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From local to global : ray tracing

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

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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 (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,

  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. 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 23 point interpolation to 43 interpolation: O(h4)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

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