From local to global : ray tracing

2. From local to global : ray tracing with grid spacing h

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

4. 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 • The derivatives enable a higher order interpolation of the data-cube. • Consider a 2 point interpolation in 1 dimension,

5. 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

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

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

8. k s Construction of data-cube allows eigenvalue iso-sufaces to be visualized Another example : LHD variant studied by Nakajima et al. ISW 2005 as eigenvalue is increased, iso-sufaces become more localized

9. Future work possibly includes . . . • compare results of ray-tracing to global stability results • investigate discrepancy between local and global stability limits • appropriate mass normalization for comparison with CAS3D / TERPSICHORE • include FLR effects / chaotic ray-dynamics as studied by MacMillan & Dewar