From local to global : ray tracing with grid spacing h

Alternatively, the eigenvalue derivatives can be determined directly using perturbation theory

The direct calculation of the derivatives is beneficial because... The rays may be integrated directly –the data-cube need not be constructed –the eigenvalue derivatives may be given directly to an o.d.e. integrator –this may be useful if only a few ray trajectories are required –simple to locally refine ray trajectories using higher numerical accuracy The calculation of the derivatives is consistent with the calculation of the eigenvalue (I need to work on this : eigenvalue and derivative are both extrapolated; are they still consistent ?) The derivatives enable a higher order interpolation of the data-cube. –Consider a 2 point interpolation in 1 dimension, without derivative 2 point interpolation with derivative2 point interpolation linearerror O(h 2 )cubicerror O(h 4 )

For example, consider a tokamak A circular cross section tokamak is simple –there is no  dependence, minimal #Fourier harmonics –note that the ballooning code, interpolation, ray tracing etc. is fully 3D Shown below are unstable ballooning contours

In 3D, 4 th order interpolation is easily obtained eigenvalue interpolation error derivative interpolation error

The use of the derivatives enables a crude-grid to give good interpolation solid : exact calculated at 100 radial points radial (VMEC) coordinate ballooning profile X : grid points dashed : 2-point interpolation X : grid points

Future work possibly includes... the eigenvalue and derivatives are calculated using Richardson’s extrapolation; –extrapolation wrt #grid points along field line for each ballooning calculation –need to check the consistency of extrapolated-eigenvalue with extrapolated-derivatives higher order interpolation on data-cube grid –eg: extend 2 3 point interpolation to 4 3 interpolation: O(h 4 )  O(h ? ) higher order derivatives can be calculated using perturbation theory –higher order derivatives can further improve the data-cube interpolation –is it worthwhile to calculate the higher order derivatives ? study some configurations of interest –need to understand the theory in more detail !!! –probably start with axisymmetric approximation, slowly add non-axisymmetry towards NCSX –appropriate mass normalization for comparison with CAS3D / TERPSICHORE –include FLR effects