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From local to global : ray tracing

Explore direct calculation benefits, eigenvalue derivatives consistency, and higher-order interpolation with 3D ray tracing. Visualize eigenvalue iso-surfaces in data cubes for improved accuracy and analysis.

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From local to global : ray tracing

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

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

  3. 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,

  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

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

  6. 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

  7. 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

  8. 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

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