radial grids are constructed and stored in them.

Constructing the ‘radial grid’ means choosing the flux surfaces that are gridded in the core, SOL, PFR, and (for disconnected double null configurations) the region between the separatrices.

The parameters controlling the radial grid are:

• the number of radial grid points in the core (nx_core), SOL (nx_sol), and for disconnected double null inter-separatrix (nx_inter_sep)

• the ‘inner’ values of \(\psi\) in the core (psi_core or psinorm_core) and PFRs (psi_pf_lower, psi_pf_upper or psinorm_pf_lower, psinorm_pf_upper)

• the ‘outer’ values of \(\psi\) in the SOL (psi_sol, psi_sol_inner or psinorm_sol, psinorm_sol_inner)

• psi_spacing_separatrix_multiplier which can be used to decrease (or increase) the grid spacing at the separatrices. Useful to increase resolution near separatrices so that the radial resolution is adequate near the X-points.

Note some of these settings are not used for single null configurations.

As the settings for the boundary value of \(\psi\) are generally different between core and PFR, and can also be different for inner and outer SOL regions in a double null configuration, the grid values of \(\psi\) are different in different regions – \(\psi\) is not a 1d radial coordinate globally, for the whole grid. This means that, for example, the SOL region connects at the separatrix to the core and PFR(s), in different poloidal locations, which have different \(\psi\) grids. To ensure that the grid spacing is continuous and smooth across the separatrix everywhere, the radial derivative of the grid spacing is forced to vanish at the separatrices, so the radial gradient of the grid spacing approaches zero on both sides of the boundary, ensuring the spacing is smooth.

Technical details

To make the radial grid spacing continuous and smooth across the boundaries between regions (the separatrices) a gradient for the spacing function is imposed at these boundaries, and the second derivative of the spacing function is constrained to vanish. At the radial boundaries (where there is no connection to another region) neither the gradient nor the second derivative of the spacing function are constrained. All these requirements would be simple to implement if the spacing function was a polynomial (they would reduce to some simultaneous equations that need to be solved for the coefficents). However, it was found that using a polynomial spacing function was not robust, as the function could become non-monotonic. A more complex algorithm has been implemented in getSmoothMonotonicGridFunc() that guarantees the function is always monotonic, with the spacing function involving the error function or trigonometric functions in some cases.