From local to global ray tracing
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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

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From local to global : ray tracing

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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

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


The direct calculation of the derivatives is beneficial because

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,


For example consider a tokamak

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

In 3D, 4th order interpolation is easily obtained

eigenvalue interpolation error derivative interpolation error


The use of the derivatives enables a crude grid to give good interpolation

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


Construction of data cube allows eigenvalue iso sufaces to be visualized

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


Future work possibly includes

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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